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Properties

Label 32.0.14421556453...4128.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{191}\cdot 11^{16}$
Root discriminant $207.72$
Ramified primes $2, 11$
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![91899459727144322, 0, 1069375531370406656, 0, 2065839094692831040, 0, 1577549854129070976, 0, 632556516184870992, 0, 153347034226635392, 0, 24396119081510176, 0, 2680892206759360, 0, 210206320757268, 0, 11990378367616, 0, 501989524960, 0, 15409359680, 0, 342599400, 0, 5366592, 0, 56144, 0, 352, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 352*x^30 + 56144*x^28 + 5366592*x^26 + 342599400*x^24 + 15409359680*x^22 + 501989524960*x^20 + 11990378367616*x^18 + 210206320757268*x^16 + 2680892206759360*x^14 + 24396119081510176*x^12 + 153347034226635392*x^10 + 632556516184870992*x^8 + 1577549854129070976*x^6 + 2065839094692831040*x^4 + 1069375531370406656*x^2 + 91899459727144322)
 
gp: K = bnfinit(x^32 + 352*x^30 + 56144*x^28 + 5366592*x^26 + 342599400*x^24 + 15409359680*x^22 + 501989524960*x^20 + 11990378367616*x^18 + 210206320757268*x^16 + 2680892206759360*x^14 + 24396119081510176*x^12 + 153347034226635392*x^10 + 632556516184870992*x^8 + 1577549854129070976*x^6 + 2065839094692831040*x^4 + 1069375531370406656*x^2 + 91899459727144322, 1)
 

Normalized defining polynomial

\( x^{32} + 352 x^{30} + 56144 x^{28} + 5366592 x^{26} + 342599400 x^{24} + 15409359680 x^{22} + 501989524960 x^{20} + 11990378367616 x^{18} + 210206320757268 x^{16} + 2680892206759360 x^{14} + 24396119081510176 x^{12} + 153347034226635392 x^{10} + 632556516184870992 x^{8} + 1577549854129070976 x^{6} + 2065839094692831040 x^{4} + 1069375531370406656 x^{2} + 91899459727144322 \)

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:  \(144215564533589000876246801170130951941346253827716612240069988132090544128=2^{191}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $207.72$
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:  $2, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1408=2^{7}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1408}(1,·)$, $\chi_{1408}(901,·)$, $\chi_{1408}(265,·)$, $\chi_{1408}(1165,·)$, $\chi_{1408}(529,·)$, $\chi_{1408}(21,·)$, $\chi_{1408}(793,·)$, $\chi_{1408}(285,·)$, $\chi_{1408}(1057,·)$, $\chi_{1408}(549,·)$, $\chi_{1408}(1321,·)$, $\chi_{1408}(813,·)$, $\chi_{1408}(177,·)$, $\chi_{1408}(1077,·)$, $\chi_{1408}(441,·)$, $\chi_{1408}(1341,·)$, $\chi_{1408}(705,·)$, $\chi_{1408}(197,·)$, $\chi_{1408}(969,·)$, $\chi_{1408}(461,·)$, $\chi_{1408}(1233,·)$, $\chi_{1408}(725,·)$, $\chi_{1408}(89,·)$, $\chi_{1408}(989,·)$, $\chi_{1408}(353,·)$, $\chi_{1408}(1253,·)$, $\chi_{1408}(617,·)$, $\chi_{1408}(109,·)$, $\chi_{1408}(881,·)$, $\chi_{1408}(373,·)$, $\chi_{1408}(1145,·)$, $\chi_{1408}(637,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{11} a^{2}$, $\frac{1}{11} a^{3}$, $\frac{1}{121} a^{4}$, $\frac{1}{121} a^{5}$, $\frac{1}{1331} a^{6}$, $\frac{1}{1331} a^{7}$, $\frac{1}{14641} a^{8}$, $\frac{1}{14641} a^{9}$, $\frac{1}{161051} a^{10}$, $\frac{1}{161051} a^{11}$, $\frac{1}{1771561} a^{12}$, $\frac{1}{1771561} a^{13}$, $\frac{1}{19487171} a^{14}$, $\frac{1}{19487171} a^{15}$, $\frac{1}{214358881} a^{16}$, $\frac{1}{214358881} a^{17}$, $\frac{1}{2357947691} a^{18}$, $\frac{1}{2357947691} a^{19}$, $\frac{1}{25937424601} a^{20}$, $\frac{1}{25937424601} a^{21}$, $\frac{1}{285311670611} a^{22}$, $\frac{1}{285311670611} a^{23}$, $\frac{1}{3138428376721} a^{24}$, $\frac{1}{3138428376721} a^{25}$, $\frac{1}{34522712143931} a^{26}$, $\frac{1}{34522712143931} a^{27}$, $\frac{1}{379749833583241} a^{28}$, $\frac{1}{379749833583241} a^{29}$, $\frac{1}{4177248169415651} a^{30}$, $\frac{1}{4177248169415651} a^{31}$

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:  $15$
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:  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 $32$ $32$ $16^{2}$ R $32$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

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
11Data not computed