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Properties

Label 32.0.524...056.1
Degree $32$
Signature $[0, 16]$
Discriminant $5.242\times 10^{51}$
Root discriminant \(41.33\)
Ramified primes $2,3,17$
Class number $40$ (GRH)
Class group [40] (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 15*x^30 + 134*x^28 - 793*x^26 + 3496*x^24 - 11638*x^22 + 30241*x^20 - 60672*x^18 + 94862*x^16 - 111861*x^14 + 99380*x^12 - 61666*x^10 + 26973*x^8 - 6636*x^6 + 1086*x^4 - 36*x^2 + 1)
 
gp: K = bnfinit(y^32 - 15*y^30 + 134*y^28 - 793*y^26 + 3496*y^24 - 11638*y^22 + 30241*y^20 - 60672*y^18 + 94862*y^16 - 111861*y^14 + 99380*y^12 - 61666*y^10 + 26973*y^8 - 6636*y^6 + 1086*y^4 - 36*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 15*x^30 + 134*x^28 - 793*x^26 + 3496*x^24 - 11638*x^22 + 30241*x^20 - 60672*x^18 + 94862*x^16 - 111861*x^14 + 99380*x^12 - 61666*x^10 + 26973*x^8 - 6636*x^6 + 1086*x^4 - 36*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 15*x^30 + 134*x^28 - 793*x^26 + 3496*x^24 - 11638*x^22 + 30241*x^20 - 60672*x^18 + 94862*x^16 - 111861*x^14 + 99380*x^12 - 61666*x^10 + 26973*x^8 - 6636*x^6 + 1086*x^4 - 36*x^2 + 1)
 

\( x^{32} - 15 x^{30} + 134 x^{28} - 793 x^{26} + 3496 x^{24} - 11638 x^{22} + 30241 x^{20} - 60672 x^{18} + \cdots + 1 \) 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:   \(5241670721450327242974152113348873388724842246701056\) \(\medspace = 2^{32}\cdot 3^{16}\cdot 17^{28}\) 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:  \(41.33\)
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:  $2\cdot 3^{1/2}17^{7/8}\approx 41.32695708412257$
Ramified primes:   \(2\), \(3\), \(17\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(204=2^{2}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{204}(1,·)$, $\chi_{204}(137,·)$, $\chi_{204}(13,·)$, $\chi_{204}(145,·)$, $\chi_{204}(19,·)$, $\chi_{204}(149,·)$, $\chi_{204}(151,·)$, $\chi_{204}(25,·)$, $\chi_{204}(155,·)$, $\chi_{204}(157,·)$, $\chi_{204}(161,·)$, $\chi_{204}(35,·)$, $\chi_{204}(169,·)$, $\chi_{204}(43,·)$, $\chi_{204}(47,·)$, $\chi_{204}(49,·)$, $\chi_{204}(179,·)$, $\chi_{204}(53,·)$, $\chi_{204}(55,·)$, $\chi_{204}(185,·)$, $\chi_{204}(59,·)$, $\chi_{204}(191,·)$, $\chi_{204}(67,·)$, $\chi_{204}(203,·)$, $\chi_{204}(77,·)$, $\chi_{204}(83,·)$, $\chi_{204}(89,·)$, $\chi_{204}(101,·)$, $\chi_{204}(103,·)$, $\chi_{204}(115,·)$, $\chi_{204}(121,·)$, $\chi_{204}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{4}a^{24}-\frac{1}{4}a^{18}+\frac{1}{4}a^{12}-\frac{1}{4}a^{6}+\frac{1}{4}$, $\frac{1}{4}a^{25}-\frac{1}{4}a^{19}+\frac{1}{4}a^{13}-\frac{1}{4}a^{7}+\frac{1}{4}a$, $\frac{1}{4}a^{26}-\frac{1}{4}a^{20}+\frac{1}{4}a^{14}-\frac{1}{4}a^{8}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{27}-\frac{1}{4}a^{21}+\frac{1}{4}a^{15}-\frac{1}{4}a^{9}+\frac{1}{4}a^{3}$, $\frac{1}{358852}a^{28}+\frac{3023}{27604}a^{26}+\frac{7889}{89713}a^{24}+\frac{3975}{358852}a^{22}+\frac{84017}{358852}a^{20}+\frac{18516}{89713}a^{18}-\frac{132151}{358852}a^{16}+\frac{62195}{358852}a^{14}+\frac{33355}{89713}a^{12}-\frac{114209}{358852}a^{10}-\frac{47527}{358852}a^{8}+\frac{30317}{89713}a^{6}-\frac{138651}{358852}a^{4}-\frac{130001}{358852}a^{2}-\frac{34733}{89713}$, $\frac{1}{358852}a^{29}+\frac{3023}{27604}a^{27}+\frac{7889}{89713}a^{25}+\frac{3975}{358852}a^{23}+\frac{84017}{358852}a^{21}+\frac{18516}{89713}a^{19}-\frac{132151}{358852}a^{17}+\frac{62195}{358852}a^{15}+\frac{33355}{89713}a^{13}-\frac{114209}{358852}a^{11}-\frac{47527}{358852}a^{9}+\frac{30317}{89713}a^{7}-\frac{138651}{358852}a^{5}-\frac{130001}{358852}a^{3}-\frac{34733}{89713}a$, $\frac{1}{17\!\cdots\!16}a^{30}+\frac{7900196601225}{87\!\cdots\!58}a^{28}-\frac{99\!\cdots\!13}{87\!\cdots\!58}a^{26}-\frac{13\!\cdots\!73}{17\!\cdots\!16}a^{24}+\frac{38\!\cdots\!23}{87\!\cdots\!58}a^{22}+\frac{36\!\cdots\!33}{87\!\cdots\!58}a^{20}+\frac{49\!\cdots\!97}{17\!\cdots\!16}a^{18}-\frac{14\!\cdots\!37}{87\!\cdots\!58}a^{16}-\frac{38\!\cdots\!23}{87\!\cdots\!58}a^{14}+\frac{48\!\cdots\!43}{17\!\cdots\!16}a^{12}+\frac{18\!\cdots\!89}{87\!\cdots\!58}a^{10}+\frac{17\!\cdots\!63}{87\!\cdots\!58}a^{8}+\frac{34\!\cdots\!45}{17\!\cdots\!16}a^{6}-\frac{32\!\cdots\!19}{87\!\cdots\!58}a^{4}+\frac{17\!\cdots\!23}{67\!\cdots\!66}a^{2}+\frac{88\!\cdots\!96}{43\!\cdots\!29}$, $\frac{1}{17\!\cdots\!16}a^{31}+\frac{7900196601225}{87\!\cdots\!58}a^{29}-\frac{99\!\cdots\!13}{87\!\cdots\!58}a^{27}-\frac{13\!\cdots\!73}{17\!\cdots\!16}a^{25}+\frac{38\!\cdots\!23}{87\!\cdots\!58}a^{23}+\frac{36\!\cdots\!33}{87\!\cdots\!58}a^{21}+\frac{49\!\cdots\!97}{17\!\cdots\!16}a^{19}-\frac{14\!\cdots\!37}{87\!\cdots\!58}a^{17}-\frac{38\!\cdots\!23}{87\!\cdots\!58}a^{15}+\frac{48\!\cdots\!43}{17\!\cdots\!16}a^{13}+\frac{18\!\cdots\!89}{87\!\cdots\!58}a^{11}+\frac{17\!\cdots\!63}{87\!\cdots\!58}a^{9}+\frac{34\!\cdots\!45}{17\!\cdots\!16}a^{7}-\frac{32\!\cdots\!19}{87\!\cdots\!58}a^{5}+\frac{17\!\cdots\!23}{67\!\cdots\!66}a^{3}+\frac{88\!\cdots\!96}{43\!\cdots\!29}a$ 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

$C_{40}$, which has order $40$ (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:   \( -\frac{2996814042794625635}{17570529103923849716} a^{31} + \frac{45062338851799879863}{17570529103923849716} a^{29} - \frac{403224884666926751635}{17570529103923849716} a^{27} + \frac{1195607463592421012995}{8785264551961924858} a^{25} - \frac{10563959124639323765711}{17570529103923849716} a^{23} + \frac{35259854287434061112435}{17570529103923849716} a^{21} - \frac{3534462243250914068935}{675789580920148066} a^{19} + \frac{185099129365526453587327}{17570529103923849716} a^{17} - \frac{290787438949352501206315}{17570529103923849716} a^{15} + \frac{172610670898559810840895}{8785264551961924858} a^{13} - \frac{309268137292545588766747}{17570529103923849716} a^{11} + \frac{194467955203039354540475}{17570529103923849716} a^{9} - \frac{43135277737412749781565}{8785264551961924858} a^{7} + \frac{1681472366991125311839}{1351579161840296132} a^{5} - \frac{3533117998425916751535}{17570529103923849716} a^{3} + \frac{117170196485415320475}{17570529103923849716} a \)  (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:   $\frac{10\!\cdots\!97}{17\!\cdots\!16}a^{31}-\frac{15\!\cdots\!87}{17\!\cdots\!16}a^{29}+\frac{14\!\cdots\!71}{17\!\cdots\!16}a^{27}-\frac{41\!\cdots\!45}{87\!\cdots\!58}a^{25}+\frac{36\!\cdots\!23}{17\!\cdots\!16}a^{23}-\frac{11\!\cdots\!55}{17\!\cdots\!16}a^{21}+\frac{11\!\cdots\!33}{67\!\cdots\!66}a^{19}-\frac{59\!\cdots\!87}{17\!\cdots\!16}a^{17}+\frac{91\!\cdots\!83}{17\!\cdots\!16}a^{15}-\frac{51\!\cdots\!73}{87\!\cdots\!58}a^{13}+\frac{87\!\cdots\!19}{17\!\cdots\!16}a^{11}-\frac{49\!\cdots\!35}{17\!\cdots\!16}a^{9}+\frac{90\!\cdots\!67}{87\!\cdots\!58}a^{7}-\frac{19\!\cdots\!55}{13\!\cdots\!32}a^{5}+\frac{83\!\cdots\!19}{17\!\cdots\!16}a^{3}+\frac{93\!\cdots\!03}{17\!\cdots\!16}a-1$, $\frac{48\!\cdots\!33}{18\!\cdots\!14}a^{30}-\frac{35\!\cdots\!55}{93\!\cdots\!07}a^{28}+\frac{12\!\cdots\!77}{37\!\cdots\!28}a^{26}-\frac{74\!\cdots\!81}{37\!\cdots\!28}a^{24}+\frac{81\!\cdots\!07}{93\!\cdots\!07}a^{22}-\frac{10\!\cdots\!05}{37\!\cdots\!28}a^{20}+\frac{27\!\cdots\!53}{37\!\cdots\!28}a^{18}-\frac{13\!\cdots\!53}{93\!\cdots\!07}a^{16}+\frac{84\!\cdots\!73}{37\!\cdots\!28}a^{14}-\frac{96\!\cdots\!21}{37\!\cdots\!28}a^{12}+\frac{15\!\cdots\!07}{71\!\cdots\!39}a^{10}-\frac{48\!\cdots\!05}{37\!\cdots\!28}a^{8}+\frac{19\!\cdots\!89}{37\!\cdots\!28}a^{6}-\frac{10\!\cdots\!90}{93\!\cdots\!07}a^{4}+\frac{73\!\cdots\!09}{37\!\cdots\!28}a^{2}-\frac{24\!\cdots\!31}{37\!\cdots\!28}$, $\frac{99\!\cdots\!37}{17\!\cdots\!16}a^{30}-\frac{14\!\cdots\!95}{17\!\cdots\!16}a^{28}+\frac{33\!\cdots\!76}{43\!\cdots\!29}a^{26}-\frac{39\!\cdots\!49}{87\!\cdots\!58}a^{24}+\frac{26\!\cdots\!79}{13\!\cdots\!32}a^{22}-\frac{22\!\cdots\!22}{33\!\cdots\!33}a^{20}+\frac{15\!\cdots\!45}{87\!\cdots\!58}a^{18}-\frac{60\!\cdots\!11}{17\!\cdots\!16}a^{16}+\frac{23\!\cdots\!34}{42\!\cdots\!43}a^{14}-\frac{56\!\cdots\!91}{87\!\cdots\!58}a^{12}+\frac{99\!\cdots\!71}{17\!\cdots\!16}a^{10}-\frac{15\!\cdots\!26}{43\!\cdots\!29}a^{8}+\frac{10\!\cdots\!33}{67\!\cdots\!66}a^{6}-\frac{61\!\cdots\!43}{17\!\cdots\!16}a^{4}+\frac{20\!\cdots\!77}{43\!\cdots\!29}a^{2}-\frac{61\!\cdots\!85}{17\!\cdots\!16}$, $\frac{19\!\cdots\!85}{67\!\cdots\!66}a^{30}-\frac{38\!\cdots\!97}{87\!\cdots\!58}a^{28}+\frac{52\!\cdots\!21}{13\!\cdots\!32}a^{26}-\frac{99\!\cdots\!71}{43\!\cdots\!29}a^{24}+\frac{86\!\cdots\!77}{87\!\cdots\!58}a^{22}-\frac{56\!\cdots\!37}{17\!\cdots\!16}a^{20}+\frac{36\!\cdots\!50}{43\!\cdots\!29}a^{18}-\frac{14\!\cdots\!79}{87\!\cdots\!58}a^{16}+\frac{42\!\cdots\!33}{17\!\cdots\!16}a^{14}-\frac{11\!\cdots\!03}{43\!\cdots\!29}a^{12}+\frac{20\!\cdots\!61}{87\!\cdots\!58}a^{10}-\frac{22\!\cdots\!61}{17\!\cdots\!16}a^{8}+\frac{21\!\cdots\!45}{43\!\cdots\!29}a^{6}-\frac{73\!\cdots\!01}{87\!\cdots\!58}a^{4}+\frac{28\!\cdots\!25}{17\!\cdots\!16}a^{2}-\frac{46\!\cdots\!23}{87\!\cdots\!58}$, $\frac{74\!\cdots\!19}{17\!\cdots\!16}a^{30}-\frac{56\!\cdots\!45}{87\!\cdots\!58}a^{28}+\frac{50\!\cdots\!35}{87\!\cdots\!58}a^{26}-\frac{29\!\cdots\!99}{87\!\cdots\!58}a^{24}+\frac{13\!\cdots\!33}{87\!\cdots\!58}a^{22}-\frac{44\!\cdots\!13}{87\!\cdots\!58}a^{20}+\frac{11\!\cdots\!39}{87\!\cdots\!58}a^{18}-\frac{23\!\cdots\!85}{87\!\cdots\!58}a^{16}+\frac{36\!\cdots\!25}{87\!\cdots\!58}a^{14}-\frac{43\!\cdots\!49}{87\!\cdots\!58}a^{12}+\frac{38\!\cdots\!71}{87\!\cdots\!58}a^{10}-\frac{18\!\cdots\!43}{67\!\cdots\!66}a^{8}+\frac{15\!\cdots\!83}{13\!\cdots\!74}a^{6}-\frac{24\!\cdots\!25}{87\!\cdots\!58}a^{4}+\frac{31\!\cdots\!21}{87\!\cdots\!58}a^{2}-\frac{48\!\cdots\!27}{17\!\cdots\!16}$, $\frac{14\!\cdots\!97}{17\!\cdots\!16}a^{30}-\frac{22\!\cdots\!65}{17\!\cdots\!16}a^{28}+\frac{20\!\cdots\!53}{17\!\cdots\!16}a^{26}-\frac{12\!\cdots\!17}{17\!\cdots\!16}a^{24}+\frac{57\!\cdots\!13}{17\!\cdots\!16}a^{22}-\frac{19\!\cdots\!41}{17\!\cdots\!16}a^{20}+\frac{53\!\cdots\!05}{17\!\cdots\!16}a^{18}-\frac{11\!\cdots\!13}{17\!\cdots\!16}a^{16}+\frac{18\!\cdots\!41}{17\!\cdots\!16}a^{14}-\frac{23\!\cdots\!09}{17\!\cdots\!16}a^{12}+\frac{22\!\cdots\!65}{17\!\cdots\!16}a^{10}-\frac{16\!\cdots\!29}{17\!\cdots\!16}a^{8}+\frac{79\!\cdots\!65}{17\!\cdots\!16}a^{6}-\frac{23\!\cdots\!41}{17\!\cdots\!16}a^{4}+\frac{36\!\cdots\!89}{17\!\cdots\!16}a^{2}-\frac{30\!\cdots\!82}{43\!\cdots\!29}$, $\frac{17\!\cdots\!73}{87\!\cdots\!58}a^{30}-\frac{13\!\cdots\!53}{43\!\cdots\!29}a^{28}+\frac{23\!\cdots\!75}{87\!\cdots\!58}a^{26}-\frac{27\!\cdots\!05}{17\!\cdots\!16}a^{24}+\frac{29\!\cdots\!97}{43\!\cdots\!29}a^{22}-\frac{19\!\cdots\!91}{87\!\cdots\!58}a^{20}+\frac{98\!\cdots\!33}{17\!\cdots\!16}a^{18}-\frac{48\!\cdots\!65}{43\!\cdots\!29}a^{16}+\frac{14\!\cdots\!03}{87\!\cdots\!58}a^{14}-\frac{32\!\cdots\!73}{17\!\cdots\!16}a^{12}+\frac{69\!\cdots\!93}{43\!\cdots\!29}a^{10}-\frac{77\!\cdots\!83}{87\!\cdots\!58}a^{8}+\frac{59\!\cdots\!09}{17\!\cdots\!16}a^{6}-\frac{19\!\cdots\!47}{43\!\cdots\!29}a^{4}+\frac{13\!\cdots\!39}{87\!\cdots\!58}a^{2}+\frac{34\!\cdots\!25}{17\!\cdots\!16}$, $\frac{78\!\cdots\!65}{17\!\cdots\!16}a^{30}-\frac{29\!\cdots\!95}{43\!\cdots\!29}a^{28}+\frac{53\!\cdots\!87}{87\!\cdots\!58}a^{26}-\frac{63\!\cdots\!33}{17\!\cdots\!16}a^{24}+\frac{69\!\cdots\!83}{43\!\cdots\!29}a^{22}-\frac{46\!\cdots\!43}{87\!\cdots\!58}a^{20}+\frac{24\!\cdots\!13}{17\!\cdots\!16}a^{18}-\frac{12\!\cdots\!21}{43\!\cdots\!29}a^{16}+\frac{38\!\cdots\!03}{87\!\cdots\!58}a^{14}-\frac{92\!\cdots\!53}{17\!\cdots\!16}a^{12}+\frac{20\!\cdots\!61}{43\!\cdots\!29}a^{10}-\frac{26\!\cdots\!45}{87\!\cdots\!58}a^{8}+\frac{23\!\cdots\!29}{17\!\cdots\!16}a^{6}-\frac{15\!\cdots\!19}{43\!\cdots\!29}a^{4}+\frac{48\!\cdots\!91}{87\!\cdots\!58}a^{2}-\frac{80\!\cdots\!01}{43\!\cdots\!29}$, $\frac{24\!\cdots\!63}{87\!\cdots\!58}a^{31}+\frac{66\!\cdots\!07}{17\!\cdots\!16}a^{30}-\frac{18\!\cdots\!76}{43\!\cdots\!29}a^{29}-\frac{99\!\cdots\!59}{17\!\cdots\!16}a^{28}+\frac{33\!\cdots\!41}{87\!\cdots\!58}a^{27}+\frac{44\!\cdots\!63}{87\!\cdots\!58}a^{26}-\frac{97\!\cdots\!73}{43\!\cdots\!29}a^{25}-\frac{52\!\cdots\!01}{17\!\cdots\!16}a^{24}+\frac{43\!\cdots\!45}{43\!\cdots\!29}a^{23}+\frac{23\!\cdots\!67}{17\!\cdots\!16}a^{22}-\frac{28\!\cdots\!65}{87\!\cdots\!58}a^{21}-\frac{38\!\cdots\!39}{87\!\cdots\!58}a^{20}+\frac{37\!\cdots\!89}{43\!\cdots\!29}a^{19}+\frac{19\!\cdots\!05}{17\!\cdots\!16}a^{18}-\frac{74\!\cdots\!86}{43\!\cdots\!29}a^{17}-\frac{30\!\cdots\!67}{13\!\cdots\!32}a^{16}+\frac{17\!\cdots\!65}{67\!\cdots\!66}a^{15}+\frac{45\!\cdots\!61}{13\!\cdots\!74}a^{14}-\frac{10\!\cdots\!49}{33\!\cdots\!33}a^{13}-\frac{71\!\cdots\!21}{17\!\cdots\!16}a^{12}+\frac{12\!\cdots\!86}{43\!\cdots\!29}a^{11}+\frac{62\!\cdots\!87}{17\!\cdots\!16}a^{10}-\frac{14\!\cdots\!61}{87\!\cdots\!58}a^{9}-\frac{19\!\cdots\!45}{87\!\cdots\!58}a^{8}+\frac{32\!\cdots\!87}{43\!\cdots\!29}a^{7}+\frac{16\!\cdots\!45}{17\!\cdots\!16}a^{6}-\frac{79\!\cdots\!98}{43\!\cdots\!29}a^{5}-\frac{38\!\cdots\!43}{17\!\cdots\!16}a^{4}+\frac{25\!\cdots\!13}{87\!\cdots\!58}a^{3}+\frac{32\!\cdots\!07}{87\!\cdots\!58}a^{2}-\frac{12\!\cdots\!53}{13\!\cdots\!74}a-\frac{19\!\cdots\!45}{87\!\cdots\!58}$, $\frac{48\!\cdots\!71}{93\!\cdots\!07}a^{31}+\frac{35\!\cdots\!01}{37\!\cdots\!28}a^{30}-\frac{14\!\cdots\!13}{18\!\cdots\!14}a^{29}-\frac{59\!\cdots\!37}{37\!\cdots\!28}a^{28}+\frac{25\!\cdots\!69}{37\!\cdots\!28}a^{27}+\frac{56\!\cdots\!97}{37\!\cdots\!28}a^{26}-\frac{15\!\cdots\!49}{37\!\cdots\!28}a^{25}-\frac{35\!\cdots\!71}{37\!\cdots\!28}a^{24}+\frac{33\!\cdots\!55}{18\!\cdots\!14}a^{23}+\frac{16\!\cdots\!65}{37\!\cdots\!28}a^{22}-\frac{22\!\cdots\!65}{37\!\cdots\!28}a^{21}-\frac{60\!\cdots\!97}{37\!\cdots\!28}a^{20}+\frac{57\!\cdots\!49}{37\!\cdots\!28}a^{19}+\frac{13\!\cdots\!23}{28\!\cdots\!56}a^{18}-\frac{57\!\cdots\!17}{18\!\cdots\!14}a^{17}-\frac{37\!\cdots\!05}{37\!\cdots\!28}a^{16}+\frac{17\!\cdots\!53}{37\!\cdots\!28}a^{15}+\frac{65\!\cdots\!45}{37\!\cdots\!28}a^{14}-\frac{20\!\cdots\!69}{37\!\cdots\!28}a^{13}-\frac{85\!\cdots\!53}{36\!\cdots\!76}a^{12}+\frac{91\!\cdots\!99}{18\!\cdots\!14}a^{11}+\frac{89\!\cdots\!29}{37\!\cdots\!28}a^{10}-\frac{11\!\cdots\!61}{37\!\cdots\!28}a^{9}-\frac{67\!\cdots\!29}{37\!\cdots\!28}a^{8}+\frac{47\!\cdots\!13}{37\!\cdots\!28}a^{7}+\frac{35\!\cdots\!55}{37\!\cdots\!28}a^{6}-\frac{56\!\cdots\!65}{18\!\cdots\!14}a^{5}-\frac{90\!\cdots\!49}{28\!\cdots\!56}a^{4}+\frac{18\!\cdots\!05}{37\!\cdots\!28}a^{3}+\frac{19\!\cdots\!89}{37\!\cdots\!28}a^{2}-\frac{40\!\cdots\!29}{37\!\cdots\!28}a-\frac{52\!\cdots\!43}{18\!\cdots\!14}$, $\frac{16\!\cdots\!79}{18\!\cdots\!14}a^{31}+\frac{39\!\cdots\!05}{37\!\cdots\!28}a^{30}-\frac{48\!\cdots\!39}{37\!\cdots\!28}a^{29}-\frac{59\!\cdots\!77}{37\!\cdots\!28}a^{28}+\frac{21\!\cdots\!87}{18\!\cdots\!14}a^{27}+\frac{13\!\cdots\!03}{93\!\cdots\!07}a^{26}-\frac{25\!\cdots\!53}{37\!\cdots\!28}a^{25}-\frac{79\!\cdots\!84}{93\!\cdots\!07}a^{24}+\frac{86\!\cdots\!23}{28\!\cdots\!56}a^{23}+\frac{13\!\cdots\!25}{37\!\cdots\!28}a^{22}-\frac{14\!\cdots\!19}{14\!\cdots\!78}a^{21}-\frac{11\!\cdots\!84}{93\!\cdots\!07}a^{20}+\frac{97\!\cdots\!97}{37\!\cdots\!28}a^{19}+\frac{30\!\cdots\!72}{93\!\cdots\!07}a^{18}-\frac{29\!\cdots\!45}{55\!\cdots\!84}a^{17}-\frac{24\!\cdots\!21}{37\!\cdots\!28}a^{16}+\frac{15\!\cdots\!11}{18\!\cdots\!14}a^{15}+\frac{95\!\cdots\!91}{93\!\cdots\!07}a^{14}-\frac{53\!\cdots\!07}{55\!\cdots\!84}a^{13}-\frac{10\!\cdots\!01}{907381176612469}a^{12}+\frac{32\!\cdots\!43}{37\!\cdots\!28}a^{11}+\frac{40\!\cdots\!05}{37\!\cdots\!28}a^{10}-\frac{99\!\cdots\!69}{18\!\cdots\!14}a^{9}-\frac{62\!\cdots\!10}{93\!\cdots\!07}a^{8}+\frac{66\!\cdots\!09}{28\!\cdots\!56}a^{7}+\frac{27\!\cdots\!83}{93\!\cdots\!07}a^{6}-\frac{21\!\cdots\!31}{37\!\cdots\!28}a^{5}-\frac{28\!\cdots\!01}{37\!\cdots\!28}a^{4}+\frac{17\!\cdots\!33}{18\!\cdots\!14}a^{3}+\frac{11\!\cdots\!57}{93\!\cdots\!07}a^{2}-\frac{90\!\cdots\!51}{37\!\cdots\!28}a-\frac{19\!\cdots\!43}{37\!\cdots\!28}$, $\frac{21\!\cdots\!64}{43\!\cdots\!29}a^{31}-\frac{99\!\cdots\!18}{43\!\cdots\!29}a^{30}-\frac{13\!\cdots\!65}{17\!\cdots\!16}a^{29}+\frac{16\!\cdots\!62}{43\!\cdots\!29}a^{28}+\frac{11\!\cdots\!09}{17\!\cdots\!16}a^{27}-\frac{15\!\cdots\!02}{43\!\cdots\!29}a^{26}-\frac{71\!\cdots\!55}{17\!\cdots\!16}a^{25}+\frac{97\!\cdots\!16}{43\!\cdots\!29}a^{24}+\frac{31\!\cdots\!45}{17\!\cdots\!16}a^{23}-\frac{45\!\cdots\!10}{43\!\cdots\!29}a^{22}-\frac{10\!\cdots\!93}{17\!\cdots\!16}a^{21}+\frac{16\!\cdots\!99}{43\!\cdots\!29}a^{20}+\frac{28\!\cdots\!87}{17\!\cdots\!16}a^{19}-\frac{44\!\cdots\!90}{43\!\cdots\!29}a^{18}-\frac{57\!\cdots\!01}{17\!\cdots\!16}a^{17}+\frac{75\!\cdots\!32}{33\!\cdots\!33}a^{16}+\frac{90\!\cdots\!61}{17\!\cdots\!16}a^{15}-\frac{16\!\cdots\!50}{43\!\cdots\!29}a^{14}-\frac{10\!\cdots\!95}{17\!\cdots\!16}a^{13}+\frac{21\!\cdots\!94}{43\!\cdots\!29}a^{12}+\frac{10\!\cdots\!29}{17\!\cdots\!16}a^{11}-\frac{21\!\cdots\!38}{43\!\cdots\!29}a^{10}-\frac{65\!\cdots\!05}{17\!\cdots\!16}a^{9}+\frac{16\!\cdots\!38}{43\!\cdots\!29}a^{8}+\frac{29\!\cdots\!11}{17\!\cdots\!16}a^{7}-\frac{83\!\cdots\!20}{43\!\cdots\!29}a^{6}-\frac{81\!\cdots\!45}{17\!\cdots\!16}a^{5}+\frac{29\!\cdots\!32}{43\!\cdots\!29}a^{4}+\frac{12\!\cdots\!25}{17\!\cdots\!16}a^{3}-\frac{54\!\cdots\!94}{43\!\cdots\!29}a^{2}-\frac{42\!\cdots\!23}{17\!\cdots\!16}a+\frac{33\!\cdots\!90}{43\!\cdots\!29}$, $\frac{64\!\cdots\!29}{87\!\cdots\!58}a^{31}-\frac{66\!\cdots\!07}{17\!\cdots\!16}a^{30}-\frac{36\!\cdots\!01}{33\!\cdots\!33}a^{29}+\frac{99\!\cdots\!59}{17\!\cdots\!16}a^{28}+\frac{16\!\cdots\!67}{17\!\cdots\!16}a^{27}-\frac{44\!\cdots\!63}{87\!\cdots\!58}a^{26}-\frac{24\!\cdots\!90}{43\!\cdots\!29}a^{25}+\frac{52\!\cdots\!01}{17\!\cdots\!16}a^{24}+\frac{10\!\cdots\!67}{43\!\cdots\!29}a^{23}-\frac{23\!\cdots\!67}{17\!\cdots\!16}a^{22}-\frac{14\!\cdots\!43}{17\!\cdots\!16}a^{21}+\frac{38\!\cdots\!39}{87\!\cdots\!58}a^{20}+\frac{90\!\cdots\!15}{43\!\cdots\!29}a^{19}-\frac{19\!\cdots\!05}{17\!\cdots\!16}a^{18}-\frac{17\!\cdots\!91}{43\!\cdots\!29}a^{17}+\frac{30\!\cdots\!67}{13\!\cdots\!32}a^{16}+\frac{10\!\cdots\!55}{17\!\cdots\!16}a^{15}-\frac{45\!\cdots\!61}{13\!\cdots\!74}a^{14}-\frac{30\!\cdots\!45}{43\!\cdots\!29}a^{13}+\frac{71\!\cdots\!21}{17\!\cdots\!16}a^{12}+\frac{25\!\cdots\!89}{43\!\cdots\!29}a^{11}-\frac{62\!\cdots\!87}{17\!\cdots\!16}a^{10}-\frac{57\!\cdots\!99}{17\!\cdots\!16}a^{9}+\frac{19\!\cdots\!45}{87\!\cdots\!58}a^{8}+\frac{53\!\cdots\!62}{43\!\cdots\!29}a^{7}-\frac{16\!\cdots\!45}{17\!\cdots\!16}a^{6}-\frac{73\!\cdots\!38}{43\!\cdots\!29}a^{5}+\frac{38\!\cdots\!43}{17\!\cdots\!16}a^{4}+\frac{97\!\cdots\!95}{17\!\cdots\!16}a^{3}-\frac{32\!\cdots\!07}{87\!\cdots\!58}a^{2}+\frac{61\!\cdots\!53}{67\!\cdots\!66}a-\frac{68\!\cdots\!29}{87\!\cdots\!58}$, $\frac{70\!\cdots\!67}{17\!\cdots\!16}a^{31}-\frac{31\!\cdots\!03}{17\!\cdots\!16}a^{30}-\frac{10\!\cdots\!13}{17\!\cdots\!16}a^{29}+\frac{12\!\cdots\!43}{43\!\cdots\!29}a^{28}+\frac{94\!\cdots\!41}{17\!\cdots\!16}a^{27}-\frac{43\!\cdots\!87}{17\!\cdots\!16}a^{26}-\frac{28\!\cdots\!35}{87\!\cdots\!58}a^{25}+\frac{65\!\cdots\!34}{43\!\cdots\!29}a^{24}+\frac{24\!\cdots\!45}{17\!\cdots\!16}a^{23}-\frac{29\!\cdots\!51}{43\!\cdots\!29}a^{22}-\frac{82\!\cdots\!25}{17\!\cdots\!16}a^{21}+\frac{39\!\cdots\!99}{17\!\cdots\!16}a^{20}+\frac{82\!\cdots\!03}{67\!\cdots\!66}a^{19}-\frac{26\!\cdots\!56}{43\!\cdots\!29}a^{18}-\frac{43\!\cdots\!41}{17\!\cdots\!16}a^{17}+\frac{54\!\cdots\!71}{43\!\cdots\!29}a^{16}+\frac{10\!\cdots\!39}{26\!\cdots\!48}a^{15}-\frac{35\!\cdots\!95}{17\!\cdots\!16}a^{14}-\frac{39\!\cdots\!63}{87\!\cdots\!58}a^{13}+\frac{11\!\cdots\!18}{43\!\cdots\!29}a^{12}+\frac{70\!\cdots\!13}{17\!\cdots\!16}a^{11}-\frac{10\!\cdots\!82}{43\!\cdots\!29}a^{10}-\frac{43\!\cdots\!85}{17\!\cdots\!16}a^{9}+\frac{27\!\cdots\!07}{17\!\cdots\!16}a^{8}+\frac{95\!\cdots\!97}{87\!\cdots\!58}a^{7}-\frac{33\!\cdots\!43}{43\!\cdots\!29}a^{6}-\frac{35\!\cdots\!33}{13\!\cdots\!32}a^{5}+\frac{95\!\cdots\!18}{43\!\cdots\!29}a^{4}+\frac{71\!\cdots\!89}{17\!\cdots\!16}a^{3}-\frac{58\!\cdots\!07}{17\!\cdots\!16}a^{2}-\frac{14\!\cdots\!47}{17\!\cdots\!16}a+\frac{19\!\cdots\!45}{17\!\cdots\!16}$, $\frac{30\!\cdots\!13}{43\!\cdots\!29}a^{31}-\frac{45\!\cdots\!41}{87\!\cdots\!58}a^{30}-\frac{92\!\cdots\!45}{87\!\cdots\!58}a^{29}+\frac{33\!\cdots\!05}{43\!\cdots\!29}a^{28}+\frac{41\!\cdots\!84}{43\!\cdots\!29}a^{27}-\frac{12\!\cdots\!07}{17\!\cdots\!16}a^{26}-\frac{97\!\cdots\!13}{17\!\cdots\!16}a^{25}+\frac{17\!\cdots\!54}{43\!\cdots\!29}a^{24}+\frac{21\!\cdots\!83}{87\!\cdots\!58}a^{23}-\frac{77\!\cdots\!50}{43\!\cdots\!29}a^{22}-\frac{35\!\cdots\!19}{43\!\cdots\!29}a^{21}+\frac{10\!\cdots\!07}{17\!\cdots\!16}a^{20}+\frac{37\!\cdots\!33}{17\!\cdots\!16}a^{19}-\frac{66\!\cdots\!08}{43\!\cdots\!29}a^{18}-\frac{37\!\cdots\!25}{87\!\cdots\!58}a^{17}+\frac{13\!\cdots\!91}{43\!\cdots\!29}a^{16}+\frac{29\!\cdots\!75}{43\!\cdots\!29}a^{15}-\frac{82\!\cdots\!59}{17\!\cdots\!16}a^{14}-\frac{13\!\cdots\!25}{17\!\cdots\!16}a^{13}+\frac{24\!\cdots\!17}{43\!\cdots\!29}a^{12}+\frac{61\!\cdots\!37}{87\!\cdots\!58}a^{11}-\frac{21\!\cdots\!93}{43\!\cdots\!29}a^{10}-\frac{19\!\cdots\!51}{43\!\cdots\!29}a^{9}+\frac{55\!\cdots\!23}{17\!\cdots\!16}a^{8}+\frac{34\!\cdots\!25}{17\!\cdots\!16}a^{7}-\frac{65\!\cdots\!81}{43\!\cdots\!29}a^{6}-\frac{43\!\cdots\!45}{87\!\cdots\!58}a^{5}+\frac{18\!\cdots\!72}{43\!\cdots\!29}a^{4}+\frac{36\!\cdots\!77}{43\!\cdots\!29}a^{3}-\frac{14\!\cdots\!19}{13\!\cdots\!32}a^{2}-\frac{38\!\cdots\!13}{17\!\cdots\!16}a+\frac{41\!\cdots\!29}{87\!\cdots\!58}$ 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:  \( 550321919281.4646 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 550321919281.4646 \cdot 40}{12\cdot\sqrt{5241670721450327242974152113348873388724842246701056}}\cr\approx \mathstrut & 0.149499101457984 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 15*x^30 + 134*x^28 - 793*x^26 + 3496*x^24 - 11638*x^22 + 30241*x^20 - 60672*x^18 + 94862*x^16 - 111861*x^14 + 99380*x^12 - 61666*x^10 + 26973*x^8 - 6636*x^6 + 1086*x^4 - 36*x^2 + 1)
 
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 - 15*x^30 + 134*x^28 - 793*x^26 + 3496*x^24 - 11638*x^22 + 30241*x^20 - 60672*x^18 + 94862*x^16 - 111861*x^14 + 99380*x^12 - 61666*x^10 + 26973*x^8 - 6636*x^6 + 1086*x^4 - 36*x^2 + 1, 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 - 15*x^30 + 134*x^28 - 793*x^26 + 3496*x^24 - 11638*x^22 + 30241*x^20 - 60672*x^18 + 94862*x^16 - 111861*x^14 + 99380*x^12 - 61666*x^10 + 26973*x^8 - 6636*x^6 + 1086*x^4 - 36*x^2 + 1);
 
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 - 15*x^30 + 134*x^28 - 793*x^26 + 3496*x^24 - 11638*x^22 + 30241*x^20 - 60672*x^18 + 94862*x^16 - 111861*x^14 + 99380*x^12 - 61666*x^10 + 26973*x^8 - 6636*x^6 + 1086*x^4 - 36*x^2 + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^2\times C_8$ (as 32T37):

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)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{51}) \), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{17})\), \(\Q(i, \sqrt{51})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{-3}, \sqrt{-17})\), \(\Q(\sqrt{3}, \sqrt{17})\), \(\Q(\sqrt{3}, \sqrt{-17})\), 4.4.4913.1, 4.0.78608.1, 4.0.44217.1, 4.4.707472.1, 8.0.1731891456.1, 8.0.6179217664.1, 8.0.500516630784.2, 8.0.1955143089.1, 8.0.500516630784.3, 8.8.500516630784.1, 8.0.500516630784.1, \(\Q(\zeta_{17})^+\), 8.0.105046700288.1, 8.0.33237432513.1, 8.8.8508782723328.1, 16.0.250516897691366976454656.1, 16.0.11034809241396899282944.1, 16.0.72399383432805056195395584.3, 16.0.1104726920056229495169.1, 16.0.72399383432805056195395584.2, 16.16.72399383432805056195395584.1, 16.0.72399383432805056195395584.1

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]
 

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 ${\href{/padicField/5.8.0.1}{8} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{16}$ R ${\href{/padicField/19.4.0.1}{4} }^{8}$ ${\href{/padicField/23.8.0.1}{8} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{4}$ ${\href{/padicField/31.8.0.1}{8} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{16}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{8}$

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 2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(3\) Copy content Toggle raw display 3.16.8.1$x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
3.16.8.1$x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
\(17\) Copy content Toggle raw display 17.16.14.1$x^{16} + 128 x^{15} + 7192 x^{14} + 232064 x^{13} + 4716796 x^{12} + 62185088 x^{11} + 525781480 x^{10} + 2696730752 x^{9} + 7365142088 x^{8} + 8090194432 x^{7} + 4732152320 x^{6} + 1682759680 x^{5} + 456414056 x^{4} + 996830464 x^{3} + 7439529968 x^{2} + 33582546688 x + 66368009604$$8$$2$$14$$C_8\times C_2$$[\ ]_{8}^{2}$
17.16.14.1$x^{16} + 128 x^{15} + 7192 x^{14} + 232064 x^{13} + 4716796 x^{12} + 62185088 x^{11} + 525781480 x^{10} + 2696730752 x^{9} + 7365142088 x^{8} + 8090194432 x^{7} + 4732152320 x^{6} + 1682759680 x^{5} + 456414056 x^{4} + 996830464 x^{3} + 7439529968 x^{2} + 33582546688 x + 66368009604$$8$$2$$14$$C_8\times C_2$$[\ ]_{8}^{2}$