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Properties

Label 32.0.55751862996...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{158}\cdot 5^{16}$
Root discriminant $68.52$
Ramified primes $2, 5$
Class number $97393$ (GRH)
Class group $[17, 5729]$ (GRH)
Galois group $C_2\times C_{16}$ (as 32T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, 384, 0, 8800, 0, 88704, 0, 475992, 0, 1524288, 0, 3129696, 0, 4305120, 0, 4067918, 0, 2670144, 0, 1217920, 0, 382560, 0, 81260, 0, 11328, 0, 984, 0, 48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 48*x^30 + 984*x^28 + 11328*x^26 + 81260*x^24 + 382560*x^22 + 1217920*x^20 + 2670144*x^18 + 4067918*x^16 + 4305120*x^14 + 3129696*x^12 + 1524288*x^10 + 475992*x^8 + 88704*x^6 + 8800*x^4 + 384*x^2 + 4)
 
gp: K = bnfinit(x^32 + 48*x^30 + 984*x^28 + 11328*x^26 + 81260*x^24 + 382560*x^22 + 1217920*x^20 + 2670144*x^18 + 4067918*x^16 + 4305120*x^14 + 3129696*x^12 + 1524288*x^10 + 475992*x^8 + 88704*x^6 + 8800*x^4 + 384*x^2 + 4, 1)
 

Normalized defining polynomial

\( x^{32} + 48 x^{30} + 984 x^{28} + 11328 x^{26} + 81260 x^{24} + 382560 x^{22} + 1217920 x^{20} + 2670144 x^{18} + 4067918 x^{16} + 4305120 x^{14} + 3129696 x^{12} + 1524288 x^{10} + 475992 x^{8} + 88704 x^{6} + 8800 x^{4} + 384 x^{2} + 4 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(55751862996326557853839295681620903764951040000000000000000=2^{158}\cdot 5^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.52$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(320=2^{6}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{320}(1,·)$, $\chi_{320}(259,·)$, $\chi_{320}(51,·)$, $\chi_{320}(129,·)$, $\chi_{320}(9,·)$, $\chi_{320}(139,·)$, $\chi_{320}(131,·)$, $\chi_{320}(281,·)$, $\chi_{320}(289,·)$, $\chi_{320}(291,·)$, $\chi_{320}(41,·)$, $\chi_{320}(171,·)$, $\chi_{320}(49,·)$, $\chi_{320}(91,·)$, $\chi_{320}(179,·)$, $\chi_{320}(59,·)$, $\chi_{320}(11,·)$, $\chi_{320}(161,·)$, $\chi_{320}(201,·)$, $\chi_{320}(81,·)$, $\chi_{320}(211,·)$, $\chi_{320}(121,·)$, $\chi_{320}(89,·)$, $\chi_{320}(219,·)$, $\chi_{320}(99,·)$, $\chi_{320}(209,·)$, $\chi_{320}(241,·)$, $\chi_{320}(19,·)$, $\chi_{320}(169,·)$, $\chi_{320}(249,·)$, $\chi_{320}(251,·)$, $\chi_{320}(299,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{2} a^{18}$, $\frac{1}{2} a^{19}$, $\frac{1}{2} a^{20}$, $\frac{1}{2} a^{21}$, $\frac{1}{62} a^{22} - \frac{11}{62} a^{20} - \frac{1}{62} a^{18} - \frac{1}{62} a^{16} - \frac{8}{31} a^{14} + \frac{2}{31} a^{12} - \frac{8}{31} a^{10} - \frac{15}{31} a^{8} - \frac{15}{31} a^{6} + \frac{8}{31} a^{4} + \frac{13}{31} a^{2} - \frac{2}{31}$, $\frac{1}{62} a^{23} - \frac{11}{62} a^{21} - \frac{1}{62} a^{19} - \frac{1}{62} a^{17} - \frac{8}{31} a^{15} + \frac{2}{31} a^{13} - \frac{8}{31} a^{11} - \frac{15}{31} a^{9} - \frac{15}{31} a^{7} + \frac{8}{31} a^{5} + \frac{13}{31} a^{3} - \frac{2}{31} a$, $\frac{1}{62} a^{24} + \frac{1}{31} a^{20} - \frac{6}{31} a^{18} + \frac{2}{31} a^{16} + \frac{7}{31} a^{14} + \frac{14}{31} a^{12} - \frac{10}{31} a^{10} + \frac{6}{31} a^{8} - \frac{2}{31} a^{6} + \frac{8}{31} a^{4} - \frac{14}{31} a^{2} + \frac{9}{31}$, $\frac{1}{62} a^{25} + \frac{1}{31} a^{21} - \frac{6}{31} a^{19} + \frac{2}{31} a^{17} + \frac{7}{31} a^{15} + \frac{14}{31} a^{13} - \frac{10}{31} a^{11} + \frac{6}{31} a^{9} - \frac{2}{31} a^{7} + \frac{8}{31} a^{5} - \frac{14}{31} a^{3} + \frac{9}{31} a$, $\frac{1}{62} a^{26} + \frac{5}{31} a^{20} + \frac{3}{31} a^{18} - \frac{15}{62} a^{16} - \frac{1}{31} a^{14} - \frac{14}{31} a^{12} - \frac{9}{31} a^{10} - \frac{3}{31} a^{8} + \frac{7}{31} a^{6} + \frac{1}{31} a^{4} + \frac{14}{31} a^{2} + \frac{4}{31}$, $\frac{1}{62} a^{27} + \frac{5}{31} a^{21} + \frac{3}{31} a^{19} - \frac{15}{62} a^{17} - \frac{1}{31} a^{15} - \frac{14}{31} a^{13} - \frac{9}{31} a^{11} - \frac{3}{31} a^{9} + \frac{7}{31} a^{7} + \frac{1}{31} a^{5} + \frac{14}{31} a^{3} + \frac{4}{31} a$, $\frac{1}{11842} a^{28} + \frac{41}{11842} a^{26} + \frac{37}{5921} a^{24} - \frac{13}{5921} a^{22} + \frac{480}{5921} a^{20} - \frac{1613}{11842} a^{18} + \frac{831}{11842} a^{16} - \frac{706}{5921} a^{14} + \frac{784}{5921} a^{12} - \frac{2095}{5921} a^{10} + \frac{46}{191} a^{8} + \frac{277}{5921} a^{6} - \frac{1222}{5921} a^{4} + \frac{283}{5921} a^{2} + \frac{871}{5921}$, $\frac{1}{11842} a^{29} + \frac{41}{11842} a^{27} + \frac{37}{5921} a^{25} - \frac{13}{5921} a^{23} + \frac{480}{5921} a^{21} - \frac{1613}{11842} a^{19} + \frac{831}{11842} a^{17} - \frac{706}{5921} a^{15} + \frac{784}{5921} a^{13} - \frac{2095}{5921} a^{11} + \frac{46}{191} a^{9} + \frac{277}{5921} a^{7} - \frac{1222}{5921} a^{5} + \frac{283}{5921} a^{3} + \frac{871}{5921} a$, $\frac{1}{67524606987778} a^{30} + \frac{949956029}{33762303493889} a^{28} + \frac{4738517787}{2178213128638} a^{26} + \frac{225376994105}{33762303493889} a^{24} - \frac{41355176117}{67524606987778} a^{22} + \frac{8720681762553}{67524606987778} a^{20} + \frac{3954328124485}{67524606987778} a^{18} + \frac{5656841571881}{33762303493889} a^{16} + \frac{9230679271387}{33762303493889} a^{14} + \frac{11015362851805}{33762303493889} a^{12} + \frac{2507023353802}{33762303493889} a^{10} - \frac{10888043620892}{33762303493889} a^{8} + \frac{7548763621264}{33762303493889} a^{6} + \frac{14930032276326}{33762303493889} a^{4} - \frac{2240503219585}{33762303493889} a^{2} + \frac{6985012363541}{33762303493889}$, $\frac{1}{67524606987778} a^{31} + \frac{949956029}{33762303493889} a^{29} + \frac{4738517787}{2178213128638} a^{27} + \frac{225376994105}{33762303493889} a^{25} - \frac{41355176117}{67524606987778} a^{23} + \frac{8720681762553}{67524606987778} a^{21} + \frac{3954328124485}{67524606987778} a^{19} + \frac{5656841571881}{33762303493889} a^{17} + \frac{9230679271387}{33762303493889} a^{15} + \frac{11015362851805}{33762303493889} a^{13} + \frac{2507023353802}{33762303493889} a^{11} - \frac{10888043620892}{33762303493889} a^{9} + \frac{7548763621264}{33762303493889} a^{7} + \frac{14930032276326}{33762303493889} a^{5} - \frac{2240503219585}{33762303493889} a^{3} + \frac{6985012363541}{33762303493889} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{17}\times C_{5729}$, which has order $97393$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 323273529801.8212 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{16}$ (as 32T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_{16}$
Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{16})^+\), 4.4.51200.1, 8.8.2621440000.1, \(\Q(\zeta_{32})^+\), 8.8.1342177280000.1, 16.16.1801439850948198400000000.1, 16.0.236118324143482260684800000000.1, 16.0.604462909807314587353088.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $16^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ $16^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ $16^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{32}$ $16^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ $16^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ $16^{2}$ $16^{2}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
5Data not computed