LMFDB - Number field 36.0.230309323306002912742337717628704661399933091124862510404769.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 3*x^35 + 3*x^34 - 3*x^32 + 6*x^31 - 8*x^30 + 6*x^29 - 12*x^28 + 36*x^27 - 42*x^26 - 6*x^25 + 82*x^24 - 168*x^23 + 177*x^22 + 27*x^21 - 207*x^20 - 54*x^19 + 397*x^18 - 108*x^17 - 828*x^16 + 216*x^15 + 2832*x^14 - 5376*x^13 + 5248*x^12 - 768*x^11 - 10752*x^10 + 18432*x^9 - 12288*x^8 + 12288*x^7 - 32768*x^6 + 49152*x^5 - 49152*x^4 + 196608*x^2 - 393216*x + 262144)

gp: K = bnfinit(y^36 - 3*y^35 + 3*y^34 - 3*y^32 + 6*y^31 - 8*y^30 + 6*y^29 - 12*y^28 + 36*y^27 - 42*y^26 - 6*y^25 + 82*y^24 - 168*y^23 + 177*y^22 + 27*y^21 - 207*y^20 - 54*y^19 + 397*y^18 - 108*y^17 - 828*y^16 + 216*y^15 + 2832*y^14 - 5376*y^13 + 5248*y^12 - 768*y^11 - 10752*y^10 + 18432*y^9 - 12288*y^8 + 12288*y^7 - 32768*y^6 + 49152*y^5 - 49152*y^4 + 196608*y^2 - 393216*y + 262144, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 3*x^35 + 3*x^34 - 3*x^32 + 6*x^31 - 8*x^30 + 6*x^29 - 12*x^28 + 36*x^27 - 42*x^26 - 6*x^25 + 82*x^24 - 168*x^23 + 177*x^22 + 27*x^21 - 207*x^20 - 54*x^19 + 397*x^18 - 108*x^17 - 828*x^16 + 216*x^15 + 2832*x^14 - 5376*x^13 + 5248*x^12 - 768*x^11 - 10752*x^10 + 18432*x^9 - 12288*x^8 + 12288*x^7 - 32768*x^6 + 49152*x^5 - 49152*x^4 + 196608*x^2 - 393216*x + 262144);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 3*x^35 + 3*x^34 - 3*x^32 + 6*x^31 - 8*x^30 + 6*x^29 - 12*x^28 + 36*x^27 - 42*x^26 - 6*x^25 + 82*x^24 - 168*x^23 + 177*x^22 + 27*x^21 - 207*x^20 - 54*x^19 + 397*x^18 - 108*x^17 - 828*x^16 + 216*x^15 + 2832*x^14 - 5376*x^13 + 5248*x^12 - 768*x^11 - 10752*x^10 + 18432*x^9 - 12288*x^8 + 12288*x^7 - 32768*x^6 + 49152*x^5 - 49152*x^4 + 196608*x^2 - 393216*x + 262144)

$ x^{36} - 3 x^{35} + 3 x^{34} - 3 x^{32} + 6 x^{31} - 8 x^{30} + 6 x^{29} - 12 x^{28} + 36 x^{27} + \cdots + 262144 $ x^36 - 3*x^35 + 3*x^34 - 3*x^32 + 6*x^31 - 8*x^30 + 6*x^29 - 12*x^28 + 36*x^27 - 42*x^26 - 6*x^25 + 82*x^24 - 168*x^23 + 177*x^22 + 27*x^21 - 207*x^20 - 54*x^19 + 397*x^18 - 108*x^17 - 828*x^16 + 216*x^15 + 2832*x^14 - 5376*x^13 + 5248*x^12 - 768*x^11 - 10752*x^10 + 18432*x^9 - 12288*x^8 + 12288*x^7 - 32768*x^6 + 49152*x^5 - 49152*x^4 + 196608*x^2 - 393216*x + 262144

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$36$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 18]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$230309323306002912742337717628704661399933091124862510404769$ 230309323306002912742337717628704661399933091124862510404769 $\medspace = 3^{48}\cdot 7^{30}\cdot 71^{6}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$44.56$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$3^{4/3}7^{5/6}71^{1/2}\approx 184.5182019723083$
Ramified primes:	$3$, $7$, $71$ 3, 7, 71	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$12$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:	unavailable$^{131072}$

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{15}-\frac{1}{2}a$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{5}$, $\frac{1}{4}a^{20}-\frac{1}{4}a^{6}$, $\frac{1}{8}a^{21}-\frac{1}{8}a^{20}-\frac{1}{8}a^{19}-\frac{1}{8}a^{17}-\frac{1}{4}a^{14}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{8}a^{7}-\frac{3}{8}a^{6}+\frac{1}{8}a^{5}+\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{22}-\frac{1}{16}a^{21}+\frac{1}{16}a^{20}-\frac{1}{8}a^{19}+\frac{1}{16}a^{18}-\frac{1}{8}a^{15}-\frac{1}{4}a^{13}-\frac{1}{8}a^{12}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{16}a^{8}-\frac{3}{16}a^{7}+\frac{3}{16}a^{6}+\frac{1}{4}a^{5}+\frac{5}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{32}a^{23}-\frac{1}{32}a^{22}+\frac{1}{32}a^{21}+\frac{1}{16}a^{20}+\frac{1}{32}a^{19}-\frac{1}{16}a^{16}+\frac{1}{8}a^{14}-\frac{1}{16}a^{13}+\frac{3}{16}a^{12}-\frac{1}{16}a^{11}+\frac{1}{8}a^{10}-\frac{7}{32}a^{9}-\frac{3}{32}a^{8}+\frac{11}{32}a^{7}-\frac{3}{32}a^{5}-\frac{5}{16}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{24}-\frac{1}{64}a^{23}+\frac{1}{64}a^{22}+\frac{1}{32}a^{21}+\frac{1}{64}a^{20}-\frac{1}{8}a^{19}-\frac{1}{8}a^{18}+\frac{3}{32}a^{17}+\frac{1}{16}a^{15}-\frac{1}{32}a^{14}-\frac{5}{32}a^{13}-\frac{1}{32}a^{12}-\frac{3}{16}a^{11}-\frac{7}{64}a^{10}+\frac{13}{64}a^{9}+\frac{11}{64}a^{8}+\frac{13}{64}a^{6}-\frac{1}{32}a^{5}+\frac{1}{4}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{25}+\frac{1}{256}a^{24}+\frac{3}{256}a^{23}-\frac{7}{256}a^{21}-\frac{15}{128}a^{20}+\frac{3}{64}a^{19}-\frac{13}{128}a^{18}+\frac{7}{64}a^{17}-\frac{1}{64}a^{16}-\frac{13}{128}a^{15}+\frac{17}{128}a^{14}+\frac{17}{128}a^{13}-\frac{3}{32}a^{12}+\frac{25}{256}a^{11}-\frac{17}{256}a^{10}-\frac{23}{256}a^{9}-\frac{11}{128}a^{8}+\frac{41}{256}a^{7}+\frac{7}{32}a^{6}+\frac{5}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}$, $\frac{1}{92672}a^{26}+\frac{57}{92672}a^{25}-\frac{373}{92672}a^{24}+\frac{111}{11584}a^{23}+\frac{1225}{92672}a^{22}+\frac{2621}{46336}a^{21}-\frac{1029}{23168}a^{20}+\frac{4019}{46336}a^{19}+\frac{307}{23168}a^{18}-\frac{1905}{23168}a^{17}-\frac{4637}{46336}a^{16}+\frac{10105}{46336}a^{15}+\frac{10041}{46336}a^{14}+\frac{879}{11584}a^{13}+\frac{22969}{92672}a^{12}+\frac{295}{92672}a^{11}+\frac{15041}{92672}a^{10}-\frac{7927}{46336}a^{9}+\frac{17033}{92672}a^{8}-\frac{1333}{5792}a^{7}-\frac{529}{5792}a^{6}+\frac{69}{5792}a^{5}-\frac{42}{181}a^{4}-\frac{317}{724}a^{3}+\frac{5}{724}a^{2}-\frac{167}{724}a+\frac{16}{181}$, $\frac{1}{185344}a^{27}-\frac{1}{185344}a^{26}-\frac{59}{185344}a^{25}+\frac{39}{92672}a^{24}-\frac{1771}{185344}a^{23}+\frac{25}{5792}a^{22}+\frac{2059}{46336}a^{21}-\frac{7661}{92672}a^{20}+\frac{221}{5792}a^{19}+\frac{199}{46336}a^{18}+\frac{3487}{92672}a^{17}+\frac{5379}{92672}a^{16}-\frac{14949}{92672}a^{15}+\frac{9219}{46336}a^{14}+\frac{22001}{185344}a^{13}-\frac{22915}{185344}a^{12}-\frac{15825}{185344}a^{11}-\frac{695}{46336}a^{10}+\frac{16361}{185344}a^{9}+\frac{18831}{92672}a^{8}+\frac{2517}{46336}a^{7}+\frac{699}{1448}a^{6}-\frac{1065}{2896}a^{5}+\frac{377}{1448}a^{4}-\frac{63}{362}a^{3}+\frac{629}{1448}a^{2}+\frac{169}{724}a-\frac{23}{362}$, $\frac{1}{370688}a^{28}-\frac{1}{370688}a^{27}+\frac{1}{370688}a^{26}+\frac{301}{185344}a^{25}+\frac{1913}{370688}a^{24}+\frac{303}{23168}a^{23}+\frac{805}{46336}a^{22}+\frac{9143}{185344}a^{21}+\frac{935}{23168}a^{20}-\frac{6655}{92672}a^{19}+\frac{20055}{185344}a^{18}-\frac{14709}{185344}a^{17}+\frac{2223}{185344}a^{16}-\frac{22119}{92672}a^{15}-\frac{87863}{370688}a^{14}-\frac{58563}{370688}a^{13}+\frac{53323}{370688}a^{12}-\frac{7185}{46336}a^{11}-\frac{68715}{370688}a^{10}-\frac{9357}{185344}a^{9}-\frac{3553}{23168}a^{8}-\frac{4341}{23168}a^{7}-\frac{1891}{11584}a^{6}-\frac{1469}{5792}a^{5}+\frac{313}{724}a^{4}-\frac{653}{2896}a^{3}+\frac{681}{1448}a^{2}-\frac{89}{181}a+\frac{59}{181}$, $\frac{1}{741376}a^{29}-\frac{1}{741376}a^{28}+\frac{1}{741376}a^{27}+\frac{1}{370688}a^{26}-\frac{431}{741376}a^{25}+\frac{707}{92672}a^{24}+\frac{1175}{92672}a^{23}+\frac{747}{370688}a^{22}-\frac{1065}{46336}a^{21}+\frac{18513}{185344}a^{20}+\frac{23919}{370688}a^{19}+\frac{13643}{370688}a^{18}-\frac{22785}{370688}a^{17}-\frac{6619}{185344}a^{16}+\frac{117305}{741376}a^{15}+\frac{49645}{741376}a^{14}+\frac{21739}{741376}a^{13}+\frac{2659}{23168}a^{12}+\frac{179997}{741376}a^{11}-\frac{54577}{370688}a^{10}-\frac{9405}{46336}a^{9}-\frac{3953}{92672}a^{8}-\frac{11423}{46336}a^{7}-\frac{311}{11584}a^{6}+\frac{783}{5792}a^{5}+\frac{959}{5792}a^{4}+\frac{575}{2896}a^{3}+\frac{88}{181}a^{2}+\frac{335}{724}a-\frac{47}{362}$, $\frac{1}{1482752}a^{30}-\frac{1}{1482752}a^{29}+\frac{1}{1482752}a^{28}+\frac{1}{741376}a^{27}+\frac{1}{1482752}a^{26}+\frac{165}{185344}a^{25}+\frac{581}{185344}a^{24}+\frac{10107}{741376}a^{23}-\frac{1371}{46336}a^{22}-\frac{13375}{370688}a^{21}+\frac{78959}{741376}a^{20}-\frac{45669}{741376}a^{19}-\frac{76097}{741376}a^{18}-\frac{12907}{370688}a^{17}+\frac{119001}{1482752}a^{16}+\frac{80781}{1482752}a^{15}+\frac{298763}{1482752}a^{14}+\frac{3109}{46336}a^{13}+\frac{47693}{1482752}a^{12}+\frac{17831}{741376}a^{11}+\frac{653}{5792}a^{10}-\frac{23849}{185344}a^{9}-\frac{669}{2896}a^{8}-\frac{1303}{46336}a^{7}-\frac{3529}{11584}a^{6}-\frac{4727}{11584}a^{5}+\frac{1211}{5792}a^{4}+\frac{31}{181}a^{3}-\frac{481}{1448}a^{2}-\frac{53}{181}a-\frac{41}{362}$, $\frac{1}{210550784}a^{31}+\frac{53}{210550784}a^{30}-\frac{21}{210550784}a^{29}-\frac{33}{26318848}a^{28}+\frac{429}{210550784}a^{27}-\frac{81}{105275392}a^{26}+\frac{47699}{26318848}a^{25}-\frac{262253}{105275392}a^{24}-\frac{503023}{52637696}a^{23}-\frac{138599}{52637696}a^{22}-\frac{6221573}{105275392}a^{21}+\frac{3921605}{105275392}a^{20}-\frac{10885071}{105275392}a^{19}+\frac{149373}{26318848}a^{18}-\frac{160495}{210550784}a^{17}-\frac{20012461}{210550784}a^{16}+\frac{43406505}{210550784}a^{15}+\frac{21930841}{105275392}a^{14}+\frac{2473037}{210550784}a^{13}-\frac{1907645}{52637696}a^{12}+\frac{9784285}{52637696}a^{11}+\frac{2434255}{26318848}a^{10}-\frac{2247855}{13159424}a^{9}+\frac{304669}{1644928}a^{8}-\frac{210217}{1644928}a^{7}+\frac{14675}{1644928}a^{6}+\frac{10667}{411232}a^{5}-\frac{170469}{411232}a^{4}-\frac{389}{205616}a^{3}-\frac{19823}{102808}a^{2}-\frac{9449}{25702}a+\frac{6343}{12851}$, $\frac{1}{842203136}a^{32}+\frac{1}{842203136}a^{31}-\frac{221}{842203136}a^{30}+\frac{17}{26318848}a^{29}-\frac{327}{842203136}a^{28}-\frac{159}{421101568}a^{27}+\frac{163}{210550784}a^{26}+\frac{364115}{421101568}a^{25}+\frac{727855}{210550784}a^{24}-\frac{1610769}{210550784}a^{23}+\frac{1465331}{421101568}a^{22}-\frac{2540175}{421101568}a^{21}+\frac{5981873}{421101568}a^{20}+\frac{596117}{105275392}a^{19}+\frac{56373145}{842203136}a^{18}-\frac{101934417}{842203136}a^{17}-\frac{79888567}{842203136}a^{16}+\frac{74635429}{421101568}a^{15}+\frac{119513641}{842203136}a^{14}+\frac{22995699}{105275392}a^{13}-\frac{614415}{13159424}a^{12}-\frac{7787895}{52637696}a^{11}+\frac{722831}{6579712}a^{10}-\frac{1100123}{13159424}a^{9}-\frac{708343}{6579712}a^{8}+\frac{1395201}{3289856}a^{7}+\frac{375635}{822464}a^{6}-\frac{7945}{102808}a^{5}-\frac{88935}{411232}a^{4}-\frac{1623}{25702}a^{3}+\frac{4649}{102808}a^{2}-\frac{5429}{51404}a+\frac{24809}{51404}$, $\frac{1}{1684406272}a^{33}-\frac{1}{1684406272}a^{32}+\frac{1}{1684406272}a^{31}+\frac{1}{4653056}a^{30}-\frac{439}{1684406272}a^{29}-\frac{27}{52637696}a^{28}+\frac{245}{210550784}a^{27}+\frac{3}{4653056}a^{26}-\frac{134171}{105275392}a^{25}-\frac{2789215}{421101568}a^{24}+\frac{5027207}{842203136}a^{23}-\frac{12692117}{842203136}a^{22}-\frac{34290353}{842203136}a^{21}+\frac{1522673}{421101568}a^{20}-\frac{50449559}{1684406272}a^{19}-\frac{59584163}{1684406272}a^{18}-\frac{51344181}{1684406272}a^{17}-\frac{20704847}{210550784}a^{16}+\frac{32958789}{1684406272}a^{15}+\frac{156337211}{842203136}a^{14}-\frac{987391}{105275392}a^{13}+\frac{2676139}{13159424}a^{12}+\frac{12088659}{52637696}a^{11}+\frac{1449169}{6579712}a^{10}+\frac{9645}{185344}a^{9}-\frac{14833}{92672}a^{8}-\frac{510753}{1644928}a^{7}+\frac{121511}{822464}a^{6}+\frac{45245}{205616}a^{5}+\frac{21763}{411232}a^{4}-\frac{1154}{12851}a^{3}-\frac{50727}{102808}a^{2}+\frac{23493}{102808}a-\frac{25671}{51404}$, $\frac{1}{3368812544}a^{34}-\frac{1}{3368812544}a^{33}+\frac{1}{3368812544}a^{32}-\frac{3}{1684406272}a^{31}+\frac{505}{3368812544}a^{30}+\frac{3}{210550784}a^{29}-\frac{391}{421101568}a^{28}+\frac{3399}{1684406272}a^{27}-\frac{231}{210550784}a^{26}+\frac{1272465}{842203136}a^{25}+\frac{12283063}{1684406272}a^{24}-\frac{12905429}{1684406272}a^{23}+\frac{39168559}{1684406272}a^{22}+\frac{14692969}{842203136}a^{21}+\frac{212000329}{3368812544}a^{20}+\frac{244318013}{3368812544}a^{19}+\frac{344747787}{3368812544}a^{18}-\frac{48799357}{421101568}a^{17}+\frac{148430805}{3368812544}a^{16}-\frac{420637293}{1684406272}a^{15}-\frac{41893039}{210550784}a^{14}+\frac{20900137}{210550784}a^{13}-\frac{4657337}{26318848}a^{12}-\frac{7805157}{52637696}a^{11}+\frac{1197949}{26318848}a^{10}+\frac{1340791}{13159424}a^{9}+\frac{204417}{3289856}a^{8}+\frac{454605}{1644928}a^{7}-\frac{391271}{1644928}a^{6}-\frac{50179}{411232}a^{5}-\frac{184295}{411232}a^{4}-\frac{14019}{205616}a^{3}-\frac{96279}{205616}a^{2}-\frac{25687}{102808}a-\frac{5877}{12851}$, $\frac{1}{6737625088}a^{35}-\frac{1}{6737625088}a^{34}+\frac{1}{6737625088}a^{33}+\frac{1}{3368812544}a^{32}+\frac{1}{6737625088}a^{31}-\frac{199}{842203136}a^{30}+\frac{361}{842203136}a^{29}+\frac{1515}{3368812544}a^{28}-\frac{243}{210550784}a^{27}+\frac{1737}{1684406272}a^{26}+\frac{5961935}{3368812544}a^{25}+\frac{5091099}{3368812544}a^{24}-\frac{38226785}{3368812544}a^{23}-\frac{19511115}{1684406272}a^{22}-\frac{244415655}{6737625088}a^{21}+\frac{219221325}{6737625088}a^{20}-\frac{598996661}{6737625088}a^{19}+\frac{23998575}{210550784}a^{18}-\frac{5621301}{94896128}a^{17}+\frac{194898167}{3368812544}a^{16}+\frac{20550567}{210550784}a^{15}-\frac{112813877}{842203136}a^{14}+\frac{17685421}{105275392}a^{13}+\frac{17089701}{105275392}a^{12}-\frac{4897631}{26318848}a^{11}+\frac{5682325}{26318848}a^{10}+\frac{1734133}{13159424}a^{9}-\frac{1621789}{6579712}a^{8}-\frac{307801}{1644928}a^{7}-\frac{57555}{822464}a^{6}-\frac{138207}{822464}a^{5}+\frac{22609}{51404}a^{4}-\frac{5855}{411232}a^{3}-\frac{53713}{205616}a^{2}+\frac{15243}{51404}a-\frac{7683}{51404}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/2*a^8 - 1/2*a, 1/2*a^9 - 1/2*a^2, 1/2*a^10 - 1/2*a^3, 1/2*a^11 - 1/2*a^4, 1/2*a^12 - 1/2*a^5, 1/2*a^13 - 1/2*a^6, 1/2*a^14 - 1/2*a^7, 1/2*a^15 - 1/2*a, 1/4*a^16 - 1/4*a^2, 1/4*a^17 - 1/4*a^3, 1/4*a^18 - 1/4*a^4, 1/4*a^19 - 1/4*a^5, 1/4*a^20 - 1/4*a^6, 1/8*a^21 - 1/8*a^20 - 1/8*a^19 - 1/8*a^17 - 1/4*a^14 - 1/4*a^11 - 1/4*a^10 - 1/4*a^9 + 1/8*a^7 - 3/8*a^6 + 1/8*a^5 + 1/4*a^4 - 1/8*a^3 - 1/4*a^2 - 1/2*a, 1/16*a^22 - 1/16*a^21 + 1/16*a^20 - 1/8*a^19 + 1/16*a^18 - 1/8*a^15 - 1/4*a^13 - 1/8*a^12 - 1/8*a^11 - 1/8*a^10 - 1/4*a^9 + 1/16*a^8 - 3/16*a^7 + 3/16*a^6 + 1/4*a^5 + 5/16*a^4 + 3/8*a^3 - 1/2*a, 1/32*a^23 - 1/32*a^22 + 1/32*a^21 + 1/16*a^20 + 1/32*a^19 - 1/16*a^16 + 1/8*a^14 - 1/16*a^13 + 3/16*a^12 - 1/16*a^11 + 1/8*a^10 - 7/32*a^9 - 3/32*a^8 + 11/32*a^7 - 3/32*a^5 - 5/16*a^4 - 1/4*a^3 - 1/2*a^2 - 1/2*a, 1/64*a^24 - 1/64*a^23 + 1/64*a^22 + 1/32*a^21 + 1/64*a^20 - 1/8*a^19 - 1/8*a^18 + 3/32*a^17 + 1/16*a^15 - 1/32*a^14 - 5/32*a^13 - 1/32*a^12 - 3/16*a^11 - 7/64*a^10 + 13/64*a^9 + 11/64*a^8 + 13/64*a^6 - 1/32*a^5 + 1/4*a^4 + 1/8*a^3 - 1/2*a^2 - 1/2*a, 1/256*a^25 + 1/256*a^24 + 3/256*a^23 - 7/256*a^21 - 15/128*a^20 + 3/64*a^19 - 13/128*a^18 + 7/64*a^17 - 1/64*a^16 - 13/128*a^15 + 17/128*a^14 + 17/128*a^13 - 3/32*a^12 + 25/256*a^11 - 17/256*a^10 - 23/256*a^9 - 11/128*a^8 + 41/256*a^7 + 7/32*a^6 + 5/16*a^4 - 1/8*a^3 - 1/2, 1/92672*a^26 + 57/92672*a^25 - 373/92672*a^24 + 111/11584*a^23 + 1225/92672*a^22 + 2621/46336*a^21 - 1029/23168*a^20 + 4019/46336*a^19 + 307/23168*a^18 - 1905/23168*a^17 - 4637/46336*a^16 + 10105/46336*a^15 + 10041/46336*a^14 + 879/11584*a^13 + 22969/92672*a^12 + 295/92672*a^11 + 15041/92672*a^10 - 7927/46336*a^9 + 17033/92672*a^8 - 1333/5792*a^7 - 529/5792*a^6 + 69/5792*a^5 - 42/181*a^4 - 317/724*a^3 + 5/724*a^2 - 167/724*a + 16/181, 1/185344*a^27 - 1/185344*a^26 - 59/185344*a^25 + 39/92672*a^24 - 1771/185344*a^23 + 25/5792*a^22 + 2059/46336*a^21 - 7661/92672*a^20 + 221/5792*a^19 + 199/46336*a^18 + 3487/92672*a^17 + 5379/92672*a^16 - 14949/92672*a^15 + 9219/46336*a^14 + 22001/185344*a^13 - 22915/185344*a^12 - 15825/185344*a^11 - 695/46336*a^10 + 16361/185344*a^9 + 18831/92672*a^8 + 2517/46336*a^7 + 699/1448*a^6 - 1065/2896*a^5 + 377/1448*a^4 - 63/362*a^3 + 629/1448*a^2 + 169/724*a - 23/362, 1/370688*a^28 - 1/370688*a^27 + 1/370688*a^26 + 301/185344*a^25 + 1913/370688*a^24 + 303/23168*a^23 + 805/46336*a^22 + 9143/185344*a^21 + 935/23168*a^20 - 6655/92672*a^19 + 20055/185344*a^18 - 14709/185344*a^17 + 2223/185344*a^16 - 22119/92672*a^15 - 87863/370688*a^14 - 58563/370688*a^13 + 53323/370688*a^12 - 7185/46336*a^11 - 68715/370688*a^10 - 9357/185344*a^9 - 3553/23168*a^8 - 4341/23168*a^7 - 1891/11584*a^6 - 1469/5792*a^5 + 313/724*a^4 - 653/2896*a^3 + 681/1448*a^2 - 89/181*a + 59/181, 1/741376*a^29 - 1/741376*a^28 + 1/741376*a^27 + 1/370688*a^26 - 431/741376*a^25 + 707/92672*a^24 + 1175/92672*a^23 + 747/370688*a^22 - 1065/46336*a^21 + 18513/185344*a^20 + 23919/370688*a^19 + 13643/370688*a^18 - 22785/370688*a^17 - 6619/185344*a^16 + 117305/741376*a^15 + 49645/741376*a^14 + 21739/741376*a^13 + 2659/23168*a^12 + 179997/741376*a^11 - 54577/370688*a^10 - 9405/46336*a^9 - 3953/92672*a^8 - 11423/46336*a^7 - 311/11584*a^6 + 783/5792*a^5 + 959/5792*a^4 + 575/2896*a^3 + 88/181*a^2 + 335/724*a - 47/362, 1/1482752*a^30 - 1/1482752*a^29 + 1/1482752*a^28 + 1/741376*a^27 + 1/1482752*a^26 + 165/185344*a^25 + 581/185344*a^24 + 10107/741376*a^23 - 1371/46336*a^22 - 13375/370688*a^21 + 78959/741376*a^20 - 45669/741376*a^19 - 76097/741376*a^18 - 12907/370688*a^17 + 119001/1482752*a^16 + 80781/1482752*a^15 + 298763/1482752*a^14 + 3109/46336*a^13 + 47693/1482752*a^12 + 17831/741376*a^11 + 653/5792*a^10 - 23849/185344*a^9 - 669/2896*a^8 - 1303/46336*a^7 - 3529/11584*a^6 - 4727/11584*a^5 + 1211/5792*a^4 + 31/181*a^3 - 481/1448*a^2 - 53/181*a - 41/362, 1/210550784*a^31 + 53/210550784*a^30 - 21/210550784*a^29 - 33/26318848*a^28 + 429/210550784*a^27 - 81/105275392*a^26 + 47699/26318848*a^25 - 262253/105275392*a^24 - 503023/52637696*a^23 - 138599/52637696*a^22 - 6221573/105275392*a^21 + 3921605/105275392*a^20 - 10885071/105275392*a^19 + 149373/26318848*a^18 - 160495/210550784*a^17 - 20012461/210550784*a^16 + 43406505/210550784*a^15 + 21930841/105275392*a^14 + 2473037/210550784*a^13 - 1907645/52637696*a^12 + 9784285/52637696*a^11 + 2434255/26318848*a^10 - 2247855/13159424*a^9 + 304669/1644928*a^8 - 210217/1644928*a^7 + 14675/1644928*a^6 + 10667/411232*a^5 - 170469/411232*a^4 - 389/205616*a^3 - 19823/102808*a^2 - 9449/25702*a + 6343/12851, 1/842203136*a^32 + 1/842203136*a^31 - 221/842203136*a^30 + 17/26318848*a^29 - 327/842203136*a^28 - 159/421101568*a^27 + 163/210550784*a^26 + 364115/421101568*a^25 + 727855/210550784*a^24 - 1610769/210550784*a^23 + 1465331/421101568*a^22 - 2540175/421101568*a^21 + 5981873/421101568*a^20 + 596117/105275392*a^19 + 56373145/842203136*a^18 - 101934417/842203136*a^17 - 79888567/842203136*a^16 + 74635429/421101568*a^15 + 119513641/842203136*a^14 + 22995699/105275392*a^13 - 614415/13159424*a^12 - 7787895/52637696*a^11 + 722831/6579712*a^10 - 1100123/13159424*a^9 - 708343/6579712*a^8 + 1395201/3289856*a^7 + 375635/822464*a^6 - 7945/102808*a^5 - 88935/411232*a^4 - 1623/25702*a^3 + 4649/102808*a^2 - 5429/51404*a + 24809/51404, 1/1684406272*a^33 - 1/1684406272*a^32 + 1/1684406272*a^31 + 1/4653056*a^30 - 439/1684406272*a^29 - 27/52637696*a^28 + 245/210550784*a^27 + 3/4653056*a^26 - 134171/105275392*a^25 - 2789215/421101568*a^24 + 5027207/842203136*a^23 - 12692117/842203136*a^22 - 34290353/842203136*a^21 + 1522673/421101568*a^20 - 50449559/1684406272*a^19 - 59584163/1684406272*a^18 - 51344181/1684406272*a^17 - 20704847/210550784*a^16 + 32958789/1684406272*a^15 + 156337211/842203136*a^14 - 987391/105275392*a^13 + 2676139/13159424*a^12 + 12088659/52637696*a^11 + 1449169/6579712*a^10 + 9645/185344*a^9 - 14833/92672*a^8 - 510753/1644928*a^7 + 121511/822464*a^6 + 45245/205616*a^5 + 21763/411232*a^4 - 1154/12851*a^3 - 50727/102808*a^2 + 23493/102808*a - 25671/51404, 1/3368812544*a^34 - 1/3368812544*a^33 + 1/3368812544*a^32 - 3/1684406272*a^31 + 505/3368812544*a^30 + 3/210550784*a^29 - 391/421101568*a^28 + 3399/1684406272*a^27 - 231/210550784*a^26 + 1272465/842203136*a^25 + 12283063/1684406272*a^24 - 12905429/1684406272*a^23 + 39168559/1684406272*a^22 + 14692969/842203136*a^21 + 212000329/3368812544*a^20 + 244318013/3368812544*a^19 + 344747787/3368812544*a^18 - 48799357/421101568*a^17 + 148430805/3368812544*a^16 - 420637293/1684406272*a^15 - 41893039/210550784*a^14 + 20900137/210550784*a^13 - 4657337/26318848*a^12 - 7805157/52637696*a^11 + 1197949/26318848*a^10 + 1340791/13159424*a^9 + 204417/3289856*a^8 + 454605/1644928*a^7 - 391271/1644928*a^6 - 50179/411232*a^5 - 184295/411232*a^4 - 14019/205616*a^3 - 96279/205616*a^2 - 25687/102808*a - 5877/12851, 1/6737625088*a^35 - 1/6737625088*a^34 + 1/6737625088*a^33 + 1/3368812544*a^32 + 1/6737625088*a^31 - 199/842203136*a^30 + 361/842203136*a^29 + 1515/3368812544*a^28 - 243/210550784*a^27 + 1737/1684406272*a^26 + 5961935/3368812544*a^25 + 5091099/3368812544*a^24 - 38226785/3368812544*a^23 - 19511115/1684406272*a^22 - 244415655/6737625088*a^21 + 219221325/6737625088*a^20 - 598996661/6737625088*a^19 + 23998575/210550784*a^18 - 5621301/94896128*a^17 + 194898167/3368812544*a^16 + 20550567/210550784*a^15 - 112813877/842203136*a^14 + 17685421/105275392*a^13 + 17089701/105275392*a^12 - 4897631/26318848*a^11 + 5682325/26318848*a^10 + 1734133/13159424*a^9 - 1621789/6579712*a^8 - 307801/1644928*a^7 - 57555/822464*a^6 - 138207/822464*a^5 + 22609/51404*a^4 - 5855/411232*a^3 - 53713/205616*a^2 + 15243/51404*a - 7683/51404

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

$C_{14}\times C_{14}$, which has order $196$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$17$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	\( \frac{7269}{3368812544} a^{35} - \frac{31503}{3368812544} a^{34} + \frac{21807}{3368812544} a^{33} - \frac{21807}{3368812544} a^{31} + \frac{21807}{1684406272} a^{30} - \frac{7269}{421101568} a^{29} + \frac{21807}{1684406272} a^{28} + \frac{40041}{842203136} a^{27} + \frac{65421}{842203136} a^{26} - \frac{152649}{1684406272} a^{25} - \frac{21807}{1684406272} a^{24} + \frac{298029}{1684406272} a^{23} - \frac{152649}{421101568} a^{22} + \frac{1286613}{3368812544} a^{21} + \frac{2954631}{3368812544} a^{20} - \frac{1504683}{3368812544} a^{19} - \frac{196263}{1684406272} a^{18} + \frac{2885793}{3368812544} a^{17} - \frac{196263}{842203136} a^{16} - \frac{1504683}{842203136} a^{15} + \frac{196263}{421101568} a^{14} + \frac{3268047}{210550784} a^{13} - \frac{152649}{13159424} a^{12} + \frac{298029}{26318848} a^{11} - \frac{21807}{13159424} a^{10} - \frac{152649}{6579712} a^{9} + \frac{65421}{1644928} a^{8} - \frac{21807}{822464} a^{7} - \frac{122713}{1644928} a^{6} - \frac{7269}{102808} a^{5} + \frac{21807}{205616} a^{4} - \frac{21807}{205616} a^{3} + \frac{21807}{51404} a - \frac{21807}{25702} \) (7269)/(3368812544)a^(35) - (31503)/(3368812544)a^(34) + (21807)/(3368812544)a^(33) - (21807)/(3368812544)a^(31) + (21807)/(1684406272)a^(30) - (7269)/(421101568)a^(29) + (21807)/(1684406272)a^(28) + (40041)/(842203136)a^(27) + (65421)/(842203136)a^(26) - (152649)/(1684406272)a^(25) - (21807)/(1684406272)a^(24) + (298029)/(1684406272)a^(23) - (152649)/(421101568)a^(22) + (1286613)/(3368812544)a^(21) + (2954631)/(3368812544)a^(20) - (1504683)/(3368812544)a^(19) - (196263)/(1684406272)a^(18) + (2885793)/(3368812544)a^(17) - (196263)/(842203136)a^(16) - (1504683)/(842203136)a^(15) + (196263)/(421101568)a^(14) + (3268047)/(210550784)a^(13) - (152649)/(13159424)a^(12) + (298029)/(26318848)a^(11) - (21807)/(13159424)a^(10) - (152649)/(6579712)a^(9) + (65421)/(1644928)a^(8) - (21807)/(822464)a^(7) - (122713)/(1644928)a^(6) - (7269)/(102808)a^(5) + (21807)/(205616)a^(4) - (21807)/(205616)a^(3) + (21807)/(51404)*a - (21807)/(25702) (order $14$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	$\frac{9399}{842203136}a^{35}-\frac{66801}{3368812544}a^{34}+\frac{29205}{3368812544}a^{33}+\frac{8391}{3368812544}a^{32}-\frac{27861}{1684406272}a^{31}+\frac{211779}{3368812544}a^{30}-\frac{30327}{842203136}a^{29}-\frac{10407}{421101568}a^{28}-\frac{196371}{1684406272}a^{27}+\frac{124203}{421101568}a^{26}-\frac{166167}{842203136}a^{25}-\frac{95207}{1684406272}a^{24}+\frac{1735017}{1684406272}a^{23}-\frac{2373951}{1684406272}a^{22}+\frac{88371}{210550784}a^{21}+\frac{5134959}{3368812544}a^{20}-\frac{2662377}{3368812544}a^{19}-\frac{7788619}{3368812544}a^{18}+\frac{1308177}{421101568}a^{17}+\frac{3097767}{3368812544}a^{16}-\frac{15846885}{1684406272}a^{15}-\frac{670299}{210550784}a^{14}+\frac{536079}{26318848}a^{13}-\frac{854537}{26318848}a^{12}+\frac{475671}{13159424}a^{11}+\frac{117147}{6579712}a^{10}-\frac{261477}{1644928}a^{9}+\frac{897045}{6579712}a^{8}+\frac{11079}{822464}a^{7}+\frac{15329}{205616}a^{6}-\frac{9399}{51404}a^{5}+\frac{66801}{205616}a^{4}-\frac{29205}{51404}a^{3}-\frac{133971}{205616}a^{2}+\frac{253971}{102808}a-\frac{22267}{12851}$, $\frac{29205}{1684406272}a^{35}-\frac{87615}{3368812544}a^{34}+\frac{29205}{3368812544}a^{33}+\frac{29205}{3368812544}a^{32}-\frac{48675}{1684406272}a^{31}+\frac{267283}{3368812544}a^{30}-\frac{29205}{1684406272}a^{29}-\frac{321255}{1684406272}a^{27}+\frac{321255}{842203136}a^{26}-\frac{9735}{52637696}a^{25}-\frac{379665}{1684406272}a^{24}+\frac{2317809}{1684406272}a^{23}-\frac{2716065}{1684406272}a^{22}+\frac{613305}{1684406272}a^{21}+\frac{6571125}{3368812544}a^{20}-\frac{2287725}{3368812544}a^{19}-\frac{10543005}{3368812544}a^{18}+\frac{87615}{26318848}a^{17}+\frac{8842431}{3368812544}a^{16}-\frac{4234725}{421101568}a^{15}-\frac{1898325}{210550784}a^{14}+\frac{827475}{26318848}a^{13}-\frac{1138995}{26318848}a^{12}+\frac{496485}{13159424}a^{11}+\frac{262845}{6579712}a^{10}-\frac{323919}{1644928}a^{9}+\frac{29205}{411232}a^{8}-\frac{9735}{822464}a^{7}+\frac{29205}{205616}a^{6}-\frac{29205}{102808}a^{5}+\frac{87615}{205616}a^{4}-\frac{29205}{51404}a^{3}-\frac{217227}{205616}a^{2}+\frac{68145}{25702}a-\frac{42056}{12851}$, $\frac{12195}{842203136}a^{35}-\frac{28455}{842203136}a^{34}-\frac{165}{842203136}a^{33}+\frac{12195}{421101568}a^{32}-\frac{36585}{842203136}a^{31}+\frac{12195}{210550784}a^{30}-\frac{12195}{210550784}a^{29}+\frac{4065}{421101568}a^{28}-\frac{12195}{105275392}a^{27}+\frac{59199}{210550784}a^{26}-\frac{109755}{421101568}a^{25}-\frac{207315}{421101568}a^{24}+\frac{475605}{421101568}a^{23}-\frac{345525}{210550784}a^{22}+\frac{792675}{842203136}a^{21}+\frac{1768275}{842203136}a^{20}-\frac{1131343}{842203136}a^{19}-\frac{36585}{13159424}a^{18}+\frac{4402395}{842203136}a^{17}+\frac{955275}{421101568}a^{16}-\frac{2743875}{210550784}a^{15}-\frac{256095}{52637696}a^{14}+\frac{1134135}{26318848}a^{13}-\frac{4065073}{105275392}a^{12}+\frac{158535}{6579712}a^{11}+\frac{4065}{102808}a^{10}-\frac{134145}{822464}a^{9}+\frac{134145}{822464}a^{8}+\frac{12195}{205616}a^{6}-\frac{88817}{822464}a^{5}+\frac{20325}{51404}a^{4}-\frac{12195}{51404}a^{3}-\frac{12195}{25702}a^{2}+\frac{36585}{12851}a-\frac{48780}{12851}$, $\frac{21807}{3368812544}a^{35}-\frac{36345}{3368812544}a^{34}+\frac{7269}{3368812544}a^{33}+\frac{2423}{421101568}a^{32}-\frac{62201}{3368812544}a^{31}+\frac{50883}{1684406272}a^{30}-\frac{21807}{842203136}a^{29}+\frac{7269}{1684406272}a^{28}-\frac{50883}{842203136}a^{27}+\frac{113881}{842203136}a^{26}-\frac{50883}{1684406272}a^{25}+\frac{60903}{1684406272}a^{24}+\frac{545175}{1684406272}a^{23}-\frac{283491}{421101568}a^{22}+\frac{1242999}{3368812544}a^{21}+\frac{2439961}{3368812544}a^{20}-\frac{1839057}{3368812544}a^{19}-\frac{1490145}{1684406272}a^{18}+\frac{8947559}{3368812544}a^{17}+\frac{225339}{210550784}a^{16}-\frac{167187}{52637696}a^{15}-\frac{462793}{105275392}a^{14}+\frac{327105}{26318848}a^{13}-\frac{21807}{1644928}a^{12}+\frac{21807}{1644928}a^{11}+\frac{929431}{26318848}a^{10}-\frac{50883}{822464}a^{9}+\frac{41191}{822464}a^{8}-\frac{7269}{411232}a^{7}+\frac{7269}{102808}a^{6}-\frac{21807}{205616}a^{5}+\frac{7269}{102808}a^{4}-\frac{36447}{51404}a^{3}-\frac{2423}{12851}a^{2}+\frac{14538}{12851}a-\frac{14538}{12851}$, $\frac{59149}{6737625088}a^{35}-\frac{224807}{6737625088}a^{34}+\frac{58899}{6737625088}a^{33}+\frac{62619}{1684406272}a^{32}-\frac{10419}{6737625088}a^{31}+\frac{90335}{3368812544}a^{30}-\frac{128675}{1684406272}a^{29}+\frac{10419}{3368812544}a^{28}-\frac{286647}{1684406272}a^{27}+\frac{552639}{1684406272}a^{26}-\frac{268233}{3368812544}a^{25}-\frac{672367}{3368812544}a^{24}+\frac{1192697}{3368812544}a^{23}-\frac{464031}{421101568}a^{22}+\frac{6388341}{6737625088}a^{21}+\frac{5957407}{6737625088}a^{20}-\frac{6482583}{6737625088}a^{19}-\frac{5625333}{3368812544}a^{18}+\frac{18478885}{6737625088}a^{17}+\frac{3822291}{1684406272}a^{16}-\frac{3920661}{421101568}a^{15}-\frac{4103817}{842203136}a^{14}+\frac{436529}{13159424}a^{13}-\frac{309523}{13159424}a^{12}+\frac{4543}{26318848}a^{11}-\frac{206991}{26318848}a^{10}-\frac{1060365}{13159424}a^{9}+\frac{1096143}{6579712}a^{8}+\frac{369}{23168}a^{7}+\frac{64203}{822464}a^{6}-\frac{33461}{102808}a^{5}+\frac{11313}{205616}a^{4}-\frac{128451}{411232}a^{3}+\frac{7773}{51404}a^{2}+\frac{115779}{51404}a-\frac{118087}{51404}$, $\frac{503201}{6737625088}a^{35}-\frac{785059}{6737625088}a^{34}+\frac{314211}{6737625088}a^{33}+\frac{7539}{105275392}a^{32}-\frac{793235}{6737625088}a^{31}+\frac{1129803}{3368812544}a^{30}-\frac{117851}{842203136}a^{29}+\frac{158595}{3368812544}a^{28}-\frac{1146735}{1684406272}a^{27}+\frac{2987409}{1684406272}a^{26}-\frac{2711893}{3368812544}a^{25}-\frac{3781299}{3368812544}a^{24}+\frac{15651369}{3368812544}a^{23}-\frac{5980173}{842203136}a^{22}+\frac{17840433}{6737625088}a^{21}+\frac{59114075}{6737625088}a^{20}-\frac{25126799}{6737625088}a^{19}-\frac{43329755}{3368812544}a^{18}+\frac{101679101}{6737625088}a^{17}+\frac{21006545}{1684406272}a^{16}-\frac{76467819}{1684406272}a^{15}-\frac{33182601}{842203136}a^{14}+\frac{31805343}{210550784}a^{13}-\frac{1267647}{6579712}a^{12}+\frac{3726455}{26318848}a^{11}+\frac{52647}{370688}a^{10}-\frac{595485}{822464}a^{9}+\frac{1289517}{3289856}a^{8}-\frac{143813}{1644928}a^{7}+\frac{1025409}{1644928}a^{6}-\frac{37471}{25702}a^{5}+\frac{350175}{205616}a^{4}-\frac{954939}{411232}a^{3}-\frac{361483}{102808}a^{2}+\frac{1203583}{102808}a-\frac{591543}{51404}$, $\frac{35533}{6737625088}a^{35}+\frac{111189}{6737625088}a^{34}-\frac{89397}{6737625088}a^{33}-\frac{46773}{1684406272}a^{32}+\frac{173553}{6737625088}a^{31}+\frac{21781}{3368812544}a^{30}+\frac{46773}{842203136}a^{29}-\frac{233185}{3368812544}a^{28}-\frac{75753}{1684406272}a^{27}-\frac{343059}{1684406272}a^{26}+\frac{481719}{3368812544}a^{25}+\frac{1327437}{3368812544}a^{24}-\frac{151671}{3368812544}a^{23}+\frac{95235}{210550784}a^{22}-\frac{9415827}{6737625088}a^{21}-\frac{4656645}{6737625088}a^{20}+\frac{4213449}{6737625088}a^{19}+\frac{1241091}{3368812544}a^{18}-\frac{21883671}{6737625088}a^{17}-\frac{96633}{421101568}a^{16}+\frac{10680075}{1684406272}a^{15}-\frac{2377195}{842203136}a^{14}-\frac{4796043}{210550784}a^{13}+\frac{428459}{26318848}a^{12}+\frac{390177}{52637696}a^{11}-\frac{64491}{6579712}a^{10}+\frac{54585}{6579712}a^{9}-\frac{882981}{6579712}a^{8}+\frac{69209}{1644928}a^{7}+\frac{110889}{1644928}a^{6}+\frac{98715}{411232}a^{5}-\frac{29355}{411232}a^{4}-\frac{110889}{411232}a^{3}-\frac{23619}{51404}a^{2}-\frac{95673}{102808}a+\frac{148491}{51404}$, $\frac{29621}{6737625088}a^{35}-\frac{53157}{6737625088}a^{34}-\frac{13035}{6737625088}a^{33}-\frac{2487}{3368812544}a^{32}-\frac{255891}{6737625088}a^{31}+\frac{6653}{421101568}a^{30}+\frac{36943}{842203136}a^{29}-\frac{71445}{3368812544}a^{28}-\frac{27453}{421101568}a^{27}+\frac{127101}{1684406272}a^{26}-\frac{122925}{3368812544}a^{25}-\frac{1312457}{3368812544}a^{24}+\frac{2201371}{3368812544}a^{23}-\frac{194619}{1684406272}a^{22}-\frac{4424163}{6737625088}a^{21}-\frac{4159599}{6737625088}a^{20}+\frac{4727751}{6737625088}a^{19}-\frac{1310319}{842203136}a^{18}+\frac{12343769}{6737625088}a^{17}+\frac{21027}{18612224}a^{16}-\frac{612783}{210550784}a^{15}-\frac{492303}{210550784}a^{14}+\frac{2076145}{210550784}a^{13}-\frac{234807}{26318848}a^{12}+\frac{12091}{3289856}a^{11}+\frac{809139}{13159424}a^{10}-\frac{231639}{3289856}a^{9}-\frac{89385}{1644928}a^{8}+\frac{2367}{51404}a^{7}-\frac{44881}{1644928}a^{6}-\frac{13729}{411232}a^{5}+\frac{5001}{25702}a^{4}+\frac{107643}{411232}a^{3}-\frac{153843}{205616}a^{2}+\frac{24}{71}a+\frac{6483}{12851}$, $\frac{27585}{1684406272}a^{35}-\frac{115479}{3368812544}a^{34}+\frac{54447}{3368812544}a^{33}+\frac{17551}{3368812544}a^{32}-\frac{8103}{210550784}a^{31}+\frac{303483}{3368812544}a^{30}-\frac{27585}{421101568}a^{29}-\frac{8193}{842203136}a^{28}-\frac{164379}{1684406272}a^{27}+\frac{183821}{421101568}a^{26}-\frac{127257}{421101568}a^{25}-\frac{1091149}{1684406272}a^{24}+\frac{2290671}{1684406272}a^{23}-\frac{3520407}{1684406272}a^{22}+\frac{1643967}{1684406272}a^{21}+\frac{9242605}{3368812544}a^{20}-\frac{5036115}{3368812544}a^{19}-\frac{9589495}{3368812544}a^{18}+\frac{5172249}{1684406272}a^{17}+\frac{4016475}{3368812544}a^{16}-\frac{21384411}{1684406272}a^{15}-\frac{2019115}{421101568}a^{14}+\frac{123849}{2965504}a^{13}-\frac{1324715}{26318848}a^{12}+\frac{1414251}{26318848}a^{11}+\frac{155713}{3289856}a^{10}-\frac{1390917}{6579712}a^{9}+\frac{1314449}{6579712}a^{8}-\frac{8799}{411232}a^{7}+\frac{54879}{1644928}a^{6}-\frac{62439}{205616}a^{5}+\frac{47739}{102808}a^{4}-\frac{17339}{51404}a^{3}-\frac{152291}{205616}a^{2}+\frac{352545}{102808}a-\frac{80081}{25702}$, $\frac{226091}{6737625088}a^{35}-\frac{79183}{6737625088}a^{34}-\frac{20665}{6737625088}a^{33}-\frac{24283}{3368812544}a^{32}-\frac{485493}{6737625088}a^{31}+\frac{119981}{1684406272}a^{30}+\frac{11773}{421101568}a^{29}+\frac{399761}{3368812544}a^{28}-\frac{175335}{842203136}a^{27}+\frac{868375}{1684406272}a^{26}+\frac{549613}{3368812544}a^{25}-\frac{1186147}{3368812544}a^{24}+\frac{3046545}{3368812544}a^{23}-\frac{3378555}{1684406272}a^{22}-\frac{12529901}{6737625088}a^{21}+\frac{8418347}{6737625088}a^{20}+\frac{18862781}{6737625088}a^{19}-\frac{3992433}{1684406272}a^{18}+\frac{29745831}{6737625088}a^{17}+\frac{27627851}{3368812544}a^{16}-\frac{54171}{6579712}a^{15}-\frac{34983883}{842203136}a^{14}+\frac{281313}{26318848}a^{13}-\frac{2366891}{52637696}a^{12}-\frac{228091}{52637696}a^{11}+\frac{2112191}{13159424}a^{10}-\frac{1648889}{13159424}a^{9}+\frac{59415}{3289856}a^{8}-\frac{12059}{411232}a^{7}+\frac{191573}{822464}a^{6}-\frac{188897}{411232}a^{5}+\frac{20123}{411232}a^{4}-\frac{197475}{411232}a^{3}-\frac{466589}{205616}a^{2}+\frac{42502}{12851}a+\frac{22831}{51404}$, $\frac{175793}{6737625088}a^{35}-\frac{144515}{6737625088}a^{34}-\frac{96505}{6737625088}a^{33}+\frac{18001}{1684406272}a^{32}-\frac{103687}{6737625088}a^{31}-\frac{6885}{3368812544}a^{30}-\frac{43221}{1684406272}a^{29}+\frac{325903}{3368812544}a^{28}-\frac{323883}{1684406272}a^{27}+\frac{390263}{1684406272}a^{26}+\frac{203715}{3368812544}a^{25}-\frac{3035259}{3368812544}a^{24}-\frac{738987}{3368812544}a^{23}-\frac{373699}{210550784}a^{22}+\frac{2130857}{6737625088}a^{21}+\frac{13458587}{6737625088}a^{20}+\frac{10830709}{6737625088}a^{19}-\frac{2133789}{3368812544}a^{18}+\frac{14150689}{6737625088}a^{17}+\frac{8460195}{1684406272}a^{16}-\frac{5566101}{842203136}a^{15}-\frac{32431535}{842203136}a^{14}+\frac{2413013}{105275392}a^{13}-\frac{795783}{52637696}a^{12}+\frac{248443}{13159424}a^{11}+\frac{1460013}{26318848}a^{10}-\frac{430881}{13159424}a^{9}+\frac{217231}{3289856}a^{8}-\frac{313051}{3289856}a^{7}+\frac{156423}{822464}a^{6}-\frac{130085}{411232}a^{5}-\frac{1203}{51404}a^{4}+\frac{2921}{411232}a^{3}+\frac{6293}{12851}a^{2}+\frac{195435}{51404}a-\frac{43659}{51404}$, $\frac{56119}{1684406272}a^{35}-\frac{252429}{3368812544}a^{34}-\frac{115583}{3368812544}a^{33}+\frac{231113}{3368812544}a^{32}+\frac{29131}{842203136}a^{31}+\frac{755841}{3368812544}a^{30}-\frac{45727}{842203136}a^{29}-\frac{223641}{842203136}a^{28}-\frac{1495009}{1684406272}a^{27}+\frac{68123}{105275392}a^{26}+\frac{210813}{421101568}a^{25}-\frac{83959}{1684406272}a^{24}+\frac{4233609}{1684406272}a^{23}-\frac{5282441}{1684406272}a^{22}-\frac{981907}{1684406272}a^{21}+\frac{12361327}{3368812544}a^{20}-\frac{3434421}{3368812544}a^{19}-\frac{30446513}{3368812544}a^{18}+\frac{5936413}{1684406272}a^{17}+\frac{37192873}{3368812544}a^{16}-\frac{35585235}{1684406272}a^{15}-\frac{11463421}{421101568}a^{14}+\frac{11477849}{105275392}a^{13}-\frac{2067291}{105275392}a^{12}+\frac{1602681}{52637696}a^{11}+\frac{166843}{13159424}a^{10}-\frac{3268511}{6579712}a^{9}-\frac{128711}{6579712}a^{8}+\frac{1031373}{3289856}a^{7}+\frac{108721}{102808}a^{6}-\frac{140197}{822464}a^{5}+\frac{44935}{411232}a^{4}-\frac{101683}{51404}a^{3}-\frac{563613}{205616}a^{2}+\frac{663965}{102808}a-\frac{29294}{12851}$, $\frac{108743}{6737625088}a^{35}-\frac{64533}{6737625088}a^{34}-\frac{199127}{6737625088}a^{33}-\frac{51777}{1684406272}a^{32}+\frac{165247}{6737625088}a^{31}+\frac{219593}{3368812544}a^{30}+\frac{178995}{1684406272}a^{29}+\frac{228157}{3368812544}a^{28}-\frac{721833}{1684406272}a^{27}-\frac{31555}{1684406272}a^{26}+\frac{13341}{3368812544}a^{25}-\frac{641853}{3368812544}a^{24}+\frac{3864107}{3368812544}a^{23}-\frac{86193}{105275392}a^{22}-\frac{4173009}{6737625088}a^{21}+\frac{4256269}{6737625088}a^{20}+\frac{7281643}{6737625088}a^{19}-\frac{3597843}{3368812544}a^{18}-\frac{20068201}{6737625088}a^{17}+\frac{15264211}{1684406272}a^{16}-\frac{2436603}{421101568}a^{15}-\frac{7850211}{421101568}a^{14}+\frac{2270125}{210550784}a^{13}+\frac{234899}{13159424}a^{12}+\frac{2294237}{52637696}a^{11}+\frac{565699}{13159424}a^{10}-\frac{115921}{1644928}a^{9}-\frac{1459271}{6579712}a^{8}-\frac{130977}{3289856}a^{7}+\frac{620799}{1644928}a^{6}+\frac{87035}{411232}a^{5}+\frac{200111}{411232}a^{4}-\frac{148023}{411232}a^{3}-\frac{196625}{102808}a^{2}+\frac{65923}{51404}a+\frac{5128}{12851}$, $\frac{200643}{6737625088}a^{35}-\frac{110641}{6737625088}a^{34}-\frac{251427}{6737625088}a^{33}+\frac{22515}{1684406272}a^{32}+\frac{28195}{6737625088}a^{31}+\frac{292729}{3368812544}a^{30}+\frac{26977}{1684406272}a^{29}-\frac{482055}{3368812544}a^{28}-\frac{738577}{1684406272}a^{27}+\frac{419181}{1684406272}a^{26}+\frac{2329937}{3368812544}a^{25}-\frac{186425}{3368812544}a^{24}+\frac{2793623}{3368812544}a^{23}-\frac{229719}{210550784}a^{22}-\frac{14930757}{6737625088}a^{21}+\frac{11590201}{6737625088}a^{20}+\frac{7677687}{6737625088}a^{19}-\frac{16994335}{3368812544}a^{18}-\frac{3649509}{6737625088}a^{17}+\frac{10124817}{1684406272}a^{16}-\frac{2173659}{842203136}a^{15}-\frac{18431883}{421101568}a^{14}+\frac{6460479}{210550784}a^{13}+\frac{2104013}{105275392}a^{12}-\frac{37285}{52637696}a^{11}+\frac{2555647}{26318848}a^{10}-\frac{2506581}{13159424}a^{9}-\frac{436893}{6579712}a^{8}+\frac{236671}{3289856}a^{7}+\frac{928451}{1644928}a^{6}-\frac{131771}{822464}a^{5}-\frac{212235}{411232}a^{4}-\frac{248745}{411232}a^{3}-\frac{31145}{12851}a^{2}+\frac{46520}{12851}a+\frac{33298}{12851}$, $\frac{68619}{3368812544}a^{35}-\frac{181915}{3368812544}a^{34}+\frac{114279}{3368812544}a^{33}+\frac{37525}{1684406272}a^{32}-\frac{83897}{3368812544}a^{31}+\frac{10821}{210550784}a^{30}-\frac{124319}{842203136}a^{29}-\frac{28959}{1684406272}a^{28}-\frac{30801}{105275392}a^{27}+\frac{673173}{842203136}a^{26}-\frac{515947}{1684406272}a^{25}-\frac{284463}{1684406272}a^{24}+\frac{952377}{1684406272}a^{23}-\frac{2320935}{842203136}a^{22}+\frac{6324891}{3368812544}a^{21}+\frac{12734927}{3368812544}a^{20}-\frac{8139755}{3368812544}a^{19}-\frac{4222065}{842203136}a^{18}+\frac{17868739}{3368812544}a^{17}+\frac{11567037}{1684406272}a^{16}-\frac{11935305}{842203136}a^{15}-\frac{1640217}{105275392}a^{14}+\frac{6300959}{105275392}a^{13}-\frac{6914683}{105275392}a^{12}+\frac{1998855}{26318848}a^{11}-\frac{734703}{26318848}a^{10}-\frac{2309601}{13159424}a^{9}+\frac{1774023}{6579712}a^{8}+\frac{296631}{3289856}a^{7}+\frac{229515}{822464}a^{6}-\frac{717489}{822464}a^{5}+\frac{112121}{205616}a^{4}-\frac{92397}{102808}a^{3}+\frac{22251}{51404}a^{2}+\frac{93371}{25702}a-\frac{116139}{25702}$, $\frac{64351}{6737625088}a^{35}+\frac{31119}{6737625088}a^{34}-\frac{98687}{6737625088}a^{33}+\frac{6981}{1684406272}a^{32}-\frac{277069}{6737625088}a^{31}+\frac{131247}{3368812544}a^{30}+\frac{35177}{842203136}a^{29}-\frac{247831}{3368812544}a^{28}+\frac{60685}{1684406272}a^{27}-\frac{44305}{1684406272}a^{26}+\frac{1850965}{3368812544}a^{25}-\frac{1438089}{3368812544}a^{24}+\frac{3007099}{3368812544}a^{23}-\frac{128333}{105275392}a^{22}-\frac{5263217}{6737625088}a^{21}+\frac{6560993}{6737625088}a^{20}+\frac{2276635}{6737625088}a^{19}-\frac{5462223}{3368812544}a^{18}-\frac{8923141}{6737625088}a^{17}+\frac{2678319}{421101568}a^{16}-\frac{1275995}{1684406272}a^{15}-\frac{532191}{210550784}a^{14}-\frac{648353}{105275392}a^{13}+\frac{62241}{3289856}a^{12}-\frac{200037}{26318848}a^{11}+\frac{2925687}{26318848}a^{10}-\frac{221571}{1644928}a^{9}+\frac{33519}{6579712}a^{8}-\frac{52035}{3289856}a^{7}-\frac{15171}{411232}a^{6}+\frac{41401}{411232}a^{5}-\frac{71687}{205616}a^{4}+\frac{147447}{411232}a^{3}-\frac{63703}{51404}a^{2}+\frac{271537}{102808}a+\frac{7197}{25702}$, $\frac{19071}{6737625088}a^{35}-\frac{126625}{6737625088}a^{34}+\frac{30753}{6737625088}a^{33}+\frac{50099}{1684406272}a^{32}-\frac{397637}{6737625088}a^{31}-\frac{133093}{3368812544}a^{30}+\frac{16035}{842203136}a^{29}+\frac{36453}{3368812544}a^{28}-\frac{396391}{1684406272}a^{27}+\frac{272783}{1684406272}a^{26}-\frac{136195}{3368812544}a^{25}-\frac{2483785}{3368812544}a^{24}+\frac{1426059}{3368812544}a^{23}+\frac{37413}{421101568}a^{22}-\frac{1214145}{6737625088}a^{21}-\frac{1036575}{6737625088}a^{20}+\frac{4271515}{6737625088}a^{19}-\frac{4235315}{3368812544}a^{18}-\frac{4674157}{6737625088}a^{17}+\frac{4281487}{842203136}a^{16}-\frac{7715475}{1684406272}a^{15}-\frac{1368281}{842203136}a^{14}+\frac{8270621}{210550784}a^{13}-\frac{149967}{13159424}a^{12}-\frac{628101}{26318848}a^{11}+\frac{981685}{13159424}a^{10}+\frac{384149}{6579712}a^{9}-\frac{99623}{3289856}a^{8}+\frac{77395}{1644928}a^{7}+\frac{422419}{1644928}a^{6}-\frac{44895}{205616}a^{5}+\frac{24483}{205616}a^{4}+\frac{225377}{411232}a^{3}-\frac{4335}{51404}a^{2}+\frac{11575}{102808}a-\frac{62291}{51404}$ 9399/842203136a^35 - 66801/3368812544a^34 + 29205/3368812544a^33 + 8391/3368812544a^32 - 27861/1684406272a^31 + 211779/3368812544a^30 - 30327/842203136a^29 - 10407/421101568a^28 - 196371/1684406272a^27 + 124203/421101568a^26 - 166167/842203136a^25 - 95207/1684406272a^24 + 1735017/1684406272a^23 - 2373951/1684406272a^22 + 88371/210550784a^21 + 5134959/3368812544a^20 - 2662377/3368812544a^19 - 7788619/3368812544a^18 + 1308177/421101568a^17 + 3097767/3368812544a^16 - 15846885/1684406272a^15 - 670299/210550784a^14 + 536079/26318848a^13 - 854537/26318848a^12 + 475671/13159424a^11 + 117147/6579712a^10 - 261477/1644928a^9 + 897045/6579712a^8 + 11079/822464a^7 + 15329/205616a^6 - 9399/51404a^5 + 66801/205616a^4 - 29205/51404a^3 - 133971/205616a^2 + 253971/102808a - 22267/12851, 29205/1684406272a^35 - 87615/3368812544a^34 + 29205/3368812544a^33 + 29205/3368812544a^32 - 48675/1684406272a^31 + 267283/3368812544a^30 - 29205/1684406272a^29 - 321255/1684406272a^27 + 321255/842203136a^26 - 9735/52637696a^25 - 379665/1684406272a^24 + 2317809/1684406272a^23 - 2716065/1684406272a^22 + 613305/1684406272a^21 + 6571125/3368812544a^20 - 2287725/3368812544a^19 - 10543005/3368812544a^18 + 87615/26318848a^17 + 8842431/3368812544a^16 - 4234725/421101568a^15 - 1898325/210550784a^14 + 827475/26318848a^13 - 1138995/26318848a^12 + 496485/13159424a^11 + 262845/6579712a^10 - 323919/1644928a^9 + 29205/411232a^8 - 9735/822464a^7 + 29205/205616a^6 - 29205/102808a^5 + 87615/205616a^4 - 29205/51404a^3 - 217227/205616a^2 + 68145/25702a - 42056/12851, 12195/842203136a^35 - 28455/842203136a^34 - 165/842203136a^33 + 12195/421101568a^32 - 36585/842203136a^31 + 12195/210550784a^30 - 12195/210550784a^29 + 4065/421101568a^28 - 12195/105275392a^27 + 59199/210550784a^26 - 109755/421101568a^25 - 207315/421101568a^24 + 475605/421101568a^23 - 345525/210550784a^22 + 792675/842203136a^21 + 1768275/842203136a^20 - 1131343/842203136a^19 - 36585/13159424a^18 + 4402395/842203136a^17 + 955275/421101568a^16 - 2743875/210550784a^15 - 256095/52637696a^14 + 1134135/26318848a^13 - 4065073/105275392a^12 + 158535/6579712a^11 + 4065/102808a^10 - 134145/822464a^9 + 134145/822464a^8 + 12195/205616a^6 - 88817/822464a^5 + 20325/51404a^4 - 12195/51404a^3 - 12195/25702a^2 + 36585/12851a - 48780/12851, 21807/3368812544a^35 - 36345/3368812544a^34 + 7269/3368812544a^33 + 2423/421101568a^32 - 62201/3368812544a^31 + 50883/1684406272a^30 - 21807/842203136a^29 + 7269/1684406272a^28 - 50883/842203136a^27 + 113881/842203136a^26 - 50883/1684406272a^25 + 60903/1684406272a^24 + 545175/1684406272a^23 - 283491/421101568a^22 + 1242999/3368812544a^21 + 2439961/3368812544a^20 - 1839057/3368812544a^19 - 1490145/1684406272a^18 + 8947559/3368812544a^17 + 225339/210550784a^16 - 167187/52637696a^15 - 462793/105275392a^14 + 327105/26318848a^13 - 21807/1644928a^12 + 21807/1644928a^11 + 929431/26318848a^10 - 50883/822464a^9 + 41191/822464a^8 - 7269/411232a^7 + 7269/102808a^6 - 21807/205616a^5 + 7269/102808a^4 - 36447/51404a^3 - 2423/12851a^2 + 14538/12851a - 14538/12851, 59149/6737625088a^35 - 224807/6737625088a^34 + 58899/6737625088a^33 + 62619/1684406272a^32 - 10419/6737625088a^31 + 90335/3368812544a^30 - 128675/1684406272a^29 + 10419/3368812544a^28 - 286647/1684406272a^27 + 552639/1684406272a^26 - 268233/3368812544a^25 - 672367/3368812544a^24 + 1192697/3368812544a^23 - 464031/421101568a^22 + 6388341/6737625088a^21 + 5957407/6737625088a^20 - 6482583/6737625088a^19 - 5625333/3368812544a^18 + 18478885/6737625088a^17 + 3822291/1684406272a^16 - 3920661/421101568a^15 - 4103817/842203136a^14 + 436529/13159424a^13 - 309523/13159424a^12 + 4543/26318848a^11 - 206991/26318848a^10 - 1060365/13159424a^9 + 1096143/6579712a^8 + 369/23168a^7 + 64203/822464a^6 - 33461/102808a^5 + 11313/205616a^4 - 128451/411232a^3 + 7773/51404a^2 + 115779/51404a - 118087/51404, 503201/6737625088a^35 - 785059/6737625088a^34 + 314211/6737625088a^33 + 7539/105275392a^32 - 793235/6737625088a^31 + 1129803/3368812544a^30 - 117851/842203136a^29 + 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152291/205616a^2 + 352545/102808a - 80081/25702, 226091/6737625088a^35 - 79183/6737625088a^34 - 20665/6737625088a^33 - 24283/3368812544a^32 - 485493/6737625088a^31 + 119981/1684406272a^30 + 11773/421101568a^29 + 399761/3368812544a^28 - 175335/842203136a^27 + 868375/1684406272a^26 + 549613/3368812544a^25 - 1186147/3368812544a^24 + 3046545/3368812544a^23 - 3378555/1684406272a^22 - 12529901/6737625088a^21 + 8418347/6737625088a^20 + 18862781/6737625088a^19 - 3992433/1684406272a^18 + 29745831/6737625088a^17 + 27627851/3368812544a^16 - 54171/6579712a^15 - 34983883/842203136a^14 + 281313/26318848a^13 - 2366891/52637696a^12 - 228091/52637696a^11 + 2112191/13159424a^10 - 1648889/13159424a^9 + 59415/3289856a^8 - 12059/411232a^7 + 191573/822464a^6 - 188897/411232a^5 + 20123/411232a^4 - 197475/411232a^3 - 466589/205616a^2 + 42502/12851a + 22831/51404, 175793/6737625088a^35 - 144515/6737625088a^34 - 96505/6737625088a^33 + 18001/1684406272a^32 - 103687/6737625088a^31 - 6885/3368812544a^30 - 43221/1684406272a^29 + 325903/3368812544a^28 - 323883/1684406272a^27 + 390263/1684406272a^26 + 203715/3368812544a^25 - 3035259/3368812544a^24 - 738987/3368812544a^23 - 373699/210550784a^22 + 2130857/6737625088a^21 + 13458587/6737625088a^20 + 10830709/6737625088a^19 - 2133789/3368812544a^18 + 14150689/6737625088a^17 + 8460195/1684406272a^16 - 5566101/842203136a^15 - 32431535/842203136a^14 + 2413013/105275392a^13 - 795783/52637696a^12 + 248443/13159424a^11 + 1460013/26318848a^10 - 430881/13159424a^9 + 217231/3289856a^8 - 313051/3289856a^7 + 156423/822464a^6 - 130085/411232a^5 - 1203/51404a^4 + 2921/411232a^3 + 6293/12851a^2 + 195435/51404a - 43659/51404, 56119/1684406272a^35 - 252429/3368812544a^34 - 115583/3368812544a^33 + 231113/3368812544a^32 + 29131/842203136a^31 + 755841/3368812544a^30 - 45727/842203136a^29 - 223641/842203136a^28 - 1495009/1684406272a^27 + 68123/105275392a^26 + 210813/421101568a^25 - 83959/1684406272a^24 + 4233609/1684406272a^23 - 5282441/1684406272a^22 - 981907/1684406272a^21 + 12361327/3368812544a^20 - 3434421/3368812544a^19 - 30446513/3368812544a^18 + 5936413/1684406272a^17 + 37192873/3368812544a^16 - 35585235/1684406272a^15 - 11463421/421101568a^14 + 11477849/105275392a^13 - 2067291/105275392a^12 + 1602681/52637696a^11 + 166843/13159424a^10 - 3268511/6579712a^9 - 128711/6579712a^8 + 1031373/3289856a^7 + 108721/102808a^6 - 140197/822464a^5 + 44935/411232a^4 - 101683/51404a^3 - 563613/205616a^2 + 663965/102808a - 29294/12851, 108743/6737625088a^35 - 64533/6737625088a^34 - 199127/6737625088a^33 - 51777/1684406272a^32 + 165247/6737625088a^31 + 219593/3368812544a^30 + 178995/1684406272a^29 + 228157/3368812544a^28 - 721833/1684406272a^27 - 31555/1684406272a^26 + 13341/3368812544a^25 - 641853/3368812544a^24 + 3864107/3368812544a^23 - 86193/105275392a^22 - 4173009/6737625088a^21 + 4256269/6737625088a^20 + 7281643/6737625088a^19 - 3597843/3368812544a^18 - 20068201/6737625088a^17 + 15264211/1684406272a^16 - 2436603/421101568a^15 - 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928451/1644928a^6 - 131771/822464a^5 - 212235/411232a^4 - 248745/411232a^3 - 31145/12851a^2 + 46520/12851a + 33298/12851, 68619/3368812544a^35 - 181915/3368812544a^34 + 114279/3368812544a^33 + 37525/1684406272a^32 - 83897/3368812544a^31 + 10821/210550784a^30 - 124319/842203136a^29 - 28959/1684406272a^28 - 30801/105275392a^27 + 673173/842203136a^26 - 515947/1684406272a^25 - 284463/1684406272a^24 + 952377/1684406272a^23 - 2320935/842203136a^22 + 6324891/3368812544a^21 + 12734927/3368812544a^20 - 8139755/3368812544a^19 - 4222065/842203136a^18 + 17868739/3368812544a^17 + 11567037/1684406272a^16 - 11935305/842203136a^15 - 1640217/105275392a^14 + 6300959/105275392a^13 - 6914683/105275392a^12 + 1998855/26318848a^11 - 734703/26318848a^10 - 2309601/13159424a^9 + 1774023/6579712a^8 + 296631/3289856a^7 + 229515/822464a^6 - 717489/822464a^5 + 112121/205616a^4 - 92397/102808a^3 + 22251/51404a^2 + 93371/25702a - 116139/25702, 64351/6737625088a^35 + 31119/6737625088a^34 - 98687/6737625088a^33 + 6981/1684406272a^32 - 277069/6737625088a^31 + 131247/3368812544a^30 + 35177/842203136a^29 - 247831/3368812544a^28 + 60685/1684406272a^27 - 44305/1684406272a^26 + 1850965/3368812544a^25 - 1438089/3368812544a^24 + 3007099/3368812544a^23 - 128333/105275392a^22 - 5263217/6737625088a^21 + 6560993/6737625088a^20 + 2276635/6737625088a^19 - 5462223/3368812544a^18 - 8923141/6737625088a^17 + 2678319/421101568a^16 - 1275995/1684406272a^15 - 532191/210550784a^14 - 648353/105275392a^13 + 62241/3289856a^12 - 200037/26318848a^11 + 2925687/26318848a^10 - 221571/1644928a^9 + 33519/6579712a^8 - 52035/3289856a^7 - 15171/411232a^6 + 41401/411232a^5 - 71687/205616a^4 + 147447/411232a^3 - 63703/51404a^2 + 271537/102808a + 7197/25702, 19071/6737625088a^35 - 126625/6737625088a^34 + 30753/6737625088a^33 + 50099/1684406272a^32 - 397637/6737625088a^31 - 133093/3368812544a^30 + 16035/842203136a^29 + 36453/3368812544a^28 - 396391/1684406272a^27 + 272783/1684406272a^26 - 136195/3368812544a^25 - 2483785/3368812544a^24 + 1426059/3368812544a^23 + 37413/421101568a^22 - 1214145/6737625088a^21 - 1036575/6737625088a^20 + 4271515/6737625088a^19 - 4235315/3368812544a^18 - 4674157/6737625088a^17 + 4281487/842203136a^16 - 7715475/1684406272a^15 - 1368281/842203136a^14 + 8270621/210550784a^13 - 149967/13159424a^12 - 628101/26318848a^11 + 981685/13159424a^10 + 384149/6579712a^9 - 99623/3289856a^8 + 77395/1644928a^7 + 422419/1644928a^6 - 44895/205616a^5 + 24483/205616a^4 + 225377/411232a^3 - 4335/51404a^2 + 11575/102808*a - 62291/51404 (assuming GRH)	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 104721329817071.11 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 104721329817071.11 \cdot 196}{14\cdot\sqrt{230309323306002912742337717628704661399933091124862510404769}}\cr\approx \mathstrut & 0.711614936230257 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^36 - 3*x^35 + 3*x^34 - 3*x^32 + 6*x^31 - 8*x^30 + 6*x^29 - 12*x^28 + 36*x^27 - 42*x^26 - 6*x^25 + 82*x^24 - 168*x^23 + 177*x^22 + 27*x^21 - 207*x^20 - 54*x^19 + 397*x^18 - 108*x^17 - 828*x^16 + 216*x^15 + 2832*x^14 - 5376*x^13 + 5248*x^12 - 768*x^11 - 10752*x^10 + 18432*x^9 - 12288*x^8 + 12288*x^7 - 32768*x^6 + 49152*x^5 - 49152*x^4 + 196608*x^2 - 393216*x + 262144)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^36 - 3*x^35 + 3*x^34 - 3*x^32 + 6*x^31 - 8*x^30 + 6*x^29 - 12*x^28 + 36*x^27 - 42*x^26 - 6*x^25 + 82*x^24 - 168*x^23 + 177*x^22 + 27*x^21 - 207*x^20 - 54*x^19 + 397*x^18 - 108*x^17 - 828*x^16 + 216*x^15 + 2832*x^14 - 5376*x^13 + 5248*x^12 - 768*x^11 - 10752*x^10 + 18432*x^9 - 12288*x^8 + 12288*x^7 - 32768*x^6 + 49152*x^5 - 49152*x^4 + 196608*x^2 - 393216*x + 262144, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^36 - 3*x^35 + 3*x^34 - 3*x^32 + 6*x^31 - 8*x^30 + 6*x^29 - 12*x^28 + 36*x^27 - 42*x^26 - 6*x^25 + 82*x^24 - 168*x^23 + 177*x^22 + 27*x^21 - 207*x^20 - 54*x^19 + 397*x^18 - 108*x^17 - 828*x^16 + 216*x^15 + 2832*x^14 - 5376*x^13 + 5248*x^12 - 768*x^11 - 10752*x^10 + 18432*x^9 - 12288*x^8 + 12288*x^7 - 32768*x^6 + 49152*x^5 - 49152*x^4 + 196608*x^2 - 393216*x + 262144);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 3*x^35 + 3*x^34 - 3*x^32 + 6*x^31 - 8*x^30 + 6*x^29 - 12*x^28 + 36*x^27 - 42*x^26 - 6*x^25 + 82*x^24 - 168*x^23 + 177*x^22 + 27*x^21 - 207*x^20 - 54*x^19 + 397*x^18 - 108*x^17 - 828*x^16 + 216*x^15 + 2832*x^14 - 5376*x^13 + 5248*x^12 - 768*x^11 - 10752*x^10 + 18432*x^9 - 12288*x^8 + 12288*x^7 - 32768*x^6 + 49152*x^5 - 49152*x^4 + 196608*x^2 - 393216*x + 262144);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_2^2:C_6^2$ (as 36T103):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 144

The 48 conjugacy class representatives for $C_2^2:C_6^2$

Character table for $C_2^2:C_6^2$ is not computed

Intermediate fields

$\Q(\sqrt{-7}) $, $\Q(\zeta_{9})^+$, $\Q(\zeta_{7})^+$, 3.3.3969.2, 3.3.3969.1, 6.6.159780033.1, 6.0.465831.1, 6.0.110270727.2, 6.0.2250423.1, $\Q(\zeta_{7})$, 6.0.110270727.1, 9.9.62523502209.1, 12.0.25529658945481089.1, 18.0.1340851596668237962730583.1, 18.18.479905535815125717478865692113.1, 18.0.1399141503834185765244506391.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Sibling fields

Degree 36 siblings:	deg 36, deg 36, some data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.3.0.1}{3} }^{12}$	R	${\href{/padicField/5.6.0.1}{6} }^{6}$	R	${\href{/padicField/11.6.0.1}{6} }^{6}$	${\href{/padicField/13.6.0.1}{6} }^{6}$	${\href{/padicField/17.6.0.1}{6} }^{6}$	${\href{/padicField/19.6.0.1}{6} }^{6}$	${\href{/padicField/23.6.0.1}{6} }^{6}$	${\href{/padicField/29.3.0.1}{3} }^{12}$	${\href{/padicField/31.6.0.1}{6} }^{6}$	${\href{/padicField/37.6.0.1}{6} }^{4}{,}\,{\href{/padicField/37.3.0.1}{3} }^{4}$	${\href{/padicField/41.6.0.1}{6} }^{6}$	${\href{/padicField/43.3.0.1}{3} }^{12}$	${\href{/padicField/47.6.0.1}{6} }^{6}$	${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{8}$	${\href{/padicField/59.6.0.1}{6} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$3$ 3		Deg $18$	$3$	$6$	$24$
$3$ 3		Deg $18$	$3$	$6$	$24$
$7$ 7		Deg $36$	$6$	$6$	$30$
$71$ 71	$\Q_{71}$	$x + 64$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{71}$	$x + 64$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{71}$	$x + 64$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{71}$	$x + 64$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{71}$	$x + 64$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{71}$	$x + 64$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{71}$	$x + 64$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{71}$	$x + 64$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{71}$	$x + 64$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{71}$	$x + 64$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{71}$	$x + 64$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{71}$	$x + 64$	$1$	$1$	$0$	Trivial	$[\ ]$
	71.2.0.1	$x^{2} + 69 x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	71.2.1.1	$x^{2} + 497$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.1	$x^{2} + 497$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.0.1	$x^{2} + 69 x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	71.2.0.1	$x^{2} + 69 x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	71.2.0.1	$x^{2} + 69 x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	71.2.1.1	$x^{2} + 497$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.1	$x^{2} + 497$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.0.1	$x^{2} + 69 x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	71.2.1.1	$x^{2} + 497$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.1.1	$x^{2} + 497$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.0.1	$x^{2} + 69 x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$

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