LMFDB - Number field 36.0.2369053316568954522538979736219298297541457293019971584.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + x^33 - 3*x^32 - x^30 - 34*x^29 + 37*x^28 + x^27 - 12*x^26 + 83*x^25 - x^24 + 40*x^23 + 554*x^22 - 720*x^21 - 68*x^20 + 48*x^19 - 1240*x^18 + 96*x^17 - 272*x^16 - 5760*x^15 + 8864*x^14 + 1280*x^13 - 64*x^12 + 10624*x^11 - 3072*x^10 + 512*x^9 + 37888*x^8 - 69632*x^7 - 4096*x^6 - 49152*x^4 + 32768*x^3 - 131072*x + 262144)

gp: K = bnfinit(y^36 - y^35 + y^33 - 3*y^32 - y^30 - 34*y^29 + 37*y^28 + y^27 - 12*y^26 + 83*y^25 - y^24 + 40*y^23 + 554*y^22 - 720*y^21 - 68*y^20 + 48*y^19 - 1240*y^18 + 96*y^17 - 272*y^16 - 5760*y^15 + 8864*y^14 + 1280*y^13 - 64*y^12 + 10624*y^11 - 3072*y^10 + 512*y^9 + 37888*y^8 - 69632*y^7 - 4096*y^6 - 49152*y^4 + 32768*y^3 - 131072*y + 262144, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - x^35 + x^33 - 3*x^32 - x^30 - 34*x^29 + 37*x^28 + x^27 - 12*x^26 + 83*x^25 - x^24 + 40*x^23 + 554*x^22 - 720*x^21 - 68*x^20 + 48*x^19 - 1240*x^18 + 96*x^17 - 272*x^16 - 5760*x^15 + 8864*x^14 + 1280*x^13 - 64*x^12 + 10624*x^11 - 3072*x^10 + 512*x^9 + 37888*x^8 - 69632*x^7 - 4096*x^6 - 49152*x^4 + 32768*x^3 - 131072*x + 262144);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - x^35 + x^33 - 3*x^32 - x^30 - 34*x^29 + 37*x^28 + x^27 - 12*x^26 + 83*x^25 - x^24 + 40*x^23 + 554*x^22 - 720*x^21 - 68*x^20 + 48*x^19 - 1240*x^18 + 96*x^17 - 272*x^16 - 5760*x^15 + 8864*x^14 + 1280*x^13 - 64*x^12 + 10624*x^11 - 3072*x^10 + 512*x^9 + 37888*x^8 - 69632*x^7 - 4096*x^6 - 49152*x^4 + 32768*x^3 - 131072*x + 262144)

$ x^{36} - x^{35} + x^{33} - 3 x^{32} - x^{30} - 34 x^{29} + 37 x^{28} + x^{27} - 12 x^{26} + \cdots + 262144 $ x^36 - x^35 + x^33 - 3*x^32 - x^30 - 34*x^29 + 37*x^28 + x^27 - 12*x^26 + 83*x^25 - x^24 + 40*x^23 + 554*x^22 - 720*x^21 - 68*x^20 + 48*x^19 - 1240*x^18 + 96*x^17 - 272*x^16 - 5760*x^15 + 8864*x^14 + 1280*x^13 - 64*x^12 + 10624*x^11 - 3072*x^10 + 512*x^9 + 37888*x^8 - 69632*x^7 - 4096*x^6 - 49152*x^4 + 32768*x^3 - 131072*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:	$2369053316568954522538979736219298297541457293019971584$ 2369053316568954522538979736219298297541457293019971584 $\medspace = 2^{18}\cdot 7^{32}\cdot 67^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$32.39$	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:	$2^{3/2}7^{11/12}67^{1/2}\approx 137.8020245618911$
Ramified primes:	$2$, $7$, $67$ 2, 7, 67	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{15}+\frac{1}{4}a^{13}+\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{15}-\frac{1}{4}a^{14}+\frac{1}{4}a^{12}+\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{4}$, $\frac{1}{56}a^{18}+\frac{1}{56}a^{17}+\frac{1}{14}a^{16}-\frac{13}{56}a^{15}+\frac{1}{8}a^{14}-\frac{27}{56}a^{12}-\frac{1}{2}a^{11}+\frac{3}{56}a^{10}-\frac{17}{56}a^{9}+\frac{5}{14}a^{8}-\frac{1}{8}a^{7}-\frac{27}{56}a^{6}-\frac{1}{2}a^{4}-\frac{5}{14}a^{3}+\frac{1}{7}a^{2}-\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{56}a^{19}+\frac{3}{56}a^{17}-\frac{3}{56}a^{16}+\frac{3}{28}a^{15}-\frac{1}{8}a^{14}-\frac{13}{56}a^{13}+\frac{13}{56}a^{12}-\frac{25}{56}a^{11}+\frac{11}{28}a^{10}+\frac{9}{56}a^{9}-\frac{13}{56}a^{8}-\frac{3}{28}a^{7}+\frac{27}{56}a^{6}+\frac{1}{4}a^{5}-\frac{3}{28}a^{4}-\frac{1}{2}a^{3}-\frac{1}{14}a^{2}-\frac{3}{7}a-\frac{1}{7}$, $\frac{1}{112}a^{20}-\frac{1}{112}a^{19}+\frac{5}{112}a^{17}-\frac{3}{112}a^{16}+\frac{3}{28}a^{15}-\frac{13}{112}a^{14}+\frac{27}{56}a^{13}-\frac{55}{112}a^{12}-\frac{51}{112}a^{11}-\frac{1}{14}a^{10}+\frac{15}{112}a^{9}-\frac{25}{112}a^{8}+\frac{3}{28}a^{7}+\frac{27}{56}a^{6}+\frac{1}{14}a^{5}-\frac{1}{14}a^{4}-\frac{1}{2}a^{3}-\frac{1}{7}a^{2}+\frac{2}{7}a-\frac{1}{7}$, $\frac{1}{112}a^{21}-\frac{1}{112}a^{19}-\frac{1}{112}a^{18}-\frac{1}{28}a^{17}+\frac{13}{112}a^{16}-\frac{1}{16}a^{15}-\frac{1}{112}a^{14}-\frac{29}{112}a^{13}+\frac{1}{4}a^{12}-\frac{3}{112}a^{11}+\frac{17}{112}a^{10}-\frac{5}{28}a^{9}-\frac{7}{16}a^{8}-\frac{2}{7}a^{7}+\frac{1}{4}a^{5}+\frac{5}{28}a^{4}+\frac{3}{7}a^{3}-\frac{2}{7}a^{2}+\frac{3}{7}a+\frac{3}{7}$, $\frac{1}{224}a^{22}-\frac{1}{224}a^{21}+\frac{1}{224}a^{19}+\frac{1}{224}a^{18}+\frac{1}{56}a^{17}+\frac{15}{224}a^{16}-\frac{11}{112}a^{15}+\frac{1}{224}a^{14}+\frac{1}{224}a^{13}-\frac{1}{4}a^{12}+\frac{3}{224}a^{11}+\frac{11}{224}a^{10}-\frac{23}{56}a^{9}-\frac{19}{112}a^{8}-\frac{27}{56}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{13}{28}a^{4}-\frac{3}{14}a^{3}-\frac{5}{14}a^{2}+\frac{2}{7}$, $\frac{1}{224}a^{23}-\frac{1}{224}a^{21}-\frac{1}{224}a^{20}+\frac{1}{224}a^{18}-\frac{1}{32}a^{17}-\frac{5}{224}a^{16}-\frac{17}{224}a^{15}+\frac{1}{8}a^{14}-\frac{111}{224}a^{13}-\frac{111}{224}a^{12}+\frac{13}{28}a^{11}+\frac{59}{224}a^{10}+\frac{3}{7}a^{9}+\frac{25}{56}a^{8}+\frac{1}{7}a^{7}-\frac{27}{56}a^{6}-\frac{2}{7}a^{5}-\frac{3}{14}a^{3}-\frac{2}{7}a^{2}-\frac{1}{7}a+\frac{3}{7}$, $\frac{1}{3136}a^{24}+\frac{5}{3136}a^{23}-\frac{1}{784}a^{22}-\frac{1}{3136}a^{21}+\frac{3}{3136}a^{20}-\frac{1}{196}a^{19}-\frac{1}{448}a^{18}-\frac{25}{392}a^{17}-\frac{185}{3136}a^{16}+\frac{11}{3136}a^{15}+\frac{57}{784}a^{14}+\frac{317}{3136}a^{13}+\frac{681}{3136}a^{12}+\frac{165}{392}a^{11}+\frac{65}{392}a^{10}-\frac{321}{784}a^{9}-\frac{19}{98}a^{8}+\frac{33}{392}a^{7}+\frac{5}{14}a^{6}+\frac{17}{49}a^{5}-\frac{1}{196}a^{4}+\frac{33}{98}a^{3}+\frac{19}{98}a^{2}+\frac{13}{49}a+\frac{15}{49}$, $\frac{1}{6272}a^{25}-\frac{1}{6272}a^{24}+\frac{1}{784}a^{23}+\frac{9}{6272}a^{22}-\frac{19}{6272}a^{21}+\frac{1}{784}a^{20}-\frac{9}{6272}a^{19}-\frac{9}{3136}a^{18}+\frac{59}{896}a^{17}-\frac{671}{6272}a^{16}-\frac{369}{1568}a^{15}-\frac{981}{6272}a^{14}+\frac{1775}{6272}a^{13}-\frac{15}{98}a^{12}+\frac{1417}{3136}a^{11}-\frac{51}{196}a^{10}-\frac{781}{1568}a^{9}-\frac{39}{392}a^{8}-\frac{359}{784}a^{7}+\frac{97}{196}a^{6}+\frac{151}{392}a^{5}+\frac{9}{49}a^{4}-\frac{95}{196}a^{3}+\frac{47}{98}a^{2}-\frac{5}{14}a-\frac{24}{49}$, $\frac{1}{12544}a^{26}-\frac{1}{12544}a^{25}+\frac{25}{12544}a^{23}+\frac{13}{12544}a^{22}-\frac{5}{1568}a^{21}+\frac{23}{12544}a^{20}-\frac{1}{6272}a^{19}+\frac{11}{1792}a^{18}+\frac{649}{12544}a^{17}+\frac{183}{3136}a^{16}-\frac{1125}{12544}a^{15}+\frac{9}{1792}a^{14}-\frac{33}{784}a^{13}-\frac{1447}{6272}a^{12}-\frac{167}{784}a^{11}+\frac{797}{3136}a^{10}+\frac{37}{196}a^{9}+\frac{655}{1568}a^{8}+\frac{31}{392}a^{7}-\frac{241}{784}a^{6}+\frac{85}{196}a^{5}+\frac{11}{56}a^{4}-\frac{9}{49}a^{3}+\frac{43}{196}a^{2}-\frac{27}{98}a+\frac{19}{49}$, $\frac{1}{25088}a^{27}-\frac{1}{25088}a^{26}+\frac{1}{25088}a^{24}-\frac{51}{25088}a^{23}+\frac{47}{25088}a^{21}-\frac{9}{12544}a^{20}+\frac{69}{25088}a^{19}+\frac{145}{25088}a^{18}+\frac{249}{6272}a^{17}+\frac{2979}{25088}a^{16}-\frac{3729}{25088}a^{15}+\frac{41}{3136}a^{14}+\frac{797}{12544}a^{13}+\frac{377}{784}a^{12}-\frac{249}{6272}a^{11}+\frac{111}{784}a^{10}+\frac{545}{3136}a^{9}-\frac{89}{392}a^{8}-\frac{67}{224}a^{7}-\frac{195}{392}a^{6}+\frac{157}{784}a^{5}-\frac{15}{196}a^{4}-\frac{45}{392}a^{3}+\frac{13}{196}a^{2}+\frac{16}{49}a-\frac{3}{49}$, $\frac{1}{50176}a^{28}-\frac{1}{50176}a^{27}+\frac{1}{50176}a^{25}-\frac{3}{50176}a^{24}-\frac{3}{1568}a^{23}-\frac{33}{50176}a^{22}+\frac{79}{25088}a^{21}-\frac{123}{50176}a^{20}+\frac{23}{7168}a^{19}+\frac{109}{12544}a^{18}+\frac{5363}{50176}a^{17}-\frac{961}{50176}a^{16}+\frac{289}{6272}a^{15}-\frac{283}{25088}a^{14}-\frac{969}{3136}a^{13}-\frac{1737}{12544}a^{12}-\frac{1327}{3136}a^{11}+\frac{2797}{6272}a^{10}-\frac{613}{1568}a^{9}-\frac{81}{3136}a^{8}+\frac{103}{392}a^{7}-\frac{515}{1568}a^{6}+\frac{22}{49}a^{5}-\frac{57}{784}a^{4}+\frac{3}{392}a^{3}-\frac{2}{49}a^{2}+\frac{5}{98}a-\frac{18}{49}$, $\frac{1}{702464}a^{29}-\frac{1}{702464}a^{28}+\frac{1}{702464}a^{26}-\frac{3}{702464}a^{25}-\frac{449}{702464}a^{23}-\frac{113}{351232}a^{22}+\frac{677}{702464}a^{21}-\frac{447}{702464}a^{20}+\frac{173}{175616}a^{19}+\frac{2003}{702464}a^{18}+\frac{447}{702464}a^{17}+\frac{10949}{87808}a^{16}-\frac{4125}{50176}a^{15}-\frac{295}{6272}a^{14}+\frac{19087}{175616}a^{13}-\frac{14769}{43904}a^{12}-\frac{9811}{87808}a^{11}-\frac{2777}{21952}a^{10}-\frac{5097}{43904}a^{9}-\frac{2551}{5488}a^{8}+\frac{6157}{21952}a^{7}-\frac{1165}{2744}a^{6}-\frac{351}{1568}a^{5}+\frac{299}{5488}a^{4}+\frac{136}{343}a^{3}+\frac{79}{196}a^{2}-\frac{269}{686}a+\frac{59}{343}$, $\frac{1}{1404928}a^{30}-\frac{1}{1404928}a^{29}+\frac{1}{1404928}a^{27}-\frac{3}{1404928}a^{26}-\frac{1}{1404928}a^{24}+\frac{1007}{702464}a^{23}-\frac{1115}{1404928}a^{22}-\frac{895}{1404928}a^{21}+\frac{509}{351232}a^{20}-\frac{5165}{1404928}a^{19}-\frac{2689}{1404928}a^{18}-\frac{251}{175616}a^{17}-\frac{205}{2048}a^{16}-\frac{251}{12544}a^{15}+\frac{44623}{351232}a^{14}-\frac{5893}{87808}a^{13}+\frac{28325}{175616}a^{12}+\frac{15703}{43904}a^{11}+\frac{9463}{87808}a^{10}+\frac{3931}{10976}a^{9}+\frac{19597}{43904}a^{8}+\frac{2041}{5488}a^{7}+\frac{769}{3136}a^{6}+\frac{4107}{10976}a^{5}+\frac{265}{1372}a^{4}+\frac{15}{392}a^{3}-\frac{3}{1372}a^{2}-\frac{51}{343}a+\frac{15}{49}$, $\frac{1}{2809856}a^{31}-\frac{1}{2809856}a^{30}+\frac{1}{2809856}a^{28}-\frac{3}{2809856}a^{27}-\frac{1}{2809856}a^{25}+\frac{111}{1404928}a^{24}-\frac{3803}{2809856}a^{23}+\frac{1}{2809856}a^{22}+\frac{957}{702464}a^{21}-\frac{4269}{2809856}a^{20}+\frac{7167}{2809856}a^{19}+\frac{1317}{351232}a^{18}-\frac{9981}{200704}a^{17}+\frac{179}{3584}a^{16}+\frac{172975}{702464}a^{15}+\frac{34819}{175616}a^{14}-\frac{133627}{351232}a^{13}+\frac{22059}{87808}a^{12}-\frac{29009}{175616}a^{11}+\frac{6451}{21952}a^{10}+\frac{13493}{87808}a^{9}+\frac{221}{10976}a^{8}-\frac{39}{128}a^{7}+\frac{7635}{21952}a^{6}-\frac{561}{2744}a^{5}-\frac{117}{784}a^{4}-\frac{479}{2744}a^{3}+\frac{173}{686}a^{2}+\frac{19}{98}a+\frac{19}{49}$, $\frac{1}{5619712}a^{32}-\frac{1}{5619712}a^{31}+\frac{1}{5619712}a^{29}-\frac{3}{5619712}a^{28}-\frac{1}{5619712}a^{26}+\frac{111}{2809856}a^{25}-\frac{219}{5619712}a^{24}-\frac{7167}{5619712}a^{23}-\frac{2627}{1404928}a^{22}+\frac{17235}{5619712}a^{21}-\frac{7169}{5619712}a^{20}+\frac{421}{702464}a^{19}+\frac{771}{401408}a^{18}+\frac{2245}{50176}a^{17}+\frac{101295}{1404928}a^{16}-\frac{48957}{351232}a^{15}+\frac{50053}{702464}a^{14}-\frac{5941}{175616}a^{13}-\frac{132049}{351232}a^{12}-\frac{17349}{43904}a^{11}-\frac{74875}{175616}a^{10}-\frac{2271}{21952}a^{9}-\frac{5319}{12544}a^{8}+\frac{4051}{43904}a^{7}-\frac{1247}{5488}a^{6}+\frac{327}{1568}a^{5}+\frac{1229}{5488}a^{4}-\frac{275}{1372}a^{3}-\frac{59}{196}a^{2}+\frac{11}{49}a-\frac{19}{49}$, $\frac{1}{11239424}a^{33}-\frac{1}{11239424}a^{32}+\frac{1}{11239424}a^{30}-\frac{3}{11239424}a^{29}-\frac{1}{11239424}a^{27}+\frac{111}{5619712}a^{26}-\frac{219}{11239424}a^{25}+\frac{1}{11239424}a^{24}+\frac{61}{2809856}a^{23}+\frac{13651}{11239424}a^{22}-\frac{14337}{11239424}a^{21}-\frac{27}{1404928}a^{20}+\frac{5123}{802816}a^{19}-\frac{891}{100352}a^{18}+\frac{100399}{2809856}a^{17}+\frac{78275}{702464}a^{16}-\frac{169019}{1404928}a^{15}+\frac{443}{351232}a^{14}+\frac{22511}{702464}a^{13}-\frac{4889}{87808}a^{12}-\frac{164139}{351232}a^{11}+\frac{3273}{43904}a^{10}+\frac{3593}{25088}a^{9}+\frac{38211}{87808}a^{8}+\frac{3541}{10976}a^{7}+\frac{1279}{3136}a^{6}+\frac{1789}{10976}a^{5}+\frac{1069}{2744}a^{4}-\frac{75}{392}a^{3}-\frac{3}{14}a^{2}+\frac{19}{98}a+\frac{16}{49}$, $\frac{1}{22478848}a^{34}-\frac{1}{22478848}a^{33}+\frac{1}{22478848}a^{31}-\frac{3}{22478848}a^{30}-\frac{1}{22478848}a^{28}+\frac{111}{11239424}a^{27}-\frac{219}{22478848}a^{26}+\frac{1}{22478848}a^{25}+\frac{61}{5619712}a^{24}+\frac{13651}{22478848}a^{23}-\frac{14337}{22478848}a^{22}-\frac{27}{2809856}a^{21}+\frac{5123}{1605632}a^{20}-\frac{891}{200704}a^{19}+\frac{47}{5619712}a^{18}+\frac{53187}{1404928}a^{17}+\frac{332741}{2809856}a^{16}-\frac{12101}{702464}a^{15}-\frac{153105}{1404928}a^{14}-\frac{48793}{175616}a^{13}-\frac{1067}{702464}a^{12}-\frac{40631}{87808}a^{11}-\frac{11639}{50176}a^{10}-\frac{84093}{175616}a^{9}-\frac{9787}{21952}a^{8}+\frac{495}{6272}a^{7}-\frac{9579}{21952}a^{6}-\frac{303}{5488}a^{5}+\frac{121}{784}a^{4}+\frac{1}{4}a^{3}+\frac{89}{196}a^{2}+\frac{9}{98}a-\frac{1}{7}$, $\frac{1}{314703872}a^{35}+\frac{3}{314703872}a^{34}+\frac{3}{78675968}a^{33}-\frac{1}{44957696}a^{32}+\frac{25}{314703872}a^{31}+\frac{25}{78675968}a^{30}-\frac{15}{44957696}a^{29}-\frac{1263}{157351936}a^{28}+\frac{5165}{314703872}a^{27}+\frac{7725}{314703872}a^{26}+\frac{2139}{39337984}a^{25}+\frac{1051}{314703872}a^{24}+\frac{669427}{314703872}a^{23}-\frac{60635}{78675968}a^{22}-\frac{68919}{157351936}a^{21}-\frac{43209}{19668992}a^{20}+\frac{302863}{78675968}a^{19}+\frac{99803}{19668992}a^{18}+\frac{2410677}{39337984}a^{17}+\frac{718059}{9834496}a^{16}+\frac{1301303}{19668992}a^{15}-\frac{280845}{2458624}a^{14}-\frac{4253339}{9834496}a^{13}-\frac{160431}{1229312}a^{12}+\frac{598391}{4917248}a^{11}-\frac{427485}{2458624}a^{10}+\frac{224363}{614656}a^{9}+\frac{20647}{76832}a^{8}+\frac{4835}{153664}a^{7}-\frac{1175}{5488}a^{6}-\frac{11887}{76832}a^{5}-\frac{1297}{38416}a^{4}+\frac{101}{392}a^{3}-\frac{775}{2401}a^{2}-\frac{1056}{2401}a+\frac{1158}{2401}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, 1/2*a^14 - 1/2*a^13 - 1/2*a^11 - 1/2*a^10 - 1/2*a^8 - 1/2*a^6 - 1/2*a^5 - 1/2*a^3 - 1/2*a^2, 1/2*a^15 - 1/2*a^13 - 1/2*a^12 - 1/2*a^10 - 1/2*a^9 - 1/2*a^8 - 1/2*a^7 - 1/2*a^5 - 1/2*a^4 - 1/2*a^2, 1/4*a^16 - 1/4*a^15 + 1/4*a^13 + 1/4*a^12 - 1/4*a^10 - 1/2*a^9 + 1/4*a^8 + 1/4*a^7 - 1/4*a^5 - 1/4*a^4 - 1/2*a^2, 1/4*a^17 - 1/4*a^15 - 1/4*a^14 + 1/4*a^12 + 1/4*a^11 - 1/4*a^10 - 1/4*a^9 + 1/4*a^7 + 1/4*a^6 - 1/4*a^4, 1/56*a^18 + 1/56*a^17 + 1/14*a^16 - 13/56*a^15 + 1/8*a^14 - 27/56*a^12 - 1/2*a^11 + 3/56*a^10 - 17/56*a^9 + 5/14*a^8 - 1/8*a^7 - 27/56*a^6 - 1/2*a^4 - 5/14*a^3 + 1/7*a^2 - 3/7*a + 1/7, 1/56*a^19 + 3/56*a^17 - 3/56*a^16 + 3/28*a^15 - 1/8*a^14 - 13/56*a^13 + 13/56*a^12 - 25/56*a^11 + 11/28*a^10 + 9/56*a^9 - 13/56*a^8 - 3/28*a^7 + 27/56*a^6 + 1/4*a^5 - 3/28*a^4 - 1/2*a^3 - 1/14*a^2 - 3/7*a - 1/7, 1/112*a^20 - 1/112*a^19 + 5/112*a^17 - 3/112*a^16 + 3/28*a^15 - 13/112*a^14 + 27/56*a^13 - 55/112*a^12 - 51/112*a^11 - 1/14*a^10 + 15/112*a^9 - 25/112*a^8 + 3/28*a^7 + 27/56*a^6 + 1/14*a^5 - 1/14*a^4 - 1/2*a^3 - 1/7*a^2 + 2/7*a - 1/7, 1/112*a^21 - 1/112*a^19 - 1/112*a^18 - 1/28*a^17 + 13/112*a^16 - 1/16*a^15 - 1/112*a^14 - 29/112*a^13 + 1/4*a^12 - 3/112*a^11 + 17/112*a^10 - 5/28*a^9 - 7/16*a^8 - 2/7*a^7 + 1/4*a^5 + 5/28*a^4 + 3/7*a^3 - 2/7*a^2 + 3/7*a + 3/7, 1/224*a^22 - 1/224*a^21 + 1/224*a^19 + 1/224*a^18 + 1/56*a^17 + 15/224*a^16 - 11/112*a^15 + 1/224*a^14 + 1/224*a^13 - 1/4*a^12 + 3/224*a^11 + 11/224*a^10 - 23/56*a^9 - 19/112*a^8 - 27/56*a^7 - 1/2*a^6 - 1/2*a^5 - 13/28*a^4 - 3/14*a^3 - 5/14*a^2 + 2/7, 1/224*a^23 - 1/224*a^21 - 1/224*a^20 + 1/224*a^18 - 1/32*a^17 - 5/224*a^16 - 17/224*a^15 + 1/8*a^14 - 111/224*a^13 - 111/224*a^12 + 13/28*a^11 + 59/224*a^10 + 3/7*a^9 + 25/56*a^8 + 1/7*a^7 - 27/56*a^6 - 2/7*a^5 - 3/14*a^3 - 2/7*a^2 - 1/7*a + 3/7, 1/3136*a^24 + 5/3136*a^23 - 1/784*a^22 - 1/3136*a^21 + 3/3136*a^20 - 1/196*a^19 - 1/448*a^18 - 25/392*a^17 - 185/3136*a^16 + 11/3136*a^15 + 57/784*a^14 + 317/3136*a^13 + 681/3136*a^12 + 165/392*a^11 + 65/392*a^10 - 321/784*a^9 - 19/98*a^8 + 33/392*a^7 + 5/14*a^6 + 17/49*a^5 - 1/196*a^4 + 33/98*a^3 + 19/98*a^2 + 13/49*a + 15/49, 1/6272*a^25 - 1/6272*a^24 + 1/784*a^23 + 9/6272*a^22 - 19/6272*a^21 + 1/784*a^20 - 9/6272*a^19 - 9/3136*a^18 + 59/896*a^17 - 671/6272*a^16 - 369/1568*a^15 - 981/6272*a^14 + 1775/6272*a^13 - 15/98*a^12 + 1417/3136*a^11 - 51/196*a^10 - 781/1568*a^9 - 39/392*a^8 - 359/784*a^7 + 97/196*a^6 + 151/392*a^5 + 9/49*a^4 - 95/196*a^3 + 47/98*a^2 - 5/14*a - 24/49, 1/12544*a^26 - 1/12544*a^25 + 25/12544*a^23 + 13/12544*a^22 - 5/1568*a^21 + 23/12544*a^20 - 1/6272*a^19 + 11/1792*a^18 + 649/12544*a^17 + 183/3136*a^16 - 1125/12544*a^15 + 9/1792*a^14 - 33/784*a^13 - 1447/6272*a^12 - 167/784*a^11 + 797/3136*a^10 + 37/196*a^9 + 655/1568*a^8 + 31/392*a^7 - 241/784*a^6 + 85/196*a^5 + 11/56*a^4 - 9/49*a^3 + 43/196*a^2 - 27/98*a + 19/49, 1/25088*a^27 - 1/25088*a^26 + 1/25088*a^24 - 51/25088*a^23 + 47/25088*a^21 - 9/12544*a^20 + 69/25088*a^19 + 145/25088*a^18 + 249/6272*a^17 + 2979/25088*a^16 - 3729/25088*a^15 + 41/3136*a^14 + 797/12544*a^13 + 377/784*a^12 - 249/6272*a^11 + 111/784*a^10 + 545/3136*a^9 - 89/392*a^8 - 67/224*a^7 - 195/392*a^6 + 157/784*a^5 - 15/196*a^4 - 45/392*a^3 + 13/196*a^2 + 16/49*a - 3/49, 1/50176*a^28 - 1/50176*a^27 + 1/50176*a^25 - 3/50176*a^24 - 3/1568*a^23 - 33/50176*a^22 + 79/25088*a^21 - 123/50176*a^20 + 23/7168*a^19 + 109/12544*a^18 + 5363/50176*a^17 - 961/50176*a^16 + 289/6272*a^15 - 283/25088*a^14 - 969/3136*a^13 - 1737/12544*a^12 - 1327/3136*a^11 + 2797/6272*a^10 - 613/1568*a^9 - 81/3136*a^8 + 103/392*a^7 - 515/1568*a^6 + 22/49*a^5 - 57/784*a^4 + 3/392*a^3 - 2/49*a^2 + 5/98*a - 18/49, 1/702464*a^29 - 1/702464*a^28 + 1/702464*a^26 - 3/702464*a^25 - 449/702464*a^23 - 113/351232*a^22 + 677/702464*a^21 - 447/702464*a^20 + 173/175616*a^19 + 2003/702464*a^18 + 447/702464*a^17 + 10949/87808*a^16 - 4125/50176*a^15 - 295/6272*a^14 + 19087/175616*a^13 - 14769/43904*a^12 - 9811/87808*a^11 - 2777/21952*a^10 - 5097/43904*a^9 - 2551/5488*a^8 + 6157/21952*a^7 - 1165/2744*a^6 - 351/1568*a^5 + 299/5488*a^4 + 136/343*a^3 + 79/196*a^2 - 269/686*a + 59/343, 1/1404928*a^30 - 1/1404928*a^29 + 1/1404928*a^27 - 3/1404928*a^26 - 1/1404928*a^24 + 1007/702464*a^23 - 1115/1404928*a^22 - 895/1404928*a^21 + 509/351232*a^20 - 5165/1404928*a^19 - 2689/1404928*a^18 - 251/175616*a^17 - 205/2048*a^16 - 251/12544*a^15 + 44623/351232*a^14 - 5893/87808*a^13 + 28325/175616*a^12 + 15703/43904*a^11 + 9463/87808*a^10 + 3931/10976*a^9 + 19597/43904*a^8 + 2041/5488*a^7 + 769/3136*a^6 + 4107/10976*a^5 + 265/1372*a^4 + 15/392*a^3 - 3/1372*a^2 - 51/343*a + 15/49, 1/2809856*a^31 - 1/2809856*a^30 + 1/2809856*a^28 - 3/2809856*a^27 - 1/2809856*a^25 + 111/1404928*a^24 - 3803/2809856*a^23 + 1/2809856*a^22 + 957/702464*a^21 - 4269/2809856*a^20 + 7167/2809856*a^19 + 1317/351232*a^18 - 9981/200704*a^17 + 179/3584*a^16 + 172975/702464*a^15 + 34819/175616*a^14 - 133627/351232*a^13 + 22059/87808*a^12 - 29009/175616*a^11 + 6451/21952*a^10 + 13493/87808*a^9 + 221/10976*a^8 - 39/128*a^7 + 7635/21952*a^6 - 561/2744*a^5 - 117/784*a^4 - 479/2744*a^3 + 173/686*a^2 + 19/98*a + 19/49, 1/5619712*a^32 - 1/5619712*a^31 + 1/5619712*a^29 - 3/5619712*a^28 - 1/5619712*a^26 + 111/2809856*a^25 - 219/5619712*a^24 - 7167/5619712*a^23 - 2627/1404928*a^22 + 17235/5619712*a^21 - 7169/5619712*a^20 + 421/702464*a^19 + 771/401408*a^18 + 2245/50176*a^17 + 101295/1404928*a^16 - 48957/351232*a^15 + 50053/702464*a^14 - 5941/175616*a^13 - 132049/351232*a^12 - 17349/43904*a^11 - 74875/175616*a^10 - 2271/21952*a^9 - 5319/12544*a^8 + 4051/43904*a^7 - 1247/5488*a^6 + 327/1568*a^5 + 1229/5488*a^4 - 275/1372*a^3 - 59/196*a^2 + 11/49*a - 19/49, 1/11239424*a^33 - 1/11239424*a^32 + 1/11239424*a^30 - 3/11239424*a^29 - 1/11239424*a^27 + 111/5619712*a^26 - 219/11239424*a^25 + 1/11239424*a^24 + 61/2809856*a^23 + 13651/11239424*a^22 - 14337/11239424*a^21 - 27/1404928*a^20 + 5123/802816*a^19 - 891/100352*a^18 + 100399/2809856*a^17 + 78275/702464*a^16 - 169019/1404928*a^15 + 443/351232*a^14 + 22511/702464*a^13 - 4889/87808*a^12 - 164139/351232*a^11 + 3273/43904*a^10 + 3593/25088*a^9 + 38211/87808*a^8 + 3541/10976*a^7 + 1279/3136*a^6 + 1789/10976*a^5 + 1069/2744*a^4 - 75/392*a^3 - 3/14*a^2 + 19/98*a + 16/49, 1/22478848*a^34 - 1/22478848*a^33 + 1/22478848*a^31 - 3/22478848*a^30 - 1/22478848*a^28 + 111/11239424*a^27 - 219/22478848*a^26 + 1/22478848*a^25 + 61/5619712*a^24 + 13651/22478848*a^23 - 14337/22478848*a^22 - 27/2809856*a^21 + 5123/1605632*a^20 - 891/200704*a^19 + 47/5619712*a^18 + 53187/1404928*a^17 + 332741/2809856*a^16 - 12101/702464*a^15 - 153105/1404928*a^14 - 48793/175616*a^13 - 1067/702464*a^12 - 40631/87808*a^11 - 11639/50176*a^10 - 84093/175616*a^9 - 9787/21952*a^8 + 495/6272*a^7 - 9579/21952*a^6 - 303/5488*a^5 + 121/784*a^4 + 1/4*a^3 + 89/196*a^2 + 9/98*a - 1/7, 1/314703872*a^35 + 3/314703872*a^34 + 3/78675968*a^33 - 1/44957696*a^32 + 25/314703872*a^31 + 25/78675968*a^30 - 15/44957696*a^29 - 1263/157351936*a^28 + 5165/314703872*a^27 + 7725/314703872*a^26 + 2139/39337984*a^25 + 1051/314703872*a^24 + 669427/314703872*a^23 - 60635/78675968*a^22 - 68919/157351936*a^21 - 43209/19668992*a^20 + 302863/78675968*a^19 + 99803/19668992*a^18 + 2410677/39337984*a^17 + 718059/9834496*a^16 + 1301303/19668992*a^15 - 280845/2458624*a^14 - 4253339/9834496*a^13 - 160431/1229312*a^12 + 598391/4917248*a^11 - 427485/2458624*a^10 + 224363/614656*a^9 + 20647/76832*a^8 + 4835/153664*a^7 - 1175/5488*a^6 - 11887/76832*a^5 - 1297/38416*a^4 + 101/392*a^3 - 775/2401*a^2 - 1056/2401*a + 1158/2401

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

$C_{6}$, which has order $6$ (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{17}{5619712} a^{35} - \frac{37}{5619712} a^{34} - \frac{17}{5619712} a^{32} + \frac{51}{5619712} a^{31} + \frac{17}{5619712} a^{29} + \frac{289}{2809856} a^{28} + \frac{305}{5619712} a^{27} - \frac{17}{5619712} a^{26} + \frac{51}{1404928} a^{25} - \frac{1411}{5619712} a^{24} + \frac{17}{5619712} a^{23} - \frac{85}{702464} a^{22} - \frac{4709}{2809856} a^{21} - \frac{4415}{2809856} a^{20} + \frac{289}{1404928} a^{19} - \frac{51}{351232} a^{18} + \frac{2635}{702464} a^{17} - \frac{51}{175616} a^{16} + \frac{289}{351232} a^{15} + \frac{765}{43904} a^{14} + \frac{2389}{175616} a^{13} - \frac{85}{21952} a^{12} + \frac{17}{87808} a^{11} - \frac{1411}{43904} a^{10} + \frac{51}{5488} a^{9} - \frac{17}{10976} a^{8} - \frac{629}{5488} a^{7} - \frac{639}{21952} a^{6} + \frac{17}{1372} a^{5} + \frac{51}{343} a^{3} - \frac{34}{343} a^{2} + \frac{136}{343} $ -(17)/(5619712)a^(35) - (37)/(5619712)a^(34) - (17)/(5619712)a^(32) + (51)/(5619712)a^(31) + (17)/(5619712)a^(29) + (289)/(2809856)a^(28) + (305)/(5619712)a^(27) - (17)/(5619712)a^(26) + (51)/(1404928)a^(25) - (1411)/(5619712)a^(24) + (17)/(5619712)a^(23) - (85)/(702464)a^(22) - (4709)/(2809856)a^(21) - (4415)/(2809856)a^(20) + (289)/(1404928)a^(19) - (51)/(351232)a^(18) + (2635)/(702464)a^(17) - (51)/(175616)a^(16) + (289)/(351232)a^(15) + (765)/(43904)a^(14) + (2389)/(175616)a^(13) - (85)/(21952)a^(12) + (17)/(87808)a^(11) - (1411)/(43904)a^(10) + (51)/(5488)a^(9) - (17)/(10976)a^(8) - (629)/(5488)a^(7) - (639)/(21952)a^(6) + (17)/(1372)a^(5) + (51)/(343)a^(3) - (34)/(343)*a^(2) + (136)/(343) (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:	$\frac{17}{5619712}a^{35}+\frac{37}{5619712}a^{34}+\frac{17}{5619712}a^{32}-\frac{51}{5619712}a^{31}-\frac{17}{5619712}a^{29}-\frac{289}{2809856}a^{28}-\frac{305}{5619712}a^{27}+\frac{17}{5619712}a^{26}-\frac{51}{1404928}a^{25}+\frac{1411}{5619712}a^{24}-\frac{17}{5619712}a^{23}+\frac{85}{702464}a^{22}+\frac{4709}{2809856}a^{21}+\frac{4415}{2809856}a^{20}-\frac{289}{1404928}a^{19}+\frac{51}{351232}a^{18}-\frac{2635}{702464}a^{17}+\frac{51}{175616}a^{16}-\frac{289}{351232}a^{15}-\frac{765}{43904}a^{14}-\frac{2389}{175616}a^{13}+\frac{85}{21952}a^{12}-\frac{17}{87808}a^{11}+\frac{1411}{43904}a^{10}-\frac{51}{5488}a^{9}+\frac{17}{10976}a^{8}+\frac{629}{5488}a^{7}+\frac{639}{21952}a^{6}-\frac{17}{1372}a^{5}-\frac{51}{343}a^{3}+\frac{34}{343}a^{2}+\frac{207}{343}$, $\frac{167}{44957696}a^{35}+\frac{1195}{44957696}a^{34}+\frac{167}{22478848}a^{33}-\frac{1537}{44957696}a^{32}-\frac{1195}{44957696}a^{31}-\frac{3107}{22478848}a^{30}-\frac{7311}{44957696}a^{29}-\frac{89}{802816}a^{28}-\frac{29489}{44957696}a^{27}-\frac{103}{917504}a^{26}+\frac{30175}{22478848}a^{25}+\frac{62949}{44957696}a^{24}+\frac{25969}{6422528}a^{23}+\frac{83323}{22478848}a^{22}+\frac{11999}{22478848}a^{21}+\frac{12961}{1605632}a^{20}-\frac{32635}{11239424}a^{19}-\frac{141577}{5619712}a^{18}-\frac{18835}{802816}a^{17}-\frac{147271}{2809856}a^{16}-\frac{114799}{2809856}a^{15}+\frac{5279}{200704}a^{14}-\frac{60145}{1404928}a^{13}+\frac{41907}{702464}a^{12}+\frac{25267}{100352}a^{11}+\frac{36401}{175616}a^{10}+\frac{63837}{175616}a^{9}+\frac{613}{3136}a^{8}-\frac{9651}{21952}a^{7}+\frac{53}{686}a^{6}-\frac{2403}{5488}a^{5}-\frac{6043}{5488}a^{4}-\frac{212}{343}a^{3}-\frac{831}{686}a^{2}-\frac{167}{343}a+\frac{1028}{343}$, $\frac{681}{44957696}a^{35}-\frac{681}{44957696}a^{34}-\frac{1}{2809856}a^{33}-\frac{2271}{44957696}a^{32}+\frac{925}{44957696}a^{31}+\frac{127}{2809856}a^{30}+\frac{4015}{44957696}a^{29}-\frac{9909}{22478848}a^{28}+\frac{23165}{44957696}a^{27}+\frac{7201}{44957696}a^{26}+\frac{13357}{11239424}a^{25}-\frac{53309}{44957696}a^{24}-\frac{75073}{44957696}a^{23}-\frac{11685}{5619712}a^{22}+\frac{143905}{22478848}a^{21}-\frac{44879}{5619712}a^{20}-\frac{529}{11239424}a^{19}-\frac{2351}{175616}a^{18}+\frac{125357}{5619712}a^{17}+\frac{21723}{702464}a^{16}+\frac{80271}{2809856}a^{15}-\frac{58153}{702464}a^{14}+\frac{89253}{1404928}a^{13}-\frac{9523}{351232}a^{12}+\frac{47479}{702464}a^{11}-\frac{83091}{351232}a^{10}-\frac{11631}{43904}a^{9}-\frac{1345}{10976}a^{8}+\frac{15773}{21952}a^{7}-\frac{1935}{10976}a^{6}+\frac{3041}{10976}a^{5}-\frac{223}{1372}a^{4}+\frac{2903}{2744}a^{3}+\frac{681}{686}a^{2}-\frac{99}{343}a-\frac{1164}{343}$, $\frac{579}{44957696}a^{35}-\frac{99}{6422528}a^{34}+\frac{167}{22478848}a^{33}-\frac{449}{44957696}a^{32}+\frac{205}{44957696}a^{31}+\frac{841}{22478848}a^{30}+\frac{449}{44957696}a^{29}-\frac{5767}{11239424}a^{28}+\frac{7575}{44957696}a^{27}-\frac{20231}{44957696}a^{26}-\frac{2733}{22478848}a^{25}-\frac{4827}{44957696}a^{24}-\frac{37841}{44957696}a^{23}-\frac{3553}{22478848}a^{22}+\frac{242701}{22478848}a^{21}+\frac{2817}{11239424}a^{20}+\frac{14725}{1605632}a^{19}+\frac{18057}{5619712}a^{18}-\frac{2467}{5619712}a^{17}+\frac{45}{8192}a^{16}-\frac{21737}{2809856}a^{15}-\frac{198601}{1404928}a^{14}-\frac{37927}{1404928}a^{13}-\frac{8641}{100352}a^{12}-\frac{6957}{702464}a^{11}+\frac{1025}{21952}a^{10}-\frac{159}{12544}a^{9}+\frac{7181}{87808}a^{8}+\frac{48149}{43904}a^{7}+\frac{381}{2744}a^{6}+\frac{4419}{10976}a^{5}-\frac{1}{686}a^{4}-\frac{557}{1372}a^{3}-\frac{4}{343}a^{2}+\frac{4}{343}a-\frac{1431}{343}$, $\frac{2647}{314703872}a^{35}-\frac{2419}{314703872}a^{34}-\frac{711}{78675968}a^{33}+\frac{407}{6422528}a^{32}-\frac{37033}{314703872}a^{31}-\frac{3797}{78675968}a^{30}+\frac{2159}{44957696}a^{29}-\frac{47477}{157351936}a^{28}+\frac{47347}{314703872}a^{27}+\frac{38147}{314703872}a^{26}-\frac{48401}{39337984}a^{25}+\frac{979077}{314703872}a^{24}+\frac{348701}{314703872}a^{23}-\frac{77113}{78675968}a^{22}+\frac{733827}{157351936}a^{21}-\frac{64195}{19668992}a^{20}-\frac{64063}{78675968}a^{19}+\frac{289825}{19668992}a^{18}-\frac{1762317}{39337984}a^{17}-\frac{142787}{9834496}a^{16}+\frac{337025}{19668992}a^{15}-\frac{98061}{2458624}a^{14}+\frac{382467}{9834496}a^{13}+\frac{22451}{1229312}a^{12}-\frac{633567}{4917248}a^{11}+\frac{974797}{2458624}a^{10}+\frac{64707}{614656}a^{9}-\frac{28643}{153664}a^{8}+\frac{74147}{307328}a^{7}-\frac{2101}{5488}a^{6}+\frac{2019}{38416}a^{5}+\frac{1476}{2401}a^{4}-\frac{5499}{2744}a^{3}-\frac{1235}{4802}a^{2}+\frac{2619}{2401}a-\frac{1733}{2401}$, $\frac{681}{44957696}a^{35}-\frac{241}{44957696}a^{34}-\frac{1}{5619712}a^{33}-\frac{1039}{44957696}a^{32}-\frac{3755}{44957696}a^{31}-\frac{797}{5619712}a^{30}+\frac{5535}{44957696}a^{29}-\frac{9473}{22478848}a^{28}+\frac{17597}{44957696}a^{27}+\frac{29961}{44957696}a^{26}+\frac{8847}{11239424}a^{25}+\frac{93299}{44957696}a^{24}+\frac{127175}{44957696}a^{23}-\frac{12379}{2809856}a^{22}+\frac{89865}{22478848}a^{21}-\frac{47037}{5619712}a^{20}-\frac{19483}{1605632}a^{19}-\frac{14645}{1404928}a^{18}-\frac{121319}{5619712}a^{17}-\frac{1361}{50176}a^{16}+\frac{204627}{2809856}a^{15}-\frac{16507}{702464}a^{14}+\frac{131217}{1404928}a^{13}+\frac{39927}{351232}a^{12}+\frac{22699}{702464}a^{11}+\frac{40465}{351232}a^{10}+\frac{14359}{87808}a^{9}-\frac{3901}{5488}a^{8}+\frac{6445}{43904}a^{7}-\frac{2235}{5488}a^{6}-\frac{5995}{10976}a^{5}+\frac{533}{2744}a^{4}-\frac{33}{343}a^{3}-\frac{681}{686}a^{2}+\frac{1857}{686}a+\frac{8}{343}$, $\frac{183}{11239424}a^{35}+\frac{1}{351232}a^{34}+\frac{107}{11239424}a^{33}-\frac{139}{11239424}a^{32}+\frac{155}{5619712}a^{31}-\frac{169}{1605632}a^{30}+\frac{683}{11239424}a^{29}-\frac{5137}{11239424}a^{28}-\frac{2209}{11239424}a^{27}-\frac{17}{5619712}a^{26}+\frac{447}{1605632}a^{25}-\frac{5325}{11239424}a^{24}+\frac{12717}{5619712}a^{23}-\frac{24077}{11239424}a^{22}+\frac{21065}{2809856}a^{21}+\frac{12865}{5619712}a^{20}-\frac{7929}{2809856}a^{19}-\frac{1249}{2809856}a^{18}+\frac{3347}{1404928}a^{17}-\frac{37743}{1404928}a^{16}+\frac{24785}{702464}a^{15}-\frac{55893}{702464}a^{14}-\frac{2899}{351232}a^{13}+\frac{11149}{351232}a^{12}-\frac{993}{175616}a^{11}-\frac{3}{3584}a^{10}+\frac{12279}{87808}a^{9}-\frac{2829}{10976}a^{8}+\frac{23515}{43904}a^{7}-\frac{467}{5488}a^{6}-\frac{15}{196}a^{5}+\frac{683}{5488}a^{4}+\frac{9}{2744}a^{3}-\frac{355}{686}a^{2}+\frac{701}{686}a-\frac{565}{343}$, $\frac{1473}{22478848}a^{35}+\frac{181}{11239424}a^{34}+\frac{461}{22478848}a^{33}+\frac{2957}{22478848}a^{32}-\frac{271}{5619712}a^{31}-\frac{2185}{22478848}a^{30}-\frac{2309}{22478848}a^{29}-\frac{51897}{22478848}a^{28}-\frac{15305}{22478848}a^{27}-\frac{5991}{5619712}a^{26}-\frac{9799}{3211264}a^{25}+\frac{39355}{22478848}a^{24}+\frac{16077}{5619712}a^{23}+\frac{99765}{22478848}a^{22}+\frac{476235}{11239424}a^{21}+\frac{121005}{11239424}a^{20}+\frac{82183}{5619712}a^{19}+\frac{214131}{5619712}a^{18}-\frac{15307}{401408}a^{17}-\frac{21545}{401408}a^{16}-\frac{12337}{200704}a^{15}-\frac{663561}{1404928}a^{14}-\frac{64759}{702464}a^{13}-\frac{71891}{702464}a^{12}-\frac{87089}{351232}a^{11}+\frac{149797}{351232}a^{10}+\frac{76165}{175616}a^{9}+\frac{43121}{87808}a^{8}+\frac{9005}{2744}a^{7}+\frac{3463}{21952}a^{6}+\frac{4149}{10976}a^{5}+\frac{5497}{5488}a^{4}-\frac{435}{196}a^{3}-\frac{307}{196}a^{2}-\frac{99}{98}a-\frac{3837}{343}$, $\frac{393}{11239424}a^{35}+\frac{89}{5619712}a^{34}+\frac{1}{5619712}a^{33}+\frac{339}{5619712}a^{32}-\frac{31}{5619712}a^{31}-\frac{5}{200704}a^{30}-\frac{223}{5619712}a^{29}-\frac{15903}{11239424}a^{28}-\frac{4097}{5619712}a^{27}-\frac{619}{2809856}a^{26}-\frac{975}{702464}a^{25}+\frac{5851}{5619712}a^{24}+\frac{11145}{5619712}a^{23}+\frac{1977}{702464}a^{22}+\frac{294955}{11239424}a^{21}+\frac{1009}{87808}a^{20}+\frac{3701}{5619712}a^{19}+\frac{521}{43904}a^{18}-\frac{74441}{2809856}a^{17}-\frac{27437}{702464}a^{16}-\frac{51559}{1404928}a^{15}-\frac{100901}{351232}a^{14}-\frac{65417}{702464}a^{13}+\frac{4689}{87808}a^{12}-\frac{6191}{351232}a^{11}+\frac{28031}{87808}a^{10}+\frac{63227}{175616}a^{9}+\frac{123}{686}a^{8}+\frac{39989}{21952}a^{7}+\frac{3735}{21952}a^{6}-\frac{6843}{10976}a^{5}-\frac{923}{5488}a^{4}-\frac{3845}{2744}a^{3}-\frac{1035}{1372}a^{2}+\frac{13}{49}a-\frac{2063}{343}$, $\frac{681}{44957696}a^{35}+\frac{179}{44957696}a^{34}-\frac{219}{11239424}a^{33}+\frac{1945}{44957696}a^{32}-\frac{169}{6422528}a^{31}-\frac{961}{11239424}a^{30}-\frac{7321}{44957696}a^{29}-\frac{7239}{22478848}a^{28}+\frac{421}{44957696}a^{27}+\frac{11061}{44957696}a^{26}-\frac{261}{351232}a^{25}+\frac{1019}{917504}a^{24}+\frac{99115}{44957696}a^{23}+\frac{46607}{11239424}a^{22}+\frac{86949}{22478848}a^{21}-\frac{1161}{802816}a^{20}-\frac{62217}{11239424}a^{19}+\frac{1257}{175616}a^{18}-\frac{97835}{5619712}a^{17}-\frac{24939}{702464}a^{16}-\frac{132497}{2809856}a^{15}-\frac{8493}{702464}a^{14}+\frac{52813}{1404928}a^{13}+\frac{19489}{351232}a^{12}-\frac{25657}{702464}a^{11}+\frac{55781}{351232}a^{10}+\frac{20031}{87808}a^{9}+\frac{32769}{87808}a^{8}-\frac{8119}{43904}a^{7}-\frac{4185}{10976}a^{6}-\frac{3833}{10976}a^{5}+\frac{591}{2744}a^{4}-\frac{1097}{2744}a^{3}-\frac{891}{686}a^{2}-\frac{685}{686}a+\frac{1028}{343}$, $\frac{681}{44957696}a^{35}+\frac{689}{44957696}a^{34}+\frac{461}{22478848}a^{33}+\frac{1693}{44957696}a^{32}-\frac{95}{6422528}a^{31}-\frac{3653}{22478848}a^{30}-\frac{45}{917504}a^{29}-\frac{6725}{11239424}a^{28}-\frac{31435}{44957696}a^{27}-\frac{25765}{44957696}a^{26}-\frac{8903}{22478848}a^{25}+\frac{73903}{44957696}a^{24}+\frac{208941}{44957696}a^{23}+\frac{55389}{22478848}a^{22}+\frac{255335}{22478848}a^{21}+\frac{98303}{11239424}a^{20}+\frac{4195}{1605632}a^{19}-\frac{21213}{5619712}a^{18}-\frac{219749}{5619712}a^{17}-\frac{29809}{401408}a^{16}-\frac{54431}{2809856}a^{15}-\frac{131195}{1404928}a^{14}-\frac{58913}{1404928}a^{13}+\frac{38139}{702464}a^{12}+\frac{83029}{702464}a^{11}+\frac{16605}{43904}a^{10}+\frac{22227}{43904}a^{9}-\frac{8553}{87808}a^{8}+\frac{7443}{21952}a^{7}-\frac{361}{3136}a^{6}-\frac{5165}{10976}a^{5}-\frac{367}{784}a^{4}-\frac{2249}{1372}a^{3}-\frac{548}{343}a^{2}+\frac{685}{343}a-\frac{335}{343}$, $\frac{257}{22478848}a^{35}-\frac{481}{22478848}a^{34}+\frac{33}{22478848}a^{32}-\frac{547}{22478848}a^{31}+\frac{303}{1404928}a^{30}-\frac{401}{22478848}a^{29}-\frac{3617}{11239424}a^{28}+\frac{12405}{22478848}a^{27}-\frac{1305}{3211264}a^{26}-\frac{2467}{5619712}a^{25}-\frac{3709}{22478848}a^{24}-\frac{108289}{22478848}a^{23}+\frac{2027}{2809856}a^{22}+\frac{75357}{11239424}a^{21}-\frac{8493}{1404928}a^{20}+\frac{51651}{5619712}a^{19}+\frac{2841}{351232}a^{18}+\frac{13949}{2809856}a^{17}+\frac{21047}{351232}a^{16}-\frac{35257}{1404928}a^{15}-\frac{31317}{351232}a^{14}+\frac{25621}{702464}a^{13}-\frac{19967}{175616}a^{12}-\frac{26673}{351232}a^{11}-\frac{277}{25088}a^{10}-\frac{4765}{10976}a^{9}+\frac{1835}{5488}a^{8}+\frac{16719}{21952}a^{7}-\frac{4751}{21952}a^{6}+\frac{6793}{10976}a^{5}+\frac{115}{2744}a^{4}-\frac{19}{392}a^{3}+\frac{548}{343}a^{2}-\frac{578}{343}a-\frac{99}{49}$, $\frac{1777}{39337984}a^{35}+\frac{2977}{157351936}a^{34}+\frac{1149}{157351936}a^{33}+\frac{459}{11239424}a^{32}-\frac{3831}{157351936}a^{31}-\frac{17221}{157351936}a^{30}-\frac{1303}{11239424}a^{29}-\frac{207309}{157351936}a^{28}-\frac{363}{614656}a^{27}-\frac{24271}{157351936}a^{26}-\frac{76505}{157351936}a^{25}+\frac{107525}{78675968}a^{24}+\frac{525667}{157351936}a^{23}+\frac{528761}{157351936}a^{22}+\frac{1656979}{78675968}a^{21}+\frac{569281}{78675968}a^{20}-\frac{52777}{39337984}a^{19}+\frac{35419}{39337984}a^{18}-\frac{457161}{19668992}a^{17}-\frac{910803}{19668992}a^{16}-\frac{327815}{9834496}a^{15}-\frac{1914921}{9834496}a^{14}-\frac{208779}{4917248}a^{13}+\frac{240001}{4917248}a^{12}+\frac{66383}{2458624}a^{11}+\frac{461603}{2458624}a^{10}+\frac{106005}{307328}a^{9}+\frac{102371}{614656}a^{8}+\frac{81537}{76832}a^{7}+\frac{37}{343}a^{6}-\frac{17345}{76832}a^{5}-\frac{4157}{19208}a^{4}-\frac{1401}{2744}a^{3}-\frac{2977}{2401}a^{2}-\frac{1149}{2401}a-\frac{5624}{2401}$, $\frac{33}{2809856}a^{35}+\frac{3}{3211264}a^{34}-\frac{717}{22478848}a^{33}-\frac{67}{1404928}a^{32}-\frac{235}{22478848}a^{31}-\frac{15}{458752}a^{30}-\frac{95}{1404928}a^{29}-\frac{7373}{22478848}a^{28}+\frac{1355}{11239424}a^{27}+\frac{3303}{3211264}a^{26}+\frac{28405}{22478848}a^{25}-\frac{599}{5619712}a^{24}+\frac{7967}{22478848}a^{23}+\frac{26219}{22478848}a^{22}+\frac{279}{87808}a^{21}-\frac{49423}{11239424}a^{20}-\frac{46841}{2809856}a^{19}-\frac{11411}{802816}a^{18}+\frac{473}{87808}a^{17}-\frac{11535}{2809856}a^{16}-\frac{2025}{351232}a^{15}-\frac{10485}{1404928}a^{14}+\frac{2809}{50176}a^{13}+\frac{93209}{702464}a^{12}+\frac{12353}{175616}a^{11}-\frac{25405}{351232}a^{10}+\frac{65}{3584}a^{9}-\frac{1143}{21952}a^{8}-\frac{317}{3136}a^{7}-\frac{5927}{21952}a^{6}-\frac{4589}{10976}a^{5}-\frac{69}{784}a^{4}+\frac{165}{343}a^{3}-\frac{293}{1372}a^{2}+\frac{185}{343}a+\frac{272}{343}$, $\frac{17135}{314703872}a^{35}+\frac{7697}{314703872}a^{34}-\frac{551}{39337984}a^{33}+\frac{1577}{44957696}a^{32}-\frac{32421}{314703872}a^{31}-\frac{3467}{39337984}a^{30}+\frac{257}{6422528}a^{29}-\frac{295747}{157351936}a^{28}-\frac{199709}{314703872}a^{27}+\frac{53735}{314703872}a^{26}-\frac{55063}{78675968}a^{25}+\frac{921437}{314703872}a^{24}+\frac{761289}{314703872}a^{23}-\frac{957}{19668992}a^{22}+\frac{5061739}{157351936}a^{21}+\frac{315087}{39337984}a^{20}-\frac{280547}{78675968}a^{19}+\frac{15711}{2458624}a^{18}-\frac{1875681}{39337984}a^{17}-\frac{183753}{4917248}a^{16}+\frac{26261}{19668992}a^{15}-\frac{1709747}{4917248}a^{14}-\frac{533625}{9834496}a^{13}+\frac{143811}{2458624}a^{12}-\frac{119955}{4917248}a^{11}+\frac{1122123}{2458624}a^{10}+\frac{184283}{614656}a^{9}-\frac{19623}{307328}a^{8}+\frac{367719}{153664}a^{7}+\frac{333}{2744}a^{6}-\frac{9383}{38416}a^{5}+\frac{2801}{19208}a^{4}-\frac{6777}{2744}a^{3}-\frac{10487}{9604}a^{2}+\frac{104}{2401}a-\frac{19498}{2401}$, $\frac{14327}{314703872}a^{35}+\frac{1933}{314703872}a^{34}+\frac{1121}{78675968}a^{33}+\frac{5057}{44957696}a^{32}-\frac{12377}{314703872}a^{31}+\frac{2407}{78675968}a^{30}+\frac{2671}{44957696}a^{29}-\frac{263901}{157351936}a^{28}-\frac{165373}{314703872}a^{27}-\frac{207005}{314703872}a^{26}-\frac{94599}{39337984}a^{25}+\frac{365797}{314703872}a^{24}-\frac{139411}{314703872}a^{23}+\frac{87131}{78675968}a^{22}+\frac{5009523}{157351936}a^{21}+\frac{53563}{4917248}a^{20}+\frac{930165}{78675968}a^{19}+\frac{542359}{19668992}a^{18}-\frac{868265}{39337984}a^{17}-\frac{21761}{9834496}a^{16}-\frac{573635}{19668992}a^{15}-\frac{235117}{614656}a^{14}-\frac{1069945}{9834496}a^{13}-\frac{51599}{614656}a^{12}-\frac{843891}{4917248}a^{11}+\frac{679529}{2458624}a^{10}+\frac{753}{19208}a^{9}+\frac{96603}{307328}a^{8}+\frac{436629}{153664}a^{7}+\frac{4161}{10976}a^{6}+\frac{27897}{76832}a^{5}+\frac{19835}{38416}a^{4}-\frac{173}{98}a^{3}+\frac{731}{2401}a^{2}-\frac{2690}{2401}a-\frac{27404}{2401}$, $\frac{15}{5619712}a^{35}-\frac{139}{11239424}a^{34}+\frac{213}{11239424}a^{33}+\frac{173}{5619712}a^{32}-\frac{1263}{11239424}a^{31}+\frac{1259}{11239424}a^{30}+\frac{459}{5619712}a^{29}-\frac{3103}{11239424}a^{28}+\frac{197}{401408}a^{27}-\frac{4479}{11239424}a^{26}-\frac{15025}{11239424}a^{25}+\frac{15601}{5619712}a^{24}-\frac{25589}{11239424}a^{23}-\frac{31783}{11239424}a^{22}+\frac{4593}{702464}a^{21}-\frac{27353}{5619712}a^{20}+\frac{16901}{2809856}a^{19}+\frac{59929}{2809856}a^{18}-\frac{48525}{1404928}a^{17}+\frac{35487}{1404928}a^{16}+\frac{21145}{702464}a^{15}-\frac{1311}{14336}a^{14}+\frac{9661}{351232}a^{13}-\frac{21725}{351232}a^{12}-\frac{36885}{175616}a^{11}+\frac{59705}{175616}a^{10}-\frac{3511}{21952}a^{9}-\frac{87}{343}a^{8}+\frac{33281}{43904}a^{7}-\frac{989}{10976}a^{6}+\frac{1053}{10976}a^{5}+\frac{2951}{2744}a^{4}-\frac{550}{343}a^{3}+\frac{351}{1372}a^{2}+\frac{899}{686}a-\frac{676}{343}$ 17/5619712a^35 + 37/5619712a^34 + 17/5619712a^32 - 51/5619712a^31 - 17/5619712a^29 - 289/2809856a^28 - 305/5619712a^27 + 17/5619712a^26 - 51/1404928a^25 + 1411/5619712a^24 - 17/5619712a^23 + 85/702464a^22 + 4709/2809856a^21 + 4415/2809856a^20 - 289/1404928a^19 + 51/351232a^18 - 2635/702464a^17 + 51/175616a^16 - 289/351232a^15 - 765/43904a^14 - 2389/175616a^13 + 85/21952a^12 - 17/87808a^11 + 1411/43904a^10 - 51/5488a^9 + 17/10976a^8 + 629/5488a^7 + 639/21952a^6 - 17/1372a^5 - 51/343a^3 + 34/343a^2 + 207/343, 167/44957696a^35 + 1195/44957696a^34 + 167/22478848a^33 - 1537/44957696a^32 - 1195/44957696a^31 - 3107/22478848a^30 - 7311/44957696a^29 - 89/802816a^28 - 29489/44957696a^27 - 103/917504a^26 + 30175/22478848a^25 + 62949/44957696a^24 + 25969/6422528a^23 + 83323/22478848a^22 + 11999/22478848a^21 + 12961/1605632a^20 - 32635/11239424a^19 - 141577/5619712a^18 - 18835/802816a^17 - 147271/2809856a^16 - 114799/2809856a^15 + 5279/200704a^14 - 60145/1404928a^13 + 41907/702464a^12 + 25267/100352a^11 + 36401/175616a^10 + 63837/175616a^9 + 613/3136a^8 - 9651/21952a^7 + 53/686a^6 - 2403/5488a^5 - 6043/5488a^4 - 212/343a^3 - 831/686a^2 - 167/343a + 1028/343, 681/44957696a^35 - 681/44957696a^34 - 1/2809856a^33 - 2271/44957696a^32 + 925/44957696a^31 + 127/2809856a^30 + 4015/44957696a^29 - 9909/22478848a^28 + 23165/44957696a^27 + 7201/44957696a^26 + 13357/11239424a^25 - 53309/44957696a^24 - 75073/44957696a^23 - 11685/5619712a^22 + 143905/22478848a^21 - 44879/5619712a^20 - 529/11239424a^19 - 2351/175616a^18 + 125357/5619712a^17 + 21723/702464a^16 + 80271/2809856a^15 - 58153/702464a^14 + 89253/1404928a^13 - 9523/351232a^12 + 47479/702464a^11 - 83091/351232a^10 - 11631/43904a^9 - 1345/10976a^8 + 15773/21952a^7 - 1935/10976a^6 + 3041/10976a^5 - 223/1372a^4 + 2903/2744a^3 + 681/686a^2 - 99/343a - 1164/343, 579/44957696a^35 - 99/6422528a^34 + 167/22478848a^33 - 449/44957696a^32 + 205/44957696a^31 + 841/22478848a^30 + 449/44957696a^29 - 5767/11239424a^28 + 7575/44957696a^27 - 20231/44957696a^26 - 2733/22478848a^25 - 4827/44957696a^24 - 37841/44957696a^23 - 3553/22478848a^22 + 242701/22478848a^21 + 2817/11239424a^20 + 14725/1605632a^19 + 18057/5619712a^18 - 2467/5619712a^17 + 45/8192a^16 - 21737/2809856a^15 - 198601/1404928a^14 - 37927/1404928a^13 - 8641/100352a^12 - 6957/702464a^11 + 1025/21952a^10 - 159/12544a^9 + 7181/87808a^8 + 48149/43904a^7 + 381/2744a^6 + 4419/10976a^5 - 1/686a^4 - 557/1372a^3 - 4/343a^2 + 4/343a - 1431/343, 2647/314703872a^35 - 2419/314703872a^34 - 711/78675968a^33 + 407/6422528a^32 - 37033/314703872a^31 - 3797/78675968a^30 + 2159/44957696a^29 - 47477/157351936a^28 + 47347/314703872a^27 + 38147/314703872a^26 - 48401/39337984a^25 + 979077/314703872a^24 + 348701/314703872a^23 - 77113/78675968a^22 + 733827/157351936a^21 - 64195/19668992a^20 - 64063/78675968a^19 + 289825/19668992a^18 - 1762317/39337984a^17 - 142787/9834496a^16 + 337025/19668992a^15 - 98061/2458624a^14 + 382467/9834496a^13 + 22451/1229312a^12 - 633567/4917248a^11 + 974797/2458624a^10 + 64707/614656a^9 - 28643/153664a^8 + 74147/307328a^7 - 2101/5488a^6 + 2019/38416a^5 + 1476/2401a^4 - 5499/2744a^3 - 1235/4802a^2 + 2619/2401a - 1733/2401, 681/44957696a^35 - 241/44957696a^34 - 1/5619712a^33 - 1039/44957696a^32 - 3755/44957696a^31 - 797/5619712a^30 + 5535/44957696a^29 - 9473/22478848a^28 + 17597/44957696a^27 + 29961/44957696a^26 + 8847/11239424a^25 + 93299/44957696a^24 + 127175/44957696a^23 - 12379/2809856a^22 + 89865/22478848a^21 - 47037/5619712a^20 - 19483/1605632a^19 - 14645/1404928a^18 - 121319/5619712a^17 - 1361/50176a^16 + 204627/2809856a^15 - 16507/702464a^14 + 131217/1404928a^13 + 39927/351232a^12 + 22699/702464a^11 + 40465/351232a^10 + 14359/87808a^9 - 3901/5488a^8 + 6445/43904a^7 - 2235/5488a^6 - 5995/10976a^5 + 533/2744a^4 - 33/343a^3 - 681/686a^2 + 1857/686a + 8/343, 183/11239424a^35 + 1/351232a^34 + 107/11239424a^33 - 139/11239424a^32 + 155/5619712a^31 - 169/1605632a^30 + 683/11239424a^29 - 5137/11239424a^28 - 2209/11239424a^27 - 17/5619712a^26 + 447/1605632a^25 - 5325/11239424a^24 + 12717/5619712a^23 - 24077/11239424a^22 + 21065/2809856a^21 + 12865/5619712a^20 - 7929/2809856a^19 - 1249/2809856a^18 + 3347/1404928a^17 - 37743/1404928a^16 + 24785/702464a^15 - 55893/702464a^14 - 2899/351232a^13 + 11149/351232a^12 - 993/175616a^11 - 3/3584a^10 + 12279/87808a^9 - 2829/10976a^8 + 23515/43904a^7 - 467/5488a^6 - 15/196a^5 + 683/5488a^4 + 9/2744a^3 - 355/686a^2 + 701/686a - 565/343, 1473/22478848a^35 + 181/11239424a^34 + 461/22478848a^33 + 2957/22478848a^32 - 271/5619712a^31 - 2185/22478848a^30 - 2309/22478848a^29 - 51897/22478848a^28 - 15305/22478848a^27 - 5991/5619712a^26 - 9799/3211264a^25 + 39355/22478848a^24 + 16077/5619712a^23 + 99765/22478848a^22 + 476235/11239424a^21 + 121005/11239424a^20 + 82183/5619712a^19 + 214131/5619712a^18 - 15307/401408a^17 - 21545/401408a^16 - 12337/200704a^15 - 663561/1404928a^14 - 64759/702464a^13 - 71891/702464a^12 - 87089/351232a^11 + 149797/351232a^10 + 76165/175616a^9 + 43121/87808a^8 + 9005/2744a^7 + 3463/21952a^6 + 4149/10976a^5 + 5497/5488a^4 - 435/196a^3 - 307/196a^2 - 99/98a - 3837/343, 393/11239424a^35 + 89/5619712a^34 + 1/5619712a^33 + 339/5619712a^32 - 31/5619712a^31 - 5/200704a^30 - 223/5619712a^29 - 15903/11239424a^28 - 4097/5619712a^27 - 619/2809856a^26 - 975/702464a^25 + 5851/5619712a^24 + 11145/5619712a^23 + 1977/702464a^22 + 294955/11239424a^21 + 1009/87808a^20 + 3701/5619712a^19 + 521/43904a^18 - 74441/2809856a^17 - 27437/702464a^16 - 51559/1404928a^15 - 100901/351232a^14 - 65417/702464a^13 + 4689/87808a^12 - 6191/351232a^11 + 28031/87808a^10 + 63227/175616a^9 + 123/686a^8 + 39989/21952a^7 + 3735/21952a^6 - 6843/10976a^5 - 923/5488a^4 - 3845/2744a^3 - 1035/1372a^2 + 13/49a - 2063/343, 681/44957696a^35 + 179/44957696a^34 - 219/11239424a^33 + 1945/44957696a^32 - 169/6422528a^31 - 961/11239424a^30 - 7321/44957696a^29 - 7239/22478848a^28 + 421/44957696a^27 + 11061/44957696a^26 - 261/351232a^25 + 1019/917504a^24 + 99115/44957696a^23 + 46607/11239424a^22 + 86949/22478848a^21 - 1161/802816a^20 - 62217/11239424a^19 + 1257/175616a^18 - 97835/5619712a^17 - 24939/702464a^16 - 132497/2809856a^15 - 8493/702464a^14 + 52813/1404928a^13 + 19489/351232a^12 - 25657/702464a^11 + 55781/351232a^10 + 20031/87808a^9 + 32769/87808a^8 - 8119/43904a^7 - 4185/10976a^6 - 3833/10976a^5 + 591/2744a^4 - 1097/2744a^3 - 891/686a^2 - 685/686a + 1028/343, 681/44957696a^35 + 689/44957696a^34 + 461/22478848a^33 + 1693/44957696a^32 - 95/6422528a^31 - 3653/22478848a^30 - 45/917504a^29 - 6725/11239424a^28 - 31435/44957696a^27 - 25765/44957696a^26 - 8903/22478848a^25 + 73903/44957696a^24 + 208941/44957696a^23 + 55389/22478848a^22 + 255335/22478848a^21 + 98303/11239424a^20 + 4195/1605632a^19 - 21213/5619712a^18 - 219749/5619712a^17 - 29809/401408a^16 - 54431/2809856a^15 - 131195/1404928a^14 - 58913/1404928a^13 + 38139/702464a^12 + 83029/702464a^11 + 16605/43904a^10 + 22227/43904a^9 - 8553/87808a^8 + 7443/21952a^7 - 361/3136a^6 - 5165/10976a^5 - 367/784a^4 - 2249/1372a^3 - 548/343a^2 + 685/343a - 335/343, 257/22478848a^35 - 481/22478848a^34 + 33/22478848a^32 - 547/22478848a^31 + 303/1404928a^30 - 401/22478848a^29 - 3617/11239424a^28 + 12405/22478848a^27 - 1305/3211264a^26 - 2467/5619712a^25 - 3709/22478848a^24 - 108289/22478848a^23 + 2027/2809856a^22 + 75357/11239424a^21 - 8493/1404928a^20 + 51651/5619712a^19 + 2841/351232a^18 + 13949/2809856a^17 + 21047/351232a^16 - 35257/1404928a^15 - 31317/351232a^14 + 25621/702464a^13 - 19967/175616a^12 - 26673/351232a^11 - 277/25088a^10 - 4765/10976a^9 + 1835/5488a^8 + 16719/21952a^7 - 4751/21952a^6 + 6793/10976a^5 + 115/2744a^4 - 19/392a^3 + 548/343a^2 - 578/343a - 99/49, 1777/39337984a^35 + 2977/157351936a^34 + 1149/157351936a^33 + 459/11239424a^32 - 3831/157351936a^31 - 17221/157351936a^30 - 1303/11239424a^29 - 207309/157351936a^28 - 363/614656a^27 - 24271/157351936a^26 - 76505/157351936a^25 + 107525/78675968a^24 + 525667/157351936a^23 + 528761/157351936a^22 + 1656979/78675968a^21 + 569281/78675968a^20 - 52777/39337984a^19 + 35419/39337984a^18 - 457161/19668992a^17 - 910803/19668992a^16 - 327815/9834496a^15 - 1914921/9834496a^14 - 208779/4917248a^13 + 240001/4917248a^12 + 66383/2458624a^11 + 461603/2458624a^10 + 106005/307328a^9 + 102371/614656a^8 + 81537/76832a^7 + 37/343a^6 - 17345/76832a^5 - 4157/19208a^4 - 1401/2744a^3 - 2977/2401a^2 - 1149/2401a - 5624/2401, 33/2809856a^35 + 3/3211264a^34 - 717/22478848a^33 - 67/1404928a^32 - 235/22478848a^31 - 15/458752a^30 - 95/1404928a^29 - 7373/22478848a^28 + 1355/11239424a^27 + 3303/3211264a^26 + 28405/22478848a^25 - 599/5619712a^24 + 7967/22478848a^23 + 26219/22478848a^22 + 279/87808a^21 - 49423/11239424a^20 - 46841/2809856a^19 - 11411/802816a^18 + 473/87808a^17 - 11535/2809856a^16 - 2025/351232a^15 - 10485/1404928a^14 + 2809/50176a^13 + 93209/702464a^12 + 12353/175616a^11 - 25405/351232a^10 + 65/3584a^9 - 1143/21952a^8 - 317/3136a^7 - 5927/21952a^6 - 4589/10976a^5 - 69/784a^4 + 165/343a^3 - 293/1372a^2 + 185/343a + 272/343, 17135/314703872a^35 + 7697/314703872a^34 - 551/39337984a^33 + 1577/44957696a^32 - 32421/314703872a^31 - 3467/39337984a^30 + 257/6422528a^29 - 295747/157351936a^28 - 199709/314703872a^27 + 53735/314703872a^26 - 55063/78675968a^25 + 921437/314703872a^24 + 761289/314703872a^23 - 957/19668992a^22 + 5061739/157351936a^21 + 315087/39337984a^20 - 280547/78675968a^19 + 15711/2458624a^18 - 1875681/39337984a^17 - 183753/4917248a^16 + 26261/19668992a^15 - 1709747/4917248a^14 - 533625/9834496a^13 + 143811/2458624a^12 - 119955/4917248a^11 + 1122123/2458624a^10 + 184283/614656a^9 - 19623/307328a^8 + 367719/153664a^7 + 333/2744a^6 - 9383/38416a^5 + 2801/19208a^4 - 6777/2744a^3 - 10487/9604a^2 + 104/2401a - 19498/2401, 14327/314703872a^35 + 1933/314703872a^34 + 1121/78675968a^33 + 5057/44957696a^32 - 12377/314703872a^31 + 2407/78675968a^30 + 2671/44957696a^29 - 263901/157351936a^28 - 165373/314703872a^27 - 207005/314703872a^26 - 94599/39337984a^25 + 365797/314703872a^24 - 139411/314703872a^23 + 87131/78675968a^22 + 5009523/157351936a^21 + 53563/4917248a^20 + 930165/78675968a^19 + 542359/19668992a^18 - 868265/39337984a^17 - 21761/9834496a^16 - 573635/19668992a^15 - 235117/614656a^14 - 1069945/9834496a^13 - 51599/614656a^12 - 843891/4917248a^11 + 679529/2458624a^10 + 753/19208a^9 + 96603/307328a^8 + 436629/153664a^7 + 4161/10976a^6 + 27897/76832a^5 + 19835/38416a^4 - 173/98a^3 + 731/2401a^2 - 2690/2401a - 27404/2401, 15/5619712a^35 - 139/11239424a^34 + 213/11239424a^33 + 173/5619712a^32 - 1263/11239424a^31 + 1259/11239424a^30 + 459/5619712a^29 - 3103/11239424a^28 + 197/401408a^27 - 4479/11239424a^26 - 15025/11239424a^25 + 15601/5619712a^24 - 25589/11239424a^23 - 31783/11239424a^22 + 4593/702464a^21 - 27353/5619712a^20 + 16901/2809856a^19 + 59929/2809856a^18 - 48525/1404928a^17 + 35487/1404928a^16 + 21145/702464a^15 - 1311/14336a^14 + 9661/351232a^13 - 21725/351232a^12 - 36885/175616a^11 + 59705/175616a^10 - 3511/21952a^9 - 87/343a^8 + 33281/43904a^7 - 989/10976a^6 + 1053/10976a^5 + 2951/2744a^4 - 550/343a^3 + 351/1372a^2 + 899/686a - 676/343 (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:	$ 4137469685705.321 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 4137469685705.321 \cdot 6}{14\cdot\sqrt{2369053316568954522538979736219298297541457293019971584}}\cr\approx \mathstrut & 0.268354110873635 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + x^33 - 3*x^32 - x^30 - 34*x^29 + 37*x^28 + x^27 - 12*x^26 + 83*x^25 - x^24 + 40*x^23 + 554*x^22 - 720*x^21 - 68*x^20 + 48*x^19 - 1240*x^18 + 96*x^17 - 272*x^16 - 5760*x^15 + 8864*x^14 + 1280*x^13 - 64*x^12 + 10624*x^11 - 3072*x^10 + 512*x^9 + 37888*x^8 - 69632*x^7 - 4096*x^6 - 49152*x^4 + 32768*x^3 - 131072*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 - x^35 + x^33 - 3*x^32 - x^30 - 34*x^29 + 37*x^28 + x^27 - 12*x^26 + 83*x^25 - x^24 + 40*x^23 + 554*x^22 - 720*x^21 - 68*x^20 + 48*x^19 - 1240*x^18 + 96*x^17 - 272*x^16 - 5760*x^15 + 8864*x^14 + 1280*x^13 - 64*x^12 + 10624*x^11 - 3072*x^10 + 512*x^9 + 37888*x^8 - 69632*x^7 - 4096*x^6 - 49152*x^4 + 32768*x^3 - 131072*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 - x^35 + x^33 - 3*x^32 - x^30 - 34*x^29 + 37*x^28 + x^27 - 12*x^26 + 83*x^25 - x^24 + 40*x^23 + 554*x^22 - 720*x^21 - 68*x^20 + 48*x^19 - 1240*x^18 + 96*x^17 - 272*x^16 - 5760*x^15 + 8864*x^14 + 1280*x^13 - 64*x^12 + 10624*x^11 - 3072*x^10 + 512*x^9 + 37888*x^8 - 69632*x^7 - 4096*x^6 - 49152*x^4 + 32768*x^3 - 131072*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 - x^35 + x^33 - 3*x^32 - x^30 - 34*x^29 + 37*x^28 + x^27 - 12*x^26 + 83*x^25 - x^24 + 40*x^23 + 554*x^22 - 720*x^21 - 68*x^20 + 48*x^19 - 1240*x^18 + 96*x^17 - 272*x^16 - 5760*x^15 + 8864*x^14 + 1280*x^13 - 64*x^12 + 10624*x^11 - 3072*x^10 + 512*x^9 + 37888*x^8 - 69632*x^7 - 4096*x^6 - 49152*x^4 + 32768*x^3 - 131072*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

$C_2\times C_6\times S_4$ (as 36T330):

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 288

The 60 conjugacy class representatives for $C_2\times C_6\times S_4$

Character table for $C_2\times C_6\times S_4$

Intermediate fields

$\Q(\sqrt{-7}) $, $\Q(\zeta_{7})^+$, 3.3.469.1, 6.6.86224712.1, 6.0.1539727.2, 6.0.12317816.1, $\Q(\zeta_{7})$, 9.9.247691263309.1, 12.0.7434700959482944.1, 18.18.1539172932639134777264419328.1, 18.0.429456733437258587406367.1, 18.0.219881847519876396752059904.1

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, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36
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	R	${\href{/padicField/3.6.0.1}{6} }^{6}$	${\href{/padicField/5.6.0.1}{6} }^{6}$	R	${\href{/padicField/11.6.0.1}{6} }^{6}$	${\href{/padicField/13.6.0.1}{6} }^{6}$	${\href{/padicField/17.12.0.1}{12} }^{2}{,}\,{\href{/padicField/17.6.0.1}{6} }^{2}$	${\href{/padicField/19.12.0.1}{12} }^{2}{,}\,{\href{/padicField/19.6.0.1}{6} }^{2}$	${\href{/padicField/23.3.0.1}{3} }^{12}$	${\href{/padicField/29.6.0.1}{6} }^{6}$	${\href{/padicField/31.6.0.1}{6} }^{6}$	${\href{/padicField/37.6.0.1}{6} }^{6}$	${\href{/padicField/41.6.0.1}{6} }^{6}$	${\href{/padicField/43.4.0.1}{4} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{12}$	${\href{/padicField/47.12.0.1}{12} }^{2}{,}\,{\href{/padicField/47.6.0.1}{6} }^{2}$	${\href{/padicField/53.12.0.1}{12} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{4}$	${\href{/padicField/59.6.0.1}{6} }^{6}$

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
$2$ 2	2.6.9.1	$x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$	$2$	$3$	$9$	$C_6$	$[3]^{3}$
	2.6.9.1	$x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$	$2$	$3$	$9$	$C_6$	$[3]^{3}$
	2.6.0.1	$x^{6} + x^{4} + x^{3} + x + 1$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	2.6.0.1	$x^{6} + x^{4} + x^{3} + x + 1$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	2.6.0.1	$x^{6} + x^{4} + x^{3} + x + 1$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	2.6.0.1	$x^{6} + x^{4} + x^{3} + x + 1$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
$7$ 7	7.6.5.5	$x^{6} + 7$	$6$	$1$	$5$	$C_6$	$[\ ]_{6}$
	7.6.5.5	$x^{6} + 7$	$6$	$1$	$5$	$C_6$	$[\ ]_{6}$
		Deg $24$	$12$	$2$	$22$
$67$ 67	67.6.0.1	$x^{6} + 63 x^{3} + 49 x^{2} + 55 x + 2$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	67.6.0.1	$x^{6} + 63 x^{3} + 49 x^{2} + 55 x + 2$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	67.12.6.1	$x^{12} + 402 x^{10} + 126 x^{9} + 67433 x^{8} + 110 x^{7} + 5992969 x^{6} - 3453840 x^{5} + 298670675 x^{4} - 304805096 x^{3} + 8036896364 x^{2} - 7423678304 x + 93605576188$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
	67.12.6.1	$x^{12} + 402 x^{10} + 126 x^{9} + 67433 x^{8} + 110 x^{7} + 5992969 x^{6} - 3453840 x^{5} + 298670675 x^{4} - 304805096 x^{3} + 8036896364 x^{2} - 7423678304 x + 93605576188$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$

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