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Properties

Label 32.0.98151526926...3536.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 19^{16}$
Root discriminant $69.74$
Ramified primes $2, 19$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![152587890625, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -697729, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 697729*x^16 + 152587890625)
 
gp: K = bnfinit(x^32 - 697729*x^16 + 152587890625, 1)
 

Normalized defining polynomial

\( x^{32} - 697729 x^{16} + 152587890625 \)

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:  \(98151526926811424140342224755366174047996680049191641153536=2^{128}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.74$
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:  $2, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(608=2^{5}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{608}(1,·)$, $\chi_{608}(265,·)$, $\chi_{608}(267,·)$, $\chi_{608}(531,·)$, $\chi_{608}(533,·)$, $\chi_{608}(151,·)$, $\chi_{608}(153,·)$, $\chi_{608}(417,·)$, $\chi_{608}(419,·)$, $\chi_{608}(37,·)$, $\chi_{608}(39,·)$, $\chi_{608}(303,·)$, $\chi_{608}(305,·)$, $\chi_{608}(569,·)$, $\chi_{608}(571,·)$, $\chi_{608}(189,·)$, $\chi_{608}(191,·)$, $\chi_{608}(455,·)$, $\chi_{608}(457,·)$, $\chi_{608}(75,·)$, $\chi_{608}(77,·)$, $\chi_{608}(341,·)$, $\chi_{608}(343,·)$, $\chi_{608}(607,·)$, $\chi_{608}(227,·)$, $\chi_{608}(229,·)$, $\chi_{608}(493,·)$, $\chi_{608}(495,·)$, $\chi_{608}(113,·)$, $\chi_{608}(115,·)$, $\chi_{608}(379,·)$, $\chi_{608}(381,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{80631} a^{16} + \frac{13975}{80631}$, $\frac{1}{403155} a^{17} + \frac{94606}{403155} a$, $\frac{1}{2015775} a^{18} - \frac{711704}{2015775} a^{2}$, $\frac{1}{10078875} a^{19} - \frac{2727479}{10078875} a^{3}$, $\frac{1}{50394375} a^{20} - \frac{22885229}{50394375} a^{4}$, $\frac{1}{251971875} a^{21} - \frac{22885229}{251971875} a^{5}$, $\frac{1}{1259859375} a^{22} - \frac{22885229}{1259859375} a^{6}$, $\frac{1}{6299296875} a^{23} - \frac{22885229}{6299296875} a^{7}$, $\frac{1}{31496484375} a^{24} - \frac{6322182104}{31496484375} a^{8}$, $\frac{1}{157482421875} a^{25} - \frac{69315150854}{157482421875} a^{9}$, $\frac{1}{787412109375} a^{26} - \frac{226797572729}{787412109375} a^{10}$, $\frac{1}{3937060546875} a^{27} - \frac{1014209682104}{3937060546875} a^{11}$, $\frac{1}{19685302734375} a^{28} + \frac{2922850864771}{19685302734375} a^{12}$, $\frac{1}{98426513671875} a^{29} - \frac{36447754603979}{98426513671875} a^{13}$, $\frac{1}{492132568359375} a^{30} - \frac{233300781947729}{492132568359375} a^{14}$, $\frac{1}{2460662841796875} a^{31} - \frac{725433350307104}{2460662841796875} a^{15}$

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:  \( \frac{284}{157482421875} a^{25} - \frac{200108161}{157482421875} a^{9} \) (order $32$)
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

$C_2^2\times C_8$ (as 32T37):

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 $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-38}) \), \(\Q(\sqrt{38}) \), \(\Q(i, \sqrt{19})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{38})\), \(\Q(\sqrt{2}, \sqrt{-19})\), \(\Q(\sqrt{-2}, \sqrt{-19})\), \(\Q(\sqrt{2}, \sqrt{19})\), \(\Q(\sqrt{-2}, \sqrt{19})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, 4.0.739328.2, 4.4.739328.1, 8.0.8540717056.1, \(\Q(\zeta_{16})\), 8.0.2186423566336.2, 8.0.546605891584.2, 8.0.546605891584.1, 8.8.2186423566336.1, 8.0.2186423566336.1, \(\Q(\zeta_{32})^+\), 8.0.2147483648.1, 8.0.279862216491008.16, 8.8.279862216491008.1, 16.0.4780448011429432992464896.1, \(\Q(\zeta_{32})\), 16.0.313291440877039320594179424256.1, 16.0.78322860219259830148544856064.1, 16.0.78322860219259830148544856064.2, 16.16.313291440877039320594179424256.1, 16.0.313291440877039320594179424256.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
19Data not computed