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Properties

Label 32.0.15693807744...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{96}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $70.77$
Ramified primes $2, 5, 7$
Class number $6400$ (GRH)
Class group $[80, 80]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 0, 0, 211309, 0, 0, 0, 0, 0, 0, 0, 297166, 0, 0, 0, 0, 0, 0, 0, 4564, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 4564*x^24 + 297166*x^16 + 211309*x^8 + 1)
 
gp: K = bnfinit(x^32 + 4564*x^24 + 297166*x^16 + 211309*x^8 + 1, 1)
 

Normalized defining polynomial

\( x^{32} + 4564 x^{24} + 297166 x^{16} + 211309 x^{8} + 1 \)

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:  \(156938077449417789520626992646455296000000000000000000000000=2^{96}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.77$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(560=2^{4}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(519,·)$, $\chi_{560}(139,·)$, $\chi_{560}(531,·)$, $\chi_{560}(533,·)$, $\chi_{560}(281,·)$, $\chi_{560}(153,·)$, $\chi_{560}(197,·)$, $\chi_{560}(419,·)$, $\chi_{560}(167,·)$, $\chi_{560}(169,·)$, $\chi_{560}(43,·)$, $\chi_{560}(349,·)$, $\chi_{560}(433,·)$, $\chi_{560}(181,·)$, $\chi_{560}(351,·)$, $\chi_{560}(447,·)$, $\chi_{560}(449,·)$, $\chi_{560}(323,·)$, $\chi_{560}(69,·)$, $\chi_{560}(71,·)$, $\chi_{560}(461,·)$, $\chi_{560}(547,·)$, $\chi_{560}(267,·)$, $\chi_{560}(477,·)$, $\chi_{560}(223,·)$, $\chi_{560}(97,·)$, $\chi_{560}(239,·)$, $\chi_{560}(503,·)$, $\chi_{560}(377,·)$, $\chi_{560}(251,·)$, $\chi_{560}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{3477926516171} a^{24} + \frac{189054487608}{3477926516171} a^{16} + \frac{973905598808}{3477926516171} a^{8} + \frac{1302644908324}{3477926516171}$, $\frac{1}{3477926516171} a^{25} + \frac{189054487608}{3477926516171} a^{17} + \frac{973905598808}{3477926516171} a^{9} + \frac{1302644908324}{3477926516171} a$, $\frac{1}{3477926516171} a^{26} + \frac{189054487608}{3477926516171} a^{18} + \frac{973905598808}{3477926516171} a^{10} + \frac{1302644908324}{3477926516171} a^{2}$, $\frac{1}{3477926516171} a^{27} + \frac{189054487608}{3477926516171} a^{19} + \frac{973905598808}{3477926516171} a^{11} + \frac{1302644908324}{3477926516171} a^{3}$, $\frac{1}{3477926516171} a^{28} + \frac{189054487608}{3477926516171} a^{20} + \frac{973905598808}{3477926516171} a^{12} + \frac{1302644908324}{3477926516171} a^{4}$, $\frac{1}{3477926516171} a^{29} + \frac{189054487608}{3477926516171} a^{21} + \frac{973905598808}{3477926516171} a^{13} + \frac{1302644908324}{3477926516171} a^{5}$, $\frac{1}{3477926516171} a^{30} + \frac{189054487608}{3477926516171} a^{22} + \frac{973905598808}{3477926516171} a^{14} + \frac{1302644908324}{3477926516171} a^{6}$, $\frac{1}{3477926516171} a^{31} + \frac{189054487608}{3477926516171} a^{23} + \frac{973905598808}{3477926516171} a^{15} + \frac{1302644908324}{3477926516171} a^{7}$

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

Class group and class number

$C_{80}\times C_{80}$, which has order $6400$ (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:  \( -\frac{8536578953}{183048764009} a^{30} - \frac{38960947080000}{183048764009} a^{22} - \frac{2536784392859420}{183048764009} a^{14} - \frac{1804080943439060}{183048764009} a^{6} \) (order $8$)
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:  \( 14712129696811.27 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

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 $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\zeta_{8})\), 4.0.2508800.1, 4.4.2508800.1, \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), 4.0.100352.5, 4.4.100352.1, \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), 4.4.6125.1, 4.0.98000.1, 4.4.392000.1, 4.0.392000.2, 4.0.256000.4, 4.4.256000.1, 4.0.256000.2, 4.4.256000.2, 8.0.25176309760000.55, 8.0.40960000.1, 8.0.40282095616.2, 8.0.6294077440000.7, 8.0.25176309760000.44, 8.8.6294077440000.1, 8.0.25176309760000.32, 8.0.9604000000.1, 8.0.2458624000000.7, 8.0.262144000000.3, 8.0.262144000000.4, 8.8.153664000000.1, 8.0.2458624000000.6, 8.0.65536000000.1, 8.8.65536000000.1, 8.0.153664000000.3, 8.0.2458624000000.5, 8.0.262144000000.1, 8.0.262144000000.2, 16.0.633846573131471257600000000.2, 16.0.6044831973376000000000000.1, 16.0.68719476736000000000000.1, 16.0.24759631762948096000000000000.3, 16.0.396154108207169536000000000000.6, 16.16.24759631762948096000000000000.1, 16.0.396154108207169536000000000000.5

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$