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Properties

Label 32.0.99276740263...8624.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{48}\cdot 3^{16}\cdot 17^{30}$
Root discriminant $69.77$
Ramified primes $2, 3, 17$
Class number Not computed
Class group Not computed
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![2821109907456, 0, 470184984576, 0, 78364164096, 0, 13060694016, 0, 2176782336, 0, 362797056, 0, 60466176, 0, 10077696, 0, 1679616, 0, 279936, 0, 46656, 0, 7776, 0, 1296, 0, 216, 0, 36, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 6*x^30 + 36*x^28 + 216*x^26 + 1296*x^24 + 7776*x^22 + 46656*x^20 + 279936*x^18 + 1679616*x^16 + 10077696*x^14 + 60466176*x^12 + 362797056*x^10 + 2176782336*x^8 + 13060694016*x^6 + 78364164096*x^4 + 470184984576*x^2 + 2821109907456)
 
gp: K = bnfinit(x^32 + 6*x^30 + 36*x^28 + 216*x^26 + 1296*x^24 + 7776*x^22 + 46656*x^20 + 279936*x^18 + 1679616*x^16 + 10077696*x^14 + 60466176*x^12 + 362797056*x^10 + 2176782336*x^8 + 13060694016*x^6 + 78364164096*x^4 + 470184984576*x^2 + 2821109907456, 1)
 

Normalized defining polynomial

\( x^{32} + 6 x^{30} + 36 x^{28} + 216 x^{26} + 1296 x^{24} + 7776 x^{22} + 46656 x^{20} + 279936 x^{18} + 1679616 x^{16} + 10077696 x^{14} + 60466176 x^{12} + 362797056 x^{10} + 2176782336 x^{8} + 13060694016 x^{6} + 78364164096 x^{4} + 470184984576 x^{2} + 2821109907456 \)

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:  \(99276740263879938750515115508224780490603194567662317338624=2^{48}\cdot 3^{16}\cdot 17^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.77$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(408=2^{3}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{408}(1,·)$, $\chi_{408}(131,·)$, $\chi_{408}(385,·)$, $\chi_{408}(265,·)$, $\chi_{408}(11,·)$, $\chi_{408}(145,·)$, $\chi_{408}(275,·)$, $\chi_{408}(25,·)$, $\chi_{408}(155,·)$, $\chi_{408}(35,·)$, $\chi_{408}(169,·)$, $\chi_{408}(299,·)$, $\chi_{408}(49,·)$, $\chi_{408}(179,·)$, $\chi_{408}(313,·)$, $\chi_{408}(59,·)$, $\chi_{408}(193,·)$, $\chi_{408}(395,·)$, $\chi_{408}(73,·)$, $\chi_{408}(203,·)$, $\chi_{408}(337,·)$, $\chi_{408}(83,·)$, $\chi_{408}(217,·)$, $\chi_{408}(347,·)$, $\chi_{408}(97,·)$, $\chi_{408}(227,·)$, $\chi_{408}(361,·)$, $\chi_{408}(107,·)$, $\chi_{408}(241,·)$, $\chi_{408}(371,·)$, $\chi_{408}(121,·)$, $\chi_{408}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{6} a^{2}$, $\frac{1}{6} a^{3}$, $\frac{1}{36} a^{4}$, $\frac{1}{36} a^{5}$, $\frac{1}{216} a^{6}$, $\frac{1}{216} a^{7}$, $\frac{1}{1296} a^{8}$, $\frac{1}{1296} a^{9}$, $\frac{1}{7776} a^{10}$, $\frac{1}{7776} a^{11}$, $\frac{1}{46656} a^{12}$, $\frac{1}{46656} a^{13}$, $\frac{1}{279936} a^{14}$, $\frac{1}{279936} a^{15}$, $\frac{1}{1679616} a^{16}$, $\frac{1}{1679616} a^{17}$, $\frac{1}{10077696} a^{18}$, $\frac{1}{10077696} a^{19}$, $\frac{1}{60466176} a^{20}$, $\frac{1}{60466176} a^{21}$, $\frac{1}{362797056} a^{22}$, $\frac{1}{362797056} a^{23}$, $\frac{1}{2176782336} a^{24}$, $\frac{1}{2176782336} a^{25}$, $\frac{1}{13060694016} a^{26}$, $\frac{1}{13060694016} a^{27}$, $\frac{1}{78364164096} a^{28}$, $\frac{1}{78364164096} a^{29}$, $\frac{1}{470184984576} a^{30}$, $\frac{1}{470184984576} a^{31}$

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:  \( -\frac{1}{36} a^{4} \) (order $34$)
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

$C_2\times C_{16}$ (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

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{102}) \), \(\Q(\sqrt{6}, \sqrt{17})\), 4.4.4913.1, 4.4.2829888.2, 8.8.8008266092544.1, \(\Q(\zeta_{17})^+\), 8.8.136140523573248.1, 16.16.18534242158798094386021269504.1, 16.0.315082116699567604562361581568.1, \(\Q(\zeta_{17})\)

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 $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
3Data not computed
17Data not computed