Properties

Label 32.32.1351043235...7008.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{191}\cdot 3^{16}$
Root discriminant $108.48$
Ramified primes $2, 3$
Class number Not computed
Class group Not computed
Galois group $C_{32}$ (as 32T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![86093442, 0, -3673320192, 0, 26019351360, 0, -72854183808, 0, 107112996432, 0, -95211552384, 0, 55540072224, 0, -22378783680, 0, 6433900308, 0, -1345652352, 0, 206569440, 0, -23250240, 0, 1895400, 0, -108864, 0, 4176, 0, -96, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 96*x^30 + 4176*x^28 - 108864*x^26 + 1895400*x^24 - 23250240*x^22 + 206569440*x^20 - 1345652352*x^18 + 6433900308*x^16 - 22378783680*x^14 + 55540072224*x^12 - 95211552384*x^10 + 107112996432*x^8 - 72854183808*x^6 + 26019351360*x^4 - 3673320192*x^2 + 86093442)
 
gp: K = bnfinit(x^32 - 96*x^30 + 4176*x^28 - 108864*x^26 + 1895400*x^24 - 23250240*x^22 + 206569440*x^20 - 1345652352*x^18 + 6433900308*x^16 - 22378783680*x^14 + 55540072224*x^12 - 95211552384*x^10 + 107112996432*x^8 - 72854183808*x^6 + 26019351360*x^4 - 3673320192*x^2 + 86093442, 1)
 

Normalized defining polynomial

\( x^{32} - 96 x^{30} + 4176 x^{28} - 108864 x^{26} + 1895400 x^{24} - 23250240 x^{22} + 206569440 x^{20} - 1345652352 x^{18} + 6433900308 x^{16} - 22378783680 x^{14} + 55540072224 x^{12} - 95211552384 x^{10} + 107112996432 x^{8} - 72854183808 x^{6} + 26019351360 x^{4} - 3673320192 x^{2} + 86093442 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(135104323545903136978453058557785670637514001130337144105502507008=2^{191}\cdot 3^{16}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $108.48$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3$
magma: PrimeDivisors(Discriminant(K));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(384=2^{7}\cdot 3\)
Dirichlet character group:    $\lbrace$$\chi_{384}(1,·)$, $\chi_{384}(131,·)$, $\chi_{384}(265,·)$, $\chi_{384}(11,·)$, $\chi_{384}(145,·)$, $\chi_{384}(275,·)$, $\chi_{384}(25,·)$, $\chi_{384}(155,·)$, $\chi_{384}(289,·)$, $\chi_{384}(35,·)$, $\chi_{384}(169,·)$, $\chi_{384}(299,·)$, $\chi_{384}(49,·)$, $\chi_{384}(179,·)$, $\chi_{384}(313,·)$, $\chi_{384}(59,·)$, $\chi_{384}(193,·)$, $\chi_{384}(323,·)$, $\chi_{384}(73,·)$, $\chi_{384}(203,·)$, $\chi_{384}(337,·)$, $\chi_{384}(83,·)$, $\chi_{384}(217,·)$, $\chi_{384}(347,·)$, $\chi_{384}(97,·)$, $\chi_{384}(227,·)$, $\chi_{384}(361,·)$, $\chi_{384}(107,·)$, $\chi_{384}(241,·)$, $\chi_{384}(371,·)$, $\chi_{384}(121,·)$, $\chi_{384}(251,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{2187} a^{14}$, $\frac{1}{2187} a^{15}$, $\frac{1}{6561} a^{16}$, $\frac{1}{6561} a^{17}$, $\frac{1}{19683} a^{18}$, $\frac{1}{19683} a^{19}$, $\frac{1}{59049} a^{20}$, $\frac{1}{59049} a^{21}$, $\frac{1}{177147} a^{22}$, $\frac{1}{177147} a^{23}$, $\frac{1}{531441} a^{24}$, $\frac{1}{531441} a^{25}$, $\frac{1}{1594323} a^{26}$, $\frac{1}{1594323} a^{27}$, $\frac{1}{4782969} a^{28}$, $\frac{1}{4782969} a^{29}$, $\frac{1}{14348907} a^{30}$, $\frac{1}{14348907} a^{31}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $31$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{32}$ (as 32T33):

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

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), \(\Q(\zeta_{64})^+\)

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 R $32$ $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

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
3Data not computed