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Properties

Label 40.0.40126953504...5664.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{110}\cdot 11^{36}$
Root discriminant $58.22$
Ramified primes $2, 11$
Class number $5731$ (GRH)
Class group $[5731]$ (GRH)
Galois group $C_2\times C_{20}$ (as 40T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, 0, -2048, 0, 3584, 0, -6144, 0, 10496, 0, -17920, 0, 30592, 0, -52224, 0, 89152, 0, -152192, 0, 259808, 0, -76096, 0, 22288, 0, -6528, 0, 1912, 0, -560, 0, 164, 0, -48, 0, 14, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 4*x^38 + 14*x^36 - 48*x^34 + 164*x^32 - 560*x^30 + 1912*x^28 - 6528*x^26 + 22288*x^24 - 76096*x^22 + 259808*x^20 - 152192*x^18 + 89152*x^16 - 52224*x^14 + 30592*x^12 - 17920*x^10 + 10496*x^8 - 6144*x^6 + 3584*x^4 - 2048*x^2 + 1024)
 
gp: K = bnfinit(x^40 - 4*x^38 + 14*x^36 - 48*x^34 + 164*x^32 - 560*x^30 + 1912*x^28 - 6528*x^26 + 22288*x^24 - 76096*x^22 + 259808*x^20 - 152192*x^18 + 89152*x^16 - 52224*x^14 + 30592*x^12 - 17920*x^10 + 10496*x^8 - 6144*x^6 + 3584*x^4 - 2048*x^2 + 1024, 1)
 

Normalized defining polynomial

\( x^{40} - 4 x^{38} + 14 x^{36} - 48 x^{34} + 164 x^{32} - 560 x^{30} + 1912 x^{28} - 6528 x^{26} + 22288 x^{24} - 76096 x^{22} + 259808 x^{20} - 152192 x^{18} + 89152 x^{16} - 52224 x^{14} + 30592 x^{12} - 17920 x^{10} + 10496 x^{8} - 6144 x^{6} + 3584 x^{4} - 2048 x^{2} + 1024 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(40126953504928778716867347648517224229883090465646395672541598175985664=2^{110}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.22$
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:  \(176=2^{4}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{176}(1,·)$, $\chi_{176}(3,·)$, $\chi_{176}(129,·)$, $\chi_{176}(9,·)$, $\chi_{176}(139,·)$, $\chi_{176}(115,·)$, $\chi_{176}(17,·)$, $\chi_{176}(19,·)$, $\chi_{176}(153,·)$, $\chi_{176}(25,·)$, $\chi_{176}(155,·)$, $\chi_{176}(161,·)$, $\chi_{176}(35,·)$, $\chi_{176}(41,·)$, $\chi_{176}(171,·)$, $\chi_{176}(27,·)$, $\chi_{176}(49,·)$, $\chi_{176}(51,·)$, $\chi_{176}(137,·)$, $\chi_{176}(57,·)$, $\chi_{176}(59,·)$, $\chi_{176}(65,·)$, $\chi_{176}(83,·)$, $\chi_{176}(67,·)$, $\chi_{176}(73,·)$, $\chi_{176}(75,·)$, $\chi_{176}(81,·)$, $\chi_{176}(163,·)$, $\chi_{176}(43,·)$, $\chi_{176}(89,·)$, $\chi_{176}(91,·)$, $\chi_{176}(97,·)$, $\chi_{176}(131,·)$, $\chi_{176}(145,·)$, $\chi_{176}(105,·)$, $\chi_{176}(107,·)$, $\chi_{176}(113,·)$, $\chi_{176}(147,·)$, $\chi_{176}(169,·)$, $\chi_{176}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{16} a^{16}$, $\frac{1}{16} a^{17}$, $\frac{1}{16} a^{18}$, $\frac{1}{16} a^{19}$, $\frac{1}{32} a^{20}$, $\frac{1}{32} a^{21}$, $\frac{1}{259808} a^{22} + \frac{3363}{8119}$, $\frac{1}{259808} a^{23} + \frac{3363}{8119} a$, $\frac{1}{519616} a^{24} - \frac{2378}{8119} a^{2}$, $\frac{1}{519616} a^{25} - \frac{2378}{8119} a^{3}$, $\frac{1}{519616} a^{26} + \frac{3363}{16238} a^{4}$, $\frac{1}{519616} a^{27} + \frac{3363}{16238} a^{5}$, $\frac{1}{1039232} a^{28} - \frac{1189}{8119} a^{6}$, $\frac{1}{1039232} a^{29} - \frac{1189}{8119} a^{7}$, $\frac{1}{1039232} a^{30} + \frac{3363}{32476} a^{8}$, $\frac{1}{1039232} a^{31} + \frac{3363}{32476} a^{9}$, $\frac{1}{2078464} a^{32} - \frac{1189}{16238} a^{10}$, $\frac{1}{2078464} a^{33} - \frac{1189}{16238} a^{11}$, $\frac{1}{2078464} a^{34} + \frac{3363}{64952} a^{12}$, $\frac{1}{2078464} a^{35} + \frac{3363}{64952} a^{13}$, $\frac{1}{4156928} a^{36} - \frac{1189}{32476} a^{14}$, $\frac{1}{4156928} a^{37} - \frac{1189}{32476} a^{15}$, $\frac{1}{4156928} a^{38} + \frac{3363}{129904} a^{16}$, $\frac{1}{4156928} a^{39} + \frac{3363}{129904} a^{17}$

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

Class group and class number

$C_{5731}$, which has order $5731$ (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:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{41}{2078464} a^{32} + \frac{117708}{8119} a^{10} \) (order $22$)
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:  \( 59973097516991576 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{20}$ (as 40T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{-22}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{2}, \sqrt{-11})\), 4.0.2048.2, 4.4.247808.1, \(\Q(\zeta_{11})^+\), 8.0.61408804864.1, 10.0.77265229938688.1, 10.10.7024111812608.1, \(\Q(\zeta_{11})\), 20.0.5969915757478328440239161344.6, 20.0.1655513490330868290261743826894848.1, 20.20.200317132330035063121671003054276608.1

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 $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R $20^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{40}$ $20^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$

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