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Properties

Label 32.0.20282409603...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 5^{24}$
Root discriminant $53.50$
Ramified primes $2, 5$
Class number $12546$ (GRH)
Class group $[3, 4182]$ (GRH)
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 192, 0, 5104, 0, 45024, 0, 200892, 0, 537120, 0, 940472, 0, 1136944, 0, 980627, 0, 615296, 0, 283360, 0, 95680, 0, 23400, 0, 4032, 0, 464, 0, 32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 32*x^30 + 464*x^28 + 4032*x^26 + 23400*x^24 + 95680*x^22 + 283360*x^20 + 615296*x^18 + 980627*x^16 + 1136944*x^14 + 940472*x^12 + 537120*x^10 + 200892*x^8 + 45024*x^6 + 5104*x^4 + 192*x^2 + 1)
 
gp: K = bnfinit(x^32 + 32*x^30 + 464*x^28 + 4032*x^26 + 23400*x^24 + 95680*x^22 + 283360*x^20 + 615296*x^18 + 980627*x^16 + 1136944*x^14 + 940472*x^12 + 537120*x^10 + 200892*x^8 + 45024*x^6 + 5104*x^4 + 192*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{32} + 32 x^{30} + 464 x^{28} + 4032 x^{26} + 23400 x^{24} + 95680 x^{22} + 283360 x^{20} + 615296 x^{18} + 980627 x^{16} + 1136944 x^{14} + 940472 x^{12} + 537120 x^{10} + 200892 x^{8} + 45024 x^{6} + 5104 x^{4} + 192 x^{2} + 1 \)

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:  \(20282409603651670423947251286016000000000000000000000000=2^{128}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.50$
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:  \(160=2^{5}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{160}(1,·)$, $\chi_{160}(131,·)$, $\chi_{160}(133,·)$, $\chi_{160}(7,·)$, $\chi_{160}(9,·)$, $\chi_{160}(11,·)$, $\chi_{160}(13,·)$, $\chi_{160}(143,·)$, $\chi_{160}(19,·)$, $\chi_{160}(23,·)$, $\chi_{160}(157,·)$, $\chi_{160}(37,·)$, $\chi_{160}(129,·)$, $\chi_{160}(41,·)$, $\chi_{160}(47,·)$, $\chi_{160}(49,·)$, $\chi_{160}(51,·)$, $\chi_{160}(53,·)$, $\chi_{160}(59,·)$, $\chi_{160}(63,·)$, $\chi_{160}(139,·)$, $\chi_{160}(77,·)$, $\chi_{160}(81,·)$, $\chi_{160}(87,·)$, $\chi_{160}(89,·)$, $\chi_{160}(91,·)$, $\chi_{160}(93,·)$, $\chi_{160}(99,·)$, $\chi_{160}(103,·)$, $\chi_{160}(117,·)$, $\chi_{160}(121,·)$, $\chi_{160}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$

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

Class group and class number

$C_{3}\times C_{4182}$, which has order $12546$ (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:  \( 24306760628.406784 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_8$ (as 32T43):

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 $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.8000.1, 4.4.51200.1, \(\Q(\zeta_{16})^+\), 4.4.256000.2, 4.4.256000.1, \(\Q(\zeta_{40})^+\), 8.8.2621440000.1, 8.8.65536000000.1, 8.0.33554432000000.2, 8.0.33554432000000.1, 8.0.2147483648.1, 8.0.1342177280000.1, \(\Q(\zeta_{80})^+\), 16.0.1125899906842624000000000000.1, 16.0.1801439850948198400000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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