Properties

Label 32.0.133...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.335\times 10^{50}$
Root discriminant \(36.85\)
Ramified primes $2,3,5,13,97$
Class number $20$ (GRH)
Class group [20] (GRH)
Galois group $D_4^2:C_2^3$ (as 32T12882)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 16*x^30 + 24*x^29 + 141*x^28 - 120*x^27 - 852*x^26 - 70*x^25 + 4216*x^24 + 4978*x^23 - 15640*x^22 - 47344*x^21 + 45797*x^20 + 270436*x^19 - 110380*x^18 - 986938*x^17 + 284105*x^16 + 1973876*x^15 - 441520*x^14 - 2163488*x^13 + 732752*x^12 + 1515008*x^11 - 1000960*x^10 - 637184*x^9 + 1079296*x^8 + 35840*x^7 - 872448*x^6 + 245760*x^5 + 577536*x^4 - 196608*x^3 - 262144*x^2 + 65536*x + 65536)
 
gp: K = bnfinit(y^32 - 2*y^31 - 16*y^30 + 24*y^29 + 141*y^28 - 120*y^27 - 852*y^26 - 70*y^25 + 4216*y^24 + 4978*y^23 - 15640*y^22 - 47344*y^21 + 45797*y^20 + 270436*y^19 - 110380*y^18 - 986938*y^17 + 284105*y^16 + 1973876*y^15 - 441520*y^14 - 2163488*y^13 + 732752*y^12 + 1515008*y^11 - 1000960*y^10 - 637184*y^9 + 1079296*y^8 + 35840*y^7 - 872448*y^6 + 245760*y^5 + 577536*y^4 - 196608*y^3 - 262144*y^2 + 65536*y + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 2*x^31 - 16*x^30 + 24*x^29 + 141*x^28 - 120*x^27 - 852*x^26 - 70*x^25 + 4216*x^24 + 4978*x^23 - 15640*x^22 - 47344*x^21 + 45797*x^20 + 270436*x^19 - 110380*x^18 - 986938*x^17 + 284105*x^16 + 1973876*x^15 - 441520*x^14 - 2163488*x^13 + 732752*x^12 + 1515008*x^11 - 1000960*x^10 - 637184*x^9 + 1079296*x^8 + 35840*x^7 - 872448*x^6 + 245760*x^5 + 577536*x^4 - 196608*x^3 - 262144*x^2 + 65536*x + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 2*x^31 - 16*x^30 + 24*x^29 + 141*x^28 - 120*x^27 - 852*x^26 - 70*x^25 + 4216*x^24 + 4978*x^23 - 15640*x^22 - 47344*x^21 + 45797*x^20 + 270436*x^19 - 110380*x^18 - 986938*x^17 + 284105*x^16 + 1973876*x^15 - 441520*x^14 - 2163488*x^13 + 732752*x^12 + 1515008*x^11 - 1000960*x^10 - 637184*x^9 + 1079296*x^8 + 35840*x^7 - 872448*x^6 + 245760*x^5 + 577536*x^4 - 196608*x^3 - 262144*x^2 + 65536*x + 65536)
 

\( x^{32} - 2 x^{31} - 16 x^{30} + 24 x^{29} + 141 x^{28} - 120 x^{27} - 852 x^{26} - 70 x^{25} + \cdots + 65536 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(133516116647406931991162570051420160000000000000000\) \(\medspace = 2^{48}\cdot 3^{16}\cdot 5^{16}\cdot 13^{8}\cdot 97^{4}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(36.85\)
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:  $2^{3/2}3^{1/2}5^{1/2}13^{1/2}97^{1/2}\approx 388.99871465083277$
Ramified primes:   \(2\), \(3\), \(5\), \(13\), \(97\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{5}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{6}+\frac{1}{4}a^{3}$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{7}+\frac{1}{4}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{17}-\frac{1}{2}a^{8}+\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{18}-\frac{1}{8}a^{17}-\frac{1}{8}a^{16}-\frac{1}{8}a^{15}-\frac{1}{4}a^{13}+\frac{1}{4}a^{12}-\frac{1}{2}a^{11}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{8}a^{6}+\frac{3}{8}a^{5}+\frac{3}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{19}-\frac{1}{8}a^{15}-\frac{1}{4}a^{12}+\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{3}{8}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{20}+\frac{1}{16}a^{16}-\frac{1}{8}a^{15}+\frac{1}{8}a^{13}+\frac{1}{4}a^{12}-\frac{3}{8}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{5}{16}a^{8}+\frac{3}{8}a^{7}-\frac{1}{2}a^{6}-\frac{1}{8}a^{5}-\frac{7}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{21}-\frac{3}{32}a^{17}+\frac{1}{16}a^{16}+\frac{1}{16}a^{14}-\frac{1}{8}a^{13}+\frac{5}{16}a^{12}+\frac{3}{8}a^{11}+\frac{1}{4}a^{10}+\frac{5}{32}a^{9}+\frac{7}{16}a^{8}+\frac{7}{16}a^{6}+\frac{5}{32}a^{5}-\frac{1}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{64}a^{22}-\frac{3}{64}a^{18}+\frac{1}{32}a^{17}+\frac{1}{32}a^{15}-\frac{1}{16}a^{14}+\frac{5}{32}a^{13}+\frac{3}{16}a^{12}+\frac{1}{8}a^{11}+\frac{5}{64}a^{10}-\frac{9}{32}a^{9}-\frac{9}{32}a^{7}+\frac{5}{64}a^{6}-\frac{1}{32}a^{5}+\frac{3}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{128}a^{23}-\frac{3}{128}a^{19}+\frac{1}{64}a^{18}+\frac{1}{64}a^{16}-\frac{1}{32}a^{15}+\frac{5}{64}a^{14}+\frac{3}{32}a^{13}+\frac{1}{16}a^{12}-\frac{59}{128}a^{11}+\frac{23}{64}a^{10}+\frac{23}{64}a^{8}-\frac{59}{128}a^{7}-\frac{1}{64}a^{6}+\frac{3}{32}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{3}$, $\frac{1}{768}a^{24}+\frac{1}{384}a^{23}+\frac{1}{192}a^{22}-\frac{19}{768}a^{20}-\frac{1}{192}a^{19}-\frac{1}{96}a^{18}-\frac{43}{384}a^{17}+\frac{1}{16}a^{16}+\frac{37}{384}a^{15}+\frac{1}{48}a^{14}-\frac{11}{96}a^{13}+\frac{133}{768}a^{12}-\frac{41}{96}a^{11}-\frac{5}{48}a^{10}+\frac{115}{384}a^{9}+\frac{91}{256}a^{8}-\frac{1}{24}a^{7}+\frac{55}{192}a^{6}-\frac{13}{48}a^{5}-\frac{19}{48}a^{4}+\frac{1}{8}a^{3}-\frac{5}{12}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{1536}a^{25}-\frac{1}{192}a^{22}-\frac{19}{1536}a^{21}+\frac{17}{768}a^{20}-\frac{35}{768}a^{18}-\frac{41}{384}a^{17}-\frac{11}{768}a^{16}-\frac{11}{128}a^{15}-\frac{5}{64}a^{14}+\frac{103}{512}a^{13}+\frac{29}{256}a^{12}-\frac{1}{8}a^{11}-\frac{63}{256}a^{10}+\frac{581}{1536}a^{9}-\frac{289}{768}a^{8}-\frac{121}{384}a^{7}+\frac{5}{64}a^{6}+\frac{31}{96}a^{5}-\frac{1}{24}a^{4}+\frac{1}{6}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{3072}a^{26}-\frac{1}{384}a^{23}-\frac{19}{3072}a^{22}+\frac{17}{1536}a^{21}-\frac{35}{1536}a^{19}-\frac{41}{768}a^{18}-\frac{11}{1536}a^{17}+\frac{21}{256}a^{16}+\frac{11}{128}a^{15}+\frac{103}{1024}a^{14}-\frac{99}{512}a^{13}+\frac{3}{16}a^{12}-\frac{63}{512}a^{11}+\frac{581}{3072}a^{10}+\frac{479}{1536}a^{9}-\frac{121}{768}a^{8}+\frac{37}{128}a^{7}+\frac{79}{192}a^{6}-\frac{1}{48}a^{5}-\frac{7}{24}a^{4}-\frac{1}{4}a^{2}+\frac{1}{3}a$, $\frac{1}{6144}a^{27}-\frac{1}{2048}a^{23}-\frac{5}{1024}a^{22}+\frac{27}{1024}a^{20}-\frac{49}{1536}a^{19}+\frac{101}{3072}a^{18}-\frac{157}{1536}a^{17}+\frac{11}{256}a^{16}+\frac{709}{6144}a^{15}-\frac{41}{3072}a^{14}+\frac{19}{96}a^{13}+\frac{1303}{3072}a^{12}-\frac{681}{2048}a^{11}+\frac{229}{1024}a^{10}-\frac{255}{512}a^{9}-\frac{3}{16}a^{8}-\frac{55}{128}a^{7}+\frac{43}{96}a^{6}-\frac{1}{96}a^{5}+\frac{17}{48}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{81051648}a^{28}-\frac{2383}{40525824}a^{27}+\frac{491}{10131456}a^{26}+\frac{2869}{10131456}a^{25}+\frac{10559}{27017216}a^{24}-\frac{39089}{20262912}a^{23}-\frac{109073}{20262912}a^{22}+\frac{619693}{40525824}a^{21}-\frac{74795}{5065728}a^{20}-\frac{89927}{40525824}a^{19}+\frac{2753}{5065728}a^{18}+\frac{139373}{1688576}a^{17}-\frac{9262459}{81051648}a^{16}+\frac{106269}{3377152}a^{15}-\frac{138745}{6754304}a^{14}+\frac{563041}{13508608}a^{13}-\frac{14338159}{81051648}a^{12}+\frac{1307137}{3377152}a^{11}-\frac{1692349}{5065728}a^{10}-\frac{2227573}{5065728}a^{9}-\frac{608027}{1266432}a^{8}-\frac{245039}{633216}a^{7}-\frac{45479}{316608}a^{6}+\frac{72257}{158304}a^{5}-\frac{1895}{6596}a^{4}-\frac{11275}{79152}a^{3}-\frac{1019}{19788}a^{2}-\frac{181}{9894}a-\frac{4943}{19788}$, $\frac{1}{162103296}a^{29}+\frac{483}{13508608}a^{27}+\frac{465}{6754304}a^{26}-\frac{6641}{54034432}a^{25}-\frac{16709}{27017216}a^{24}+\frac{44743}{13508608}a^{23}+\frac{137563}{27017216}a^{22}-\frac{547081}{40525824}a^{21}+\frac{52025}{81051648}a^{20}+\frac{2063771}{40525824}a^{19}+\frac{19997}{3377152}a^{18}+\frac{8013733}{162103296}a^{17}+\frac{7283239}{81051648}a^{16}+\frac{710461}{40525824}a^{15}+\frac{3029455}{81051648}a^{14}-\frac{588633}{3178496}a^{13}-\frac{10759767}{27017216}a^{12}+\frac{1627743}{3377152}a^{11}-\frac{870521}{3377152}a^{10}-\frac{102157}{1688576}a^{9}-\frac{122875}{1266432}a^{8}-\frac{22225}{158304}a^{7}+\frac{75265}{158304}a^{6}-\frac{25219}{52768}a^{5}+\frac{10545}{52768}a^{4}+\frac{5039}{79152}a^{3}+\frac{3817}{19788}a^{2}-\frac{2893}{13192}a+\frac{429}{6596}$, $\frac{1}{59\!\cdots\!88}a^{30}+\frac{32\!\cdots\!51}{29\!\cdots\!44}a^{29}+\frac{15\!\cdots\!39}{49\!\cdots\!24}a^{28}+\frac{29\!\cdots\!47}{37\!\cdots\!68}a^{27}+\frac{13\!\cdots\!25}{59\!\cdots\!88}a^{26}-\frac{17\!\cdots\!67}{12\!\cdots\!56}a^{25}-\frac{53\!\cdots\!99}{12\!\cdots\!56}a^{24}-\frac{42\!\cdots\!67}{29\!\cdots\!44}a^{23}-\frac{15\!\cdots\!93}{24\!\cdots\!12}a^{22}-\frac{19\!\cdots\!19}{29\!\cdots\!44}a^{21}-\frac{10\!\cdots\!13}{74\!\cdots\!36}a^{20}+\frac{59\!\cdots\!83}{74\!\cdots\!36}a^{19}-\frac{25\!\cdots\!67}{59\!\cdots\!88}a^{18}+\frac{53\!\cdots\!23}{14\!\cdots\!72}a^{17}+\frac{12\!\cdots\!11}{74\!\cdots\!36}a^{16}+\frac{19\!\cdots\!75}{29\!\cdots\!44}a^{15}+\frac{44\!\cdots\!83}{19\!\cdots\!96}a^{14}+\frac{18\!\cdots\!13}{14\!\cdots\!72}a^{13}-\frac{39\!\cdots\!71}{14\!\cdots\!72}a^{12}-\frac{55\!\cdots\!29}{12\!\cdots\!56}a^{11}-\frac{13\!\cdots\!01}{93\!\cdots\!92}a^{10}+\frac{11\!\cdots\!87}{31\!\cdots\!64}a^{9}-\frac{68\!\cdots\!95}{14\!\cdots\!28}a^{8}+\frac{40\!\cdots\!55}{11\!\cdots\!24}a^{7}-\frac{16\!\cdots\!95}{58\!\cdots\!12}a^{6}+\frac{61\!\cdots\!79}{19\!\cdots\!04}a^{5}-\frac{30\!\cdots\!49}{73\!\cdots\!64}a^{4}-\frac{14\!\cdots\!07}{14\!\cdots\!28}a^{3}-\frac{33\!\cdots\!35}{14\!\cdots\!28}a^{2}+\frac{33\!\cdots\!55}{12\!\cdots\!44}a-\frac{31\!\cdots\!31}{21\!\cdots\!96}$, $\frac{1}{12\!\cdots\!52}a^{31}+\frac{73171}{77\!\cdots\!72}a^{30}+\frac{29\!\cdots\!05}{45\!\cdots\!16}a^{29}-\frac{68\!\cdots\!89}{15\!\cdots\!44}a^{28}-\frac{96\!\cdots\!87}{12\!\cdots\!52}a^{27}-\frac{32\!\cdots\!09}{20\!\cdots\!92}a^{26}+\frac{63\!\cdots\!49}{25\!\cdots\!24}a^{25}+\frac{88\!\cdots\!43}{20\!\cdots\!92}a^{24}+\frac{44\!\cdots\!31}{31\!\cdots\!88}a^{23}-\frac{34\!\cdots\!81}{20\!\cdots\!92}a^{22}+\frac{39\!\cdots\!07}{31\!\cdots\!88}a^{21}-\frac{43\!\cdots\!79}{15\!\cdots\!44}a^{20}-\frac{26\!\cdots\!27}{12\!\cdots\!52}a^{19}+\frac{21\!\cdots\!31}{62\!\cdots\!76}a^{18}-\frac{64\!\cdots\!75}{25\!\cdots\!24}a^{17}+\frac{84\!\cdots\!11}{62\!\cdots\!76}a^{16}-\frac{82\!\cdots\!19}{73\!\cdots\!56}a^{15}+\frac{25\!\cdots\!05}{20\!\cdots\!92}a^{14}+\frac{75\!\cdots\!83}{31\!\cdots\!88}a^{13}+\frac{56\!\cdots\!07}{15\!\cdots\!44}a^{12}+\frac{20\!\cdots\!53}{76\!\cdots\!36}a^{11}-\frac{46\!\cdots\!53}{19\!\cdots\!68}a^{10}+\frac{11\!\cdots\!33}{32\!\cdots\!28}a^{9}+\frac{45\!\cdots\!63}{12\!\cdots\!48}a^{8}-\frac{17\!\cdots\!65}{60\!\cdots\!24}a^{7}-\frac{90\!\cdots\!23}{40\!\cdots\!16}a^{6}+\frac{23\!\cdots\!49}{60\!\cdots\!24}a^{5}+\frac{10\!\cdots\!41}{30\!\cdots\!12}a^{4}-\frac{18\!\cdots\!43}{10\!\cdots\!04}a^{3}-\frac{27\!\cdots\!49}{15\!\cdots\!56}a^{2}+\frac{18\!\cdots\!51}{75\!\cdots\!28}a+\frac{15\!\cdots\!61}{71\!\cdots\!88}$ Copy content Toggle raw display

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

$C_{20}$, which has order $20$ (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:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{6280699354319001510588492377}{126794282073490438265545608134656} a^{31} - \frac{9666981736606408222611246545}{95095711555117828699159206100992} a^{30} - \frac{792684464762217849236039051}{932310897599194399011364765696} a^{29} + \frac{3776509980154824485176227601}{2796932692797583197034094297088} a^{28} + \frac{991231811694610691032390474229}{126794282073490438265545608134656} a^{27} - \frac{1419743233517758053587809909967}{190191423110235657398318412201984} a^{26} - \frac{389217338599865810255463165941}{7924642629593152391596600508416} a^{25} + \frac{163873099324473363393333622925}{63397141036745219132772804067328} a^{24} + \frac{23626843745437979365889952194855}{95095711555117828699159206100992} a^{23} + \frac{16379170780041305952300553922989}{63397141036745219132772804067328} a^{22} - \frac{30643988209742607661689418981999}{31698570518372609566386402033664} a^{21} - \frac{125000682581242613497601485871161}{47547855777558914349579603050496} a^{20} + \frac{1113020852071040759576079056392967}{380382846220471314796636824403968} a^{19} + \frac{3005008846211023299312023226307939}{190191423110235657398318412201984} a^{18} - \frac{28810609678747878190473689318623}{3962321314796576195798300254208} a^{17} - \frac{11577031543418779028247072928613735}{190191423110235657398318412201984} a^{16} + \frac{6659902887413078207172578036770159}{380382846220471314796636824403968} a^{15} + \frac{25268200394982357586660559524821971}{190191423110235657398318412201984} a^{14} - \frac{2623129379045932299512595663183163}{95095711555117828699159206100992} a^{13} - \frac{2271658881534620108179115289352629}{15849285259186304783193201016832} a^{12} + \frac{2033582569420434774760692371797}{74760779524463701807515099136} a^{11} + \frac{464448343730734343162312167692421}{5943481972194864293697450381312} a^{10} - \frac{59214493029518640273303198564893}{990580328699144048949575063552} a^{9} - \frac{255403408873581749831009551823}{7283678887493706242276287232} a^{8} + \frac{87795450643314410778669672941}{1451045403367886790453479097} a^{7} - \frac{844333857272749798470419049659}{371467623262179018356090648832} a^{6} - \frac{9400875130023257148110769082211}{185733811631089509178045324416} a^{5} + \frac{417484008346877247147965935805}{30955635271848251529674220736} a^{4} + \frac{2848138562285835296864712607325}{92866905815544754589022662208} a^{3} - \frac{475593885459481042048592501521}{46433452907772377294511331104} a^{2} - \frac{346738523038436140883932360955}{23216726453886188647255665552} a - \frac{6306616162422131411344170487}{3869454408981031441209277592} \)  (order $12$) Copy content Toggle raw display
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:   $\frac{10\!\cdots\!93}{12\!\cdots\!12}a^{31}-\frac{25\!\cdots\!01}{15\!\cdots\!44}a^{30}-\frac{89\!\cdots\!95}{64\!\cdots\!56}a^{29}+\frac{21\!\cdots\!67}{97\!\cdots\!84}a^{28}+\frac{49\!\cdots\!07}{38\!\cdots\!36}a^{27}-\frac{18\!\cdots\!21}{15\!\cdots\!44}a^{26}-\frac{20\!\cdots\!51}{25\!\cdots\!24}a^{25}+\frac{27\!\cdots\!17}{64\!\cdots\!56}a^{24}+\frac{31\!\cdots\!07}{77\!\cdots\!72}a^{23}+\frac{54\!\cdots\!69}{12\!\cdots\!12}a^{22}-\frac{40\!\cdots\!35}{25\!\cdots\!24}a^{21}-\frac{55\!\cdots\!69}{12\!\cdots\!12}a^{20}+\frac{18\!\cdots\!57}{38\!\cdots\!36}a^{19}+\frac{13\!\cdots\!13}{51\!\cdots\!48}a^{18}-\frac{91\!\cdots\!11}{77\!\cdots\!72}a^{17}-\frac{63\!\cdots\!61}{64\!\cdots\!56}a^{16}+\frac{22\!\cdots\!31}{77\!\cdots\!72}a^{15}+\frac{11\!\cdots\!85}{51\!\cdots\!48}a^{14}-\frac{11\!\cdots\!85}{25\!\cdots\!24}a^{13}-\frac{90\!\cdots\!47}{38\!\cdots\!36}a^{12}+\frac{13\!\cdots\!77}{30\!\cdots\!88}a^{11}+\frac{61\!\cdots\!79}{48\!\cdots\!92}a^{10}-\frac{78\!\cdots\!65}{80\!\cdots\!32}a^{9}-\frac{22\!\cdots\!71}{40\!\cdots\!16}a^{8}+\frac{49\!\cdots\!77}{50\!\cdots\!52}a^{7}-\frac{55\!\cdots\!95}{15\!\cdots\!56}a^{6}-\frac{38\!\cdots\!37}{47\!\cdots\!33}a^{5}+\frac{16\!\cdots\!55}{75\!\cdots\!28}a^{4}+\frac{18\!\cdots\!47}{37\!\cdots\!64}a^{3}-\frac{62\!\cdots\!49}{37\!\cdots\!64}a^{2}-\frac{45\!\cdots\!63}{18\!\cdots\!32}a-\frac{25\!\cdots\!11}{94\!\cdots\!66}$, $\frac{62\!\cdots\!77}{12\!\cdots\!56}a^{31}-\frac{96\!\cdots\!45}{95\!\cdots\!92}a^{30}-\frac{79\!\cdots\!51}{93\!\cdots\!96}a^{29}+\frac{37\!\cdots\!01}{27\!\cdots\!88}a^{28}+\frac{99\!\cdots\!29}{12\!\cdots\!56}a^{27}-\frac{14\!\cdots\!67}{19\!\cdots\!84}a^{26}-\frac{38\!\cdots\!41}{79\!\cdots\!16}a^{25}+\frac{16\!\cdots\!25}{63\!\cdots\!28}a^{24}+\frac{23\!\cdots\!55}{95\!\cdots\!92}a^{23}+\frac{16\!\cdots\!89}{63\!\cdots\!28}a^{22}-\frac{30\!\cdots\!99}{31\!\cdots\!64}a^{21}-\frac{12\!\cdots\!61}{47\!\cdots\!96}a^{20}+\frac{11\!\cdots\!67}{38\!\cdots\!68}a^{19}+\frac{30\!\cdots\!39}{19\!\cdots\!84}a^{18}-\frac{28\!\cdots\!23}{39\!\cdots\!08}a^{17}-\frac{11\!\cdots\!35}{19\!\cdots\!84}a^{16}+\frac{66\!\cdots\!59}{38\!\cdots\!68}a^{15}+\frac{25\!\cdots\!71}{19\!\cdots\!84}a^{14}-\frac{26\!\cdots\!63}{95\!\cdots\!92}a^{13}-\frac{22\!\cdots\!29}{15\!\cdots\!32}a^{12}+\frac{20\!\cdots\!97}{74\!\cdots\!36}a^{11}+\frac{46\!\cdots\!21}{59\!\cdots\!12}a^{10}-\frac{59\!\cdots\!93}{99\!\cdots\!52}a^{9}-\frac{25\!\cdots\!23}{72\!\cdots\!32}a^{8}+\frac{87\!\cdots\!41}{14\!\cdots\!97}a^{7}-\frac{84\!\cdots\!59}{37\!\cdots\!32}a^{6}-\frac{94\!\cdots\!11}{18\!\cdots\!16}a^{5}+\frac{41\!\cdots\!05}{30\!\cdots\!36}a^{4}+\frac{28\!\cdots\!25}{92\!\cdots\!08}a^{3}-\frac{47\!\cdots\!21}{46\!\cdots\!04}a^{2}-\frac{34\!\cdots\!55}{23\!\cdots\!52}a-\frac{10\!\cdots\!79}{38\!\cdots\!92}$, $\frac{61\!\cdots\!93}{36\!\cdots\!96}a^{31}-\frac{25\!\cdots\!29}{30\!\cdots\!08}a^{30}-\frac{63\!\cdots\!15}{46\!\cdots\!12}a^{29}+\frac{52\!\cdots\!53}{46\!\cdots\!12}a^{28}+\frac{29\!\cdots\!17}{36\!\cdots\!96}a^{27}-\frac{15\!\cdots\!21}{18\!\cdots\!48}a^{26}-\frac{42\!\cdots\!61}{76\!\cdots\!52}a^{25}+\frac{67\!\cdots\!33}{18\!\cdots\!48}a^{24}+\frac{10\!\cdots\!47}{18\!\cdots\!24}a^{23}-\frac{72\!\cdots\!57}{61\!\cdots\!16}a^{22}-\frac{39\!\cdots\!19}{92\!\cdots\!24}a^{21}+\frac{55\!\cdots\!87}{15\!\cdots\!04}a^{20}+\frac{10\!\cdots\!45}{36\!\cdots\!96}a^{19}+\frac{26\!\cdots\!13}{18\!\cdots\!48}a^{18}-\frac{79\!\cdots\!75}{57\!\cdots\!64}a^{17}-\frac{11\!\cdots\!25}{18\!\cdots\!48}a^{16}+\frac{17\!\cdots\!89}{38\!\cdots\!68}a^{15}+\frac{47\!\cdots\!97}{18\!\cdots\!48}a^{14}-\frac{74\!\cdots\!89}{92\!\cdots\!24}a^{13}+\frac{10\!\cdots\!75}{15\!\cdots\!04}a^{12}+\frac{89\!\cdots\!57}{11\!\cdots\!28}a^{11}-\frac{40\!\cdots\!47}{19\!\cdots\!88}a^{10}-\frac{81\!\cdots\!29}{18\!\cdots\!48}a^{9}+\frac{13\!\cdots\!63}{36\!\cdots\!04}a^{8}+\frac{10\!\cdots\!91}{60\!\cdots\!84}a^{7}-\frac{13\!\cdots\!41}{36\!\cdots\!04}a^{6}+\frac{25\!\cdots\!31}{18\!\cdots\!52}a^{5}+\frac{25\!\cdots\!05}{90\!\cdots\!76}a^{4}-\frac{39\!\cdots\!53}{52\!\cdots\!28}a^{3}-\frac{23\!\cdots\!69}{15\!\cdots\!96}a^{2}+\frac{10\!\cdots\!47}{22\!\cdots\!44}a+\frac{20\!\cdots\!45}{37\!\cdots\!24}$, $\frac{79\!\cdots\!69}{41\!\cdots\!84}a^{31}-\frac{53\!\cdots\!61}{10\!\cdots\!96}a^{30}-\frac{43\!\cdots\!59}{15\!\cdots\!44}a^{29}+\frac{10\!\cdots\!41}{16\!\cdots\!52}a^{28}+\frac{17\!\cdots\!51}{73\!\cdots\!56}a^{27}-\frac{78\!\cdots\!37}{20\!\cdots\!92}a^{26}-\frac{11\!\cdots\!97}{77\!\cdots\!72}a^{25}+\frac{14\!\cdots\!73}{20\!\cdots\!92}a^{24}+\frac{82\!\cdots\!77}{10\!\cdots\!96}a^{23}+\frac{33\!\cdots\!99}{62\!\cdots\!76}a^{22}-\frac{10\!\cdots\!09}{31\!\cdots\!88}a^{21}-\frac{38\!\cdots\!75}{51\!\cdots\!48}a^{20}+\frac{55\!\cdots\!13}{41\!\cdots\!84}a^{19}+\frac{28\!\cdots\!95}{62\!\cdots\!76}a^{18}-\frac{54\!\cdots\!45}{11\!\cdots\!28}a^{17}-\frac{35\!\cdots\!77}{20\!\cdots\!92}a^{16}+\frac{58\!\cdots\!21}{41\!\cdots\!84}a^{15}+\frac{64\!\cdots\!57}{20\!\cdots\!92}a^{14}-\frac{63\!\cdots\!15}{31\!\cdots\!88}a^{13}-\frac{45\!\cdots\!87}{15\!\cdots\!44}a^{12}+\frac{26\!\cdots\!65}{12\!\cdots\!12}a^{11}+\frac{79\!\cdots\!51}{64\!\cdots\!56}a^{10}-\frac{21\!\cdots\!71}{97\!\cdots\!84}a^{9}-\frac{98\!\cdots\!33}{12\!\cdots\!48}a^{8}+\frac{97\!\cdots\!89}{59\!\cdots\!12}a^{7}-\frac{29\!\cdots\!81}{40\!\cdots\!16}a^{6}-\frac{63\!\cdots\!75}{60\!\cdots\!24}a^{5}+\frac{79\!\cdots\!45}{10\!\cdots\!04}a^{4}+\frac{16\!\cdots\!41}{30\!\cdots\!12}a^{3}-\frac{69\!\cdots\!13}{15\!\cdots\!56}a^{2}-\frac{17\!\cdots\!39}{75\!\cdots\!28}a+\frac{68\!\cdots\!25}{12\!\cdots\!88}$, $\frac{94\!\cdots\!27}{12\!\cdots\!32}a^{31}-\frac{19\!\cdots\!93}{15\!\cdots\!44}a^{30}-\frac{19\!\cdots\!11}{15\!\cdots\!44}a^{29}+\frac{67\!\cdots\!95}{51\!\cdots\!48}a^{28}+\frac{13\!\cdots\!17}{12\!\cdots\!52}a^{27}-\frac{26\!\cdots\!63}{62\!\cdots\!76}a^{26}-\frac{33\!\cdots\!87}{51\!\cdots\!48}a^{25}-\frac{21\!\cdots\!39}{62\!\cdots\!76}a^{24}+\frac{30\!\cdots\!41}{10\!\cdots\!96}a^{23}+\frac{31\!\cdots\!13}{62\!\cdots\!76}a^{22}-\frac{28\!\cdots\!09}{31\!\cdots\!88}a^{21}-\frac{61\!\cdots\!21}{15\!\cdots\!44}a^{20}+\frac{20\!\cdots\!17}{12\!\cdots\!52}a^{19}+\frac{12\!\cdots\!55}{62\!\cdots\!76}a^{18}+\frac{15\!\cdots\!51}{15\!\cdots\!44}a^{17}-\frac{14\!\cdots\!95}{20\!\cdots\!92}a^{16}-\frac{46\!\cdots\!53}{41\!\cdots\!84}a^{15}+\frac{80\!\cdots\!47}{62\!\cdots\!76}a^{14}+\frac{35\!\cdots\!91}{10\!\cdots\!96}a^{13}-\frac{61\!\cdots\!75}{51\!\cdots\!48}a^{12}-\frac{60\!\cdots\!65}{38\!\cdots\!36}a^{11}+\frac{49\!\cdots\!47}{64\!\cdots\!56}a^{10}-\frac{15\!\cdots\!81}{57\!\cdots\!52}a^{9}-\frac{14\!\cdots\!57}{30\!\cdots\!12}a^{8}+\frac{25\!\cdots\!71}{60\!\cdots\!24}a^{7}+\frac{22\!\cdots\!79}{12\!\cdots\!48}a^{6}-\frac{86\!\cdots\!97}{20\!\cdots\!08}a^{5}-\frac{57\!\cdots\!11}{10\!\cdots\!04}a^{4}+\frac{30\!\cdots\!09}{10\!\cdots\!04}a^{3}+\frac{76\!\cdots\!59}{15\!\cdots\!56}a^{2}-\frac{91\!\cdots\!15}{75\!\cdots\!28}a-\frac{56\!\cdots\!37}{12\!\cdots\!88}$, $\frac{18\!\cdots\!43}{62\!\cdots\!76}a^{31}-\frac{83\!\cdots\!51}{97\!\cdots\!84}a^{30}-\frac{76\!\cdots\!93}{19\!\cdots\!68}a^{29}+\frac{52\!\cdots\!69}{48\!\cdots\!92}a^{28}+\frac{39\!\cdots\!41}{12\!\cdots\!76}a^{27}-\frac{66\!\cdots\!23}{10\!\cdots\!96}a^{26}-\frac{12\!\cdots\!81}{64\!\cdots\!56}a^{25}+\frac{15\!\cdots\!93}{10\!\cdots\!96}a^{24}+\frac{17\!\cdots\!41}{15\!\cdots\!44}a^{23}+\frac{51\!\cdots\!13}{10\!\cdots\!96}a^{22}-\frac{79\!\cdots\!27}{15\!\cdots\!44}a^{21}-\frac{74\!\cdots\!93}{77\!\cdots\!72}a^{20}+\frac{13\!\cdots\!03}{62\!\cdots\!76}a^{19}+\frac{18\!\cdots\!41}{31\!\cdots\!88}a^{18}-\frac{49\!\cdots\!79}{57\!\cdots\!64}a^{17}-\frac{67\!\cdots\!23}{31\!\cdots\!88}a^{16}+\frac{16\!\cdots\!59}{62\!\cdots\!76}a^{15}+\frac{10\!\cdots\!65}{31\!\cdots\!88}a^{14}-\frac{60\!\cdots\!19}{15\!\cdots\!44}a^{13}-\frac{17\!\cdots\!37}{64\!\cdots\!56}a^{12}+\frac{23\!\cdots\!85}{64\!\cdots\!56}a^{11}+\frac{55\!\cdots\!45}{97\!\cdots\!84}a^{10}-\frac{12\!\cdots\!47}{40\!\cdots\!16}a^{9}+\frac{68\!\cdots\!45}{75\!\cdots\!28}a^{8}+\frac{68\!\cdots\!93}{35\!\cdots\!72}a^{7}-\frac{92\!\cdots\!91}{60\!\cdots\!24}a^{6}-\frac{99\!\cdots\!73}{10\!\cdots\!04}a^{5}+\frac{68\!\cdots\!57}{50\!\cdots\!52}a^{4}+\frac{18\!\cdots\!69}{50\!\cdots\!52}a^{3}-\frac{51\!\cdots\!47}{75\!\cdots\!28}a^{2}-\frac{43\!\cdots\!03}{37\!\cdots\!64}a+\frac{36\!\cdots\!15}{31\!\cdots\!22}$, $\frac{31\!\cdots\!01}{15\!\cdots\!44}a^{31}-\frac{58\!\cdots\!11}{77\!\cdots\!72}a^{30}-\frac{17\!\cdots\!35}{77\!\cdots\!72}a^{29}+\frac{15\!\cdots\!37}{16\!\cdots\!64}a^{28}+\frac{25\!\cdots\!37}{15\!\cdots\!44}a^{27}-\frac{25\!\cdots\!75}{38\!\cdots\!36}a^{26}-\frac{25\!\cdots\!63}{25\!\cdots\!24}a^{25}+\frac{18\!\cdots\!71}{77\!\cdots\!72}a^{24}+\frac{13\!\cdots\!81}{19\!\cdots\!68}a^{23}-\frac{31\!\cdots\!85}{77\!\cdots\!72}a^{22}-\frac{25\!\cdots\!97}{64\!\cdots\!56}a^{21}-\frac{13\!\cdots\!19}{38\!\cdots\!36}a^{20}+\frac{34\!\cdots\!29}{15\!\cdots\!44}a^{19}+\frac{28\!\cdots\!95}{97\!\cdots\!84}a^{18}-\frac{26\!\cdots\!19}{25\!\cdots\!24}a^{17}-\frac{82\!\cdots\!19}{77\!\cdots\!72}a^{16}+\frac{18\!\cdots\!55}{51\!\cdots\!48}a^{15}+\frac{45\!\cdots\!79}{38\!\cdots\!36}a^{14}-\frac{15\!\cdots\!61}{25\!\cdots\!24}a^{13}-\frac{19\!\cdots\!81}{12\!\cdots\!12}a^{12}+\frac{29\!\cdots\!83}{48\!\cdots\!92}a^{11}-\frac{55\!\cdots\!25}{48\!\cdots\!92}a^{10}-\frac{91\!\cdots\!85}{24\!\cdots\!96}a^{9}+\frac{53\!\cdots\!79}{20\!\cdots\!08}a^{8}+\frac{54\!\cdots\!05}{30\!\cdots\!12}a^{7}-\frac{41\!\cdots\!71}{15\!\cdots\!56}a^{6}-\frac{13\!\cdots\!79}{75\!\cdots\!28}a^{5}+\frac{16\!\cdots\!25}{75\!\cdots\!28}a^{4}-\frac{50\!\cdots\!21}{12\!\cdots\!88}a^{3}-\frac{11\!\cdots\!35}{94\!\cdots\!66}a^{2}+\frac{62\!\cdots\!93}{18\!\cdots\!32}a+\frac{13\!\cdots\!27}{31\!\cdots\!22}$, $\frac{60\!\cdots\!07}{46\!\cdots\!52}a^{31}-\frac{60\!\cdots\!81}{35\!\cdots\!64}a^{30}-\frac{14\!\cdots\!67}{58\!\cdots\!44}a^{29}-\frac{13\!\cdots\!47}{17\!\cdots\!32}a^{28}+\frac{31\!\cdots\!53}{14\!\cdots\!56}a^{27}+\frac{42\!\cdots\!93}{23\!\cdots\!76}a^{26}-\frac{35\!\cdots\!15}{29\!\cdots\!72}a^{25}-\frac{47\!\cdots\!93}{23\!\cdots\!76}a^{24}+\frac{50\!\cdots\!07}{11\!\cdots\!88}a^{23}+\frac{10\!\cdots\!09}{70\!\cdots\!28}a^{22}-\frac{16\!\cdots\!31}{35\!\cdots\!64}a^{21}-\frac{51\!\cdots\!49}{58\!\cdots\!44}a^{20}-\frac{28\!\cdots\!45}{46\!\cdots\!52}a^{19}+\frac{27\!\cdots\!69}{70\!\cdots\!28}a^{18}+\frac{17\!\cdots\!93}{36\!\cdots\!84}a^{17}-\frac{29\!\cdots\!27}{23\!\cdots\!76}a^{16}-\frac{24\!\cdots\!95}{14\!\cdots\!48}a^{15}+\frac{53\!\cdots\!79}{23\!\cdots\!76}a^{14}+\frac{36\!\cdots\!49}{11\!\cdots\!88}a^{13}-\frac{35\!\cdots\!27}{17\!\cdots\!32}a^{12}-\frac{59\!\cdots\!41}{27\!\cdots\!12}a^{11}+\frac{13\!\cdots\!09}{73\!\cdots\!68}a^{10}+\frac{80\!\cdots\!17}{11\!\cdots\!52}a^{9}-\frac{23\!\cdots\!93}{13\!\cdots\!44}a^{8}+\frac{58\!\cdots\!99}{22\!\cdots\!24}a^{7}+\frac{59\!\cdots\!93}{45\!\cdots\!48}a^{6}-\frac{16\!\cdots\!99}{22\!\cdots\!24}a^{5}-\frac{95\!\cdots\!13}{11\!\cdots\!12}a^{4}+\frac{23\!\cdots\!63}{34\!\cdots\!36}a^{3}+\frac{91\!\cdots\!33}{17\!\cdots\!68}a^{2}-\frac{67\!\cdots\!03}{28\!\cdots\!28}a-\frac{77\!\cdots\!69}{42\!\cdots\!92}$, $\frac{21\!\cdots\!27}{41\!\cdots\!84}a^{31}-\frac{85\!\cdots\!95}{77\!\cdots\!72}a^{30}-\frac{71\!\cdots\!29}{91\!\cdots\!32}a^{29}+\frac{39\!\cdots\!73}{30\!\cdots\!44}a^{28}+\frac{82\!\cdots\!49}{12\!\cdots\!52}a^{27}-\frac{13\!\cdots\!61}{20\!\cdots\!92}a^{26}-\frac{20\!\cdots\!07}{51\!\cdots\!48}a^{25}-\frac{91\!\cdots\!39}{62\!\cdots\!76}a^{24}+\frac{60\!\cdots\!15}{31\!\cdots\!88}a^{23}+\frac{14\!\cdots\!73}{62\!\cdots\!76}a^{22}-\frac{22\!\cdots\!33}{31\!\cdots\!88}a^{21}-\frac{33\!\cdots\!45}{15\!\cdots\!44}a^{20}+\frac{92\!\cdots\!87}{41\!\cdots\!84}a^{19}+\frac{76\!\cdots\!27}{62\!\cdots\!76}a^{18}-\frac{96\!\cdots\!25}{15\!\cdots\!44}a^{17}-\frac{26\!\cdots\!81}{62\!\cdots\!76}a^{16}+\frac{22\!\cdots\!01}{12\!\cdots\!52}a^{15}+\frac{45\!\cdots\!99}{62\!\cdots\!76}a^{14}-\frac{28\!\cdots\!09}{10\!\cdots\!96}a^{13}-\frac{31\!\cdots\!77}{51\!\cdots\!48}a^{12}+\frac{15\!\cdots\!95}{38\!\cdots\!36}a^{11}+\frac{60\!\cdots\!77}{19\!\cdots\!68}a^{10}-\frac{41\!\cdots\!95}{97\!\cdots\!84}a^{9}-\frac{26\!\cdots\!57}{71\!\cdots\!44}a^{8}+\frac{10\!\cdots\!35}{30\!\cdots\!12}a^{7}-\frac{45\!\cdots\!43}{40\!\cdots\!16}a^{6}-\frac{14\!\cdots\!19}{62\!\cdots\!92}a^{5}+\frac{42\!\cdots\!47}{30\!\cdots\!12}a^{4}+\frac{37\!\cdots\!83}{30\!\cdots\!12}a^{3}-\frac{35\!\cdots\!83}{50\!\cdots\!52}a^{2}-\frac{82\!\cdots\!69}{25\!\cdots\!76}a+\frac{21\!\cdots\!25}{12\!\cdots\!88}$, $\frac{99\!\cdots\!45}{41\!\cdots\!84}a^{31}+\frac{71\!\cdots\!15}{32\!\cdots\!28}a^{30}-\frac{71\!\cdots\!21}{15\!\cdots\!44}a^{29}-\frac{36\!\cdots\!27}{15\!\cdots\!44}a^{28}+\frac{16\!\cdots\!93}{41\!\cdots\!84}a^{27}+\frac{25\!\cdots\!11}{62\!\cdots\!76}a^{26}-\frac{34\!\cdots\!69}{15\!\cdots\!44}a^{25}-\frac{86\!\cdots\!11}{20\!\cdots\!92}a^{24}+\frac{22\!\cdots\!25}{31\!\cdots\!88}a^{23}+\frac{10\!\cdots\!95}{36\!\cdots\!28}a^{22}-\frac{33\!\cdots\!57}{10\!\cdots\!96}a^{21}-\frac{84\!\cdots\!93}{51\!\cdots\!48}a^{20}-\frac{17\!\cdots\!85}{12\!\cdots\!52}a^{19}+\frac{43\!\cdots\!93}{62\!\cdots\!76}a^{18}+\frac{51\!\cdots\!69}{51\!\cdots\!48}a^{17}-\frac{43\!\cdots\!29}{20\!\cdots\!92}a^{16}-\frac{43\!\cdots\!97}{12\!\cdots\!52}a^{15}+\frac{22\!\cdots\!37}{62\!\cdots\!76}a^{14}+\frac{18\!\cdots\!75}{31\!\cdots\!88}a^{13}-\frac{13\!\cdots\!43}{51\!\cdots\!48}a^{12}-\frac{29\!\cdots\!21}{76\!\cdots\!36}a^{11}+\frac{58\!\cdots\!91}{19\!\cdots\!68}a^{10}+\frac{15\!\cdots\!23}{97\!\cdots\!84}a^{9}-\frac{11\!\cdots\!55}{40\!\cdots\!16}a^{8}+\frac{67\!\cdots\!29}{20\!\cdots\!08}a^{7}+\frac{27\!\cdots\!53}{12\!\cdots\!48}a^{6}-\frac{62\!\cdots\!73}{60\!\cdots\!24}a^{5}-\frac{46\!\cdots\!03}{30\!\cdots\!12}a^{4}+\frac{30\!\cdots\!29}{30\!\cdots\!12}a^{3}+\frac{15\!\cdots\!13}{15\!\cdots\!56}a^{2}-\frac{59\!\cdots\!63}{25\!\cdots\!76}a-\frac{10\!\cdots\!87}{37\!\cdots\!64}$, $\frac{37\!\cdots\!37}{41\!\cdots\!84}a^{31}-\frac{27\!\cdots\!95}{10\!\cdots\!96}a^{30}-\frac{31\!\cdots\!53}{25\!\cdots\!24}a^{29}+\frac{50\!\cdots\!17}{15\!\cdots\!44}a^{28}+\frac{12\!\cdots\!59}{12\!\cdots\!52}a^{27}-\frac{40\!\cdots\!01}{20\!\cdots\!92}a^{26}-\frac{31\!\cdots\!09}{51\!\cdots\!48}a^{25}+\frac{96\!\cdots\!65}{20\!\cdots\!92}a^{24}+\frac{35\!\cdots\!85}{10\!\cdots\!96}a^{23}+\frac{32\!\cdots\!29}{20\!\cdots\!92}a^{22}-\frac{16\!\cdots\!95}{10\!\cdots\!96}a^{21}-\frac{87\!\cdots\!91}{29\!\cdots\!48}a^{20}+\frac{84\!\cdots\!55}{12\!\cdots\!52}a^{19}+\frac{11\!\cdots\!75}{62\!\cdots\!76}a^{18}-\frac{41\!\cdots\!33}{15\!\cdots\!44}a^{17}-\frac{41\!\cdots\!19}{62\!\cdots\!76}a^{16}+\frac{10\!\cdots\!43}{12\!\cdots\!52}a^{15}+\frac{19\!\cdots\!39}{18\!\cdots\!48}a^{14}-\frac{41\!\cdots\!41}{31\!\cdots\!88}a^{13}-\frac{12\!\cdots\!37}{15\!\cdots\!44}a^{12}+\frac{17\!\cdots\!29}{12\!\cdots\!12}a^{11}+\frac{12\!\cdots\!81}{64\!\cdots\!56}a^{10}-\frac{34\!\cdots\!79}{32\!\cdots\!28}a^{9}+\frac{34\!\cdots\!33}{10\!\cdots\!04}a^{8}+\frac{41\!\cdots\!07}{60\!\cdots\!24}a^{7}-\frac{67\!\cdots\!83}{12\!\cdots\!48}a^{6}-\frac{19\!\cdots\!23}{60\!\cdots\!24}a^{5}+\frac{14\!\cdots\!37}{30\!\cdots\!12}a^{4}+\frac{10\!\cdots\!35}{10\!\cdots\!04}a^{3}-\frac{39\!\cdots\!53}{15\!\cdots\!56}a^{2}-\frac{10\!\cdots\!61}{75\!\cdots\!28}a+\frac{14\!\cdots\!77}{22\!\cdots\!92}$, $\frac{95\!\cdots\!47}{19\!\cdots\!68}a^{31}+\frac{35\!\cdots\!89}{31\!\cdots\!88}a^{30}-\frac{19\!\cdots\!59}{15\!\cdots\!44}a^{29}-\frac{26\!\cdots\!73}{12\!\cdots\!12}a^{28}+\frac{15\!\cdots\!73}{12\!\cdots\!12}a^{27}+\frac{66\!\cdots\!53}{31\!\cdots\!88}a^{26}-\frac{52\!\cdots\!49}{77\!\cdots\!72}a^{25}-\frac{12\!\cdots\!33}{77\!\cdots\!72}a^{24}+\frac{10\!\cdots\!47}{51\!\cdots\!48}a^{23}+\frac{67\!\cdots\!33}{64\!\cdots\!56}a^{22}+\frac{10\!\cdots\!35}{51\!\cdots\!48}a^{21}-\frac{64\!\cdots\!05}{12\!\cdots\!52}a^{20}-\frac{13\!\cdots\!81}{20\!\cdots\!08}a^{19}+\frac{68\!\cdots\!69}{31\!\cdots\!88}a^{18}+\frac{17\!\cdots\!71}{38\!\cdots\!36}a^{17}-\frac{55\!\cdots\!95}{77\!\cdots\!72}a^{16}-\frac{24\!\cdots\!01}{15\!\cdots\!44}a^{15}+\frac{16\!\cdots\!79}{10\!\cdots\!96}a^{14}+\frac{10\!\cdots\!07}{38\!\cdots\!36}a^{13}-\frac{16\!\cdots\!13}{97\!\cdots\!84}a^{12}-\frac{38\!\cdots\!69}{19\!\cdots\!68}a^{11}+\frac{13\!\cdots\!19}{80\!\cdots\!32}a^{10}+\frac{51\!\cdots\!19}{80\!\cdots\!32}a^{9}-\frac{15\!\cdots\!89}{10\!\cdots\!04}a^{8}+\frac{82\!\cdots\!69}{31\!\cdots\!22}a^{7}+\frac{35\!\cdots\!23}{30\!\cdots\!12}a^{6}-\frac{12\!\cdots\!85}{17\!\cdots\!36}a^{5}-\frac{89\!\cdots\!05}{12\!\cdots\!88}a^{4}+\frac{23\!\cdots\!03}{37\!\cdots\!64}a^{3}+\frac{97\!\cdots\!15}{25\!\cdots\!76}a^{2}-\frac{21\!\cdots\!53}{94\!\cdots\!66}a-\frac{63\!\cdots\!17}{47\!\cdots\!33}$, $\frac{79\!\cdots\!75}{41\!\cdots\!84}a^{31}-\frac{16\!\cdots\!07}{31\!\cdots\!88}a^{30}-\frac{41\!\cdots\!29}{15\!\cdots\!44}a^{29}+\frac{32\!\cdots\!87}{51\!\cdots\!48}a^{28}+\frac{27\!\cdots\!57}{12\!\cdots\!52}a^{27}-\frac{75\!\cdots\!15}{20\!\cdots\!92}a^{26}-\frac{52\!\cdots\!61}{38\!\cdots\!36}a^{25}+\frac{40\!\cdots\!33}{62\!\cdots\!76}a^{24}+\frac{22\!\cdots\!21}{31\!\cdots\!88}a^{23}+\frac{30\!\cdots\!89}{62\!\cdots\!76}a^{22}-\frac{98\!\cdots\!23}{31\!\cdots\!88}a^{21}-\frac{10\!\cdots\!75}{15\!\cdots\!44}a^{20}+\frac{15\!\cdots\!57}{12\!\cdots\!52}a^{19}+\frac{26\!\cdots\!33}{62\!\cdots\!76}a^{18}-\frac{20\!\cdots\!75}{45\!\cdots\!16}a^{17}-\frac{92\!\cdots\!33}{62\!\cdots\!76}a^{16}+\frac{32\!\cdots\!17}{23\!\cdots\!84}a^{15}+\frac{15\!\cdots\!25}{62\!\cdots\!76}a^{14}-\frac{60\!\cdots\!09}{31\!\cdots\!88}a^{13}-\frac{33\!\cdots\!27}{15\!\cdots\!44}a^{12}+\frac{80\!\cdots\!37}{38\!\cdots\!36}a^{11}+\frac{15\!\cdots\!59}{19\!\cdots\!68}a^{10}-\frac{18\!\cdots\!43}{97\!\cdots\!84}a^{9}+\frac{31\!\cdots\!17}{12\!\cdots\!48}a^{8}+\frac{41\!\cdots\!69}{30\!\cdots\!12}a^{7}-\frac{91\!\cdots\!01}{12\!\cdots\!48}a^{6}-\frac{48\!\cdots\!61}{60\!\cdots\!24}a^{5}+\frac{76\!\cdots\!63}{10\!\cdots\!04}a^{4}+\frac{11\!\cdots\!99}{30\!\cdots\!12}a^{3}-\frac{21\!\cdots\!53}{50\!\cdots\!52}a^{2}-\frac{88\!\cdots\!57}{75\!\cdots\!28}a+\frac{32\!\cdots\!67}{37\!\cdots\!64}$, $\frac{71\!\cdots\!65}{62\!\cdots\!76}a^{31}-\frac{24\!\cdots\!03}{15\!\cdots\!44}a^{30}-\frac{94\!\cdots\!99}{50\!\cdots\!52}a^{29}+\frac{11\!\cdots\!19}{77\!\cdots\!72}a^{28}+\frac{10\!\cdots\!45}{62\!\cdots\!76}a^{27}-\frac{61\!\cdots\!05}{31\!\cdots\!88}a^{26}-\frac{71\!\cdots\!75}{77\!\cdots\!72}a^{25}-\frac{22\!\cdots\!79}{31\!\cdots\!88}a^{24}+\frac{61\!\cdots\!45}{15\!\cdots\!44}a^{23}+\frac{85\!\cdots\!15}{10\!\cdots\!96}a^{22}-\frac{16\!\cdots\!31}{15\!\cdots\!44}a^{21}-\frac{15\!\cdots\!67}{25\!\cdots\!24}a^{20}+\frac{44\!\cdots\!49}{62\!\cdots\!76}a^{19}+\frac{90\!\cdots\!21}{31\!\cdots\!88}a^{18}+\frac{66\!\cdots\!63}{77\!\cdots\!72}a^{17}-\frac{97\!\cdots\!03}{10\!\cdots\!96}a^{16}-\frac{74\!\cdots\!85}{20\!\cdots\!92}a^{15}+\frac{49\!\cdots\!21}{31\!\cdots\!88}a^{14}+\frac{11\!\cdots\!89}{15\!\cdots\!44}a^{13}-\frac{32\!\cdots\!89}{25\!\cdots\!24}a^{12}-\frac{49\!\cdots\!87}{19\!\cdots\!68}a^{11}+\frac{88\!\cdots\!25}{97\!\cdots\!84}a^{10}-\frac{34\!\cdots\!37}{16\!\cdots\!64}a^{9}-\frac{68\!\cdots\!61}{12\!\cdots\!48}a^{8}+\frac{14\!\cdots\!57}{31\!\cdots\!22}a^{7}+\frac{59\!\cdots\!79}{20\!\cdots\!08}a^{6}-\frac{48\!\cdots\!89}{10\!\cdots\!04}a^{5}-\frac{19\!\cdots\!53}{15\!\cdots\!56}a^{4}+\frac{52\!\cdots\!11}{15\!\cdots\!56}a^{3}+\frac{90\!\cdots\!53}{75\!\cdots\!28}a^{2}-\frac{34\!\cdots\!55}{37\!\cdots\!64}a-\frac{76\!\cdots\!33}{18\!\cdots\!32}$, $\frac{43\!\cdots\!67}{41\!\cdots\!84}a^{31}-\frac{12\!\cdots\!83}{60\!\cdots\!88}a^{30}-\frac{41\!\cdots\!43}{25\!\cdots\!24}a^{29}+\frac{35\!\cdots\!73}{15\!\cdots\!44}a^{28}+\frac{16\!\cdots\!69}{12\!\cdots\!52}a^{27}-\frac{65\!\cdots\!09}{62\!\cdots\!76}a^{26}-\frac{11\!\cdots\!39}{15\!\cdots\!44}a^{25}-\frac{31\!\cdots\!65}{20\!\cdots\!92}a^{24}+\frac{11\!\cdots\!05}{31\!\cdots\!88}a^{23}+\frac{30\!\cdots\!13}{62\!\cdots\!76}a^{22}-\frac{39\!\cdots\!55}{31\!\cdots\!88}a^{21}-\frac{68\!\cdots\!77}{15\!\cdots\!44}a^{20}+\frac{43\!\cdots\!97}{12\!\cdots\!52}a^{19}+\frac{49\!\cdots\!15}{20\!\cdots\!92}a^{18}-\frac{13\!\cdots\!33}{15\!\cdots\!44}a^{17}-\frac{49\!\cdots\!09}{62\!\cdots\!76}a^{16}+\frac{34\!\cdots\!69}{12\!\cdots\!52}a^{15}+\frac{76\!\cdots\!61}{62\!\cdots\!76}a^{14}-\frac{15\!\cdots\!23}{31\!\cdots\!88}a^{13}-\frac{96\!\cdots\!53}{15\!\cdots\!44}a^{12}+\frac{96\!\cdots\!71}{12\!\cdots\!12}a^{11}+\frac{28\!\cdots\!01}{19\!\cdots\!68}a^{10}-\frac{53\!\cdots\!21}{97\!\cdots\!84}a^{9}+\frac{17\!\cdots\!61}{12\!\cdots\!48}a^{8}+\frac{27\!\cdots\!67}{62\!\cdots\!92}a^{7}-\frac{31\!\cdots\!65}{12\!\cdots\!48}a^{6}-\frac{44\!\cdots\!87}{20\!\cdots\!08}a^{5}+\frac{69\!\cdots\!27}{30\!\cdots\!12}a^{4}+\frac{18\!\cdots\!91}{30\!\cdots\!12}a^{3}-\frac{24\!\cdots\!99}{89\!\cdots\!68}a^{2}+\frac{14\!\cdots\!69}{75\!\cdots\!28}a-\frac{53\!\cdots\!61}{74\!\cdots\!64}$ Copy content Toggle raw display (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:  \( 564327334704.2394 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 564327334704.2394 \cdot 20}{12\cdot\sqrt{133516116647406931991162570051420160000000000000000}}\cr\approx \mathstrut & 0.480275999186225 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 16*x^30 + 24*x^29 + 141*x^28 - 120*x^27 - 852*x^26 - 70*x^25 + 4216*x^24 + 4978*x^23 - 15640*x^22 - 47344*x^21 + 45797*x^20 + 270436*x^19 - 110380*x^18 - 986938*x^17 + 284105*x^16 + 1973876*x^15 - 441520*x^14 - 2163488*x^13 + 732752*x^12 + 1515008*x^11 - 1000960*x^10 - 637184*x^9 + 1079296*x^8 + 35840*x^7 - 872448*x^6 + 245760*x^5 + 577536*x^4 - 196608*x^3 - 262144*x^2 + 65536*x + 65536)
 
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^32 - 2*x^31 - 16*x^30 + 24*x^29 + 141*x^28 - 120*x^27 - 852*x^26 - 70*x^25 + 4216*x^24 + 4978*x^23 - 15640*x^22 - 47344*x^21 + 45797*x^20 + 270436*x^19 - 110380*x^18 - 986938*x^17 + 284105*x^16 + 1973876*x^15 - 441520*x^14 - 2163488*x^13 + 732752*x^12 + 1515008*x^11 - 1000960*x^10 - 637184*x^9 + 1079296*x^8 + 35840*x^7 - 872448*x^6 + 245760*x^5 + 577536*x^4 - 196608*x^3 - 262144*x^2 + 65536*x + 65536, 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^32 - 2*x^31 - 16*x^30 + 24*x^29 + 141*x^28 - 120*x^27 - 852*x^26 - 70*x^25 + 4216*x^24 + 4978*x^23 - 15640*x^22 - 47344*x^21 + 45797*x^20 + 270436*x^19 - 110380*x^18 - 986938*x^17 + 284105*x^16 + 1973876*x^15 - 441520*x^14 - 2163488*x^13 + 732752*x^12 + 1515008*x^11 - 1000960*x^10 - 637184*x^9 + 1079296*x^8 + 35840*x^7 - 872448*x^6 + 245760*x^5 + 577536*x^4 - 196608*x^3 - 262144*x^2 + 65536*x + 65536);
 
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^32 - 2*x^31 - 16*x^30 + 24*x^29 + 141*x^28 - 120*x^27 - 852*x^26 - 70*x^25 + 4216*x^24 + 4978*x^23 - 15640*x^22 - 47344*x^21 + 45797*x^20 + 270436*x^19 - 110380*x^18 - 986938*x^17 + 284105*x^16 + 1973876*x^15 - 441520*x^14 - 2163488*x^13 + 732752*x^12 + 1515008*x^11 - 1000960*x^10 - 637184*x^9 + 1079296*x^8 + 35840*x^7 - 872448*x^6 + 245760*x^5 + 577536*x^4 - 196608*x^3 - 262144*x^2 + 65536*x + 65536);
 
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

$D_4^2:C_2^3$ (as 32T12882):

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 512
The 80 conjugacy class representatives for $D_4^2:C_2^3$
Character table for $D_4^2:C_2^3$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), 4.4.7488.1, 4.0.7488.1, 4.4.187200.1, 4.0.187200.1, \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{-5})\), 8.0.3399252480000.1, 8.8.3399252480000.1, 8.0.5438803968.1, 8.8.5438803968.1, 8.0.12960000.1, 8.0.56070144.2, 8.0.35043840000.18, 8.8.35043840000.1, 8.0.35043840000.41, 8.0.35043840000.39, 8.0.35043840000.36, 16.0.1228070721945600000000.2, 16.0.11554917422786150400000000.4, 16.0.29580588602332545024.3, 16.0.11554917422786150400000000.2, 16.16.11554917422786150400000000.1, 16.0.11554917422786150400000000.3, 16.0.11554917422786150400000000.1

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 32 siblings: data not computed
Minimal sibling: not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.8.0.1}{8} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{8}$ R ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.8.0.1}{8} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{16}$ ${\href{/padicField/41.8.0.1}{8} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{16}$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.8.12.14$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$$4$$2$$12$$D_4$$[2, 2]^{2}$
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.16.8.1$x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
5.16.8.1$x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
\(13\) Copy content Toggle raw display 13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(97\) Copy content Toggle raw display 97.2.0.1$x^{2} + 96 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} + 96 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} + 96 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} + 96 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} + 6 x^{2} + 80 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 10474 x^{3} + 27919909 x^{2} + 2585716380 x + 183392493$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} + 6 x^{2} + 80 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} + 6 x^{2} + 80 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 10474 x^{3} + 27919909 x^{2} + 2585716380 x + 183392493$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} + 6 x^{2} + 80 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$