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Properties

Label 40.0.93132257461...0000.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 5^{70}$
Root discriminant $33.44$
Ramified primes $2, 5$
Class number $55$ (GRH)
Class group $[55]$ (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![1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^30 + x^20 - x^10 + 1)
 
gp: K = bnfinit(x^40 - x^30 + x^20 - x^10 + 1, 1)
 

Normalized defining polynomial

\( x^{40} - x^{30} + x^{20} - x^{10} + 1 \)

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:  \(9313225746154785156250000000000000000000000000000000000000000=2^{40}\cdot 5^{70}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.44$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(100=2^{2}\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{100}(1,·)$, $\chi_{100}(3,·)$, $\chi_{100}(7,·)$, $\chi_{100}(9,·)$, $\chi_{100}(11,·)$, $\chi_{100}(13,·)$, $\chi_{100}(17,·)$, $\chi_{100}(19,·)$, $\chi_{100}(21,·)$, $\chi_{100}(23,·)$, $\chi_{100}(27,·)$, $\chi_{100}(29,·)$, $\chi_{100}(31,·)$, $\chi_{100}(33,·)$, $\chi_{100}(37,·)$, $\chi_{100}(39,·)$, $\chi_{100}(41,·)$, $\chi_{100}(43,·)$, $\chi_{100}(47,·)$, $\chi_{100}(49,·)$, $\chi_{100}(51,·)$, $\chi_{100}(53,·)$, $\chi_{100}(57,·)$, $\chi_{100}(59,·)$, $\chi_{100}(61,·)$, $\chi_{100}(63,·)$, $\chi_{100}(67,·)$, $\chi_{100}(69,·)$, $\chi_{100}(71,·)$, $\chi_{100}(73,·)$, $\chi_{100}(77,·)$, $\chi_{100}(79,·)$, $\chi_{100}(81,·)$, $\chi_{100}(83,·)$, $\chi_{100}(87,·)$, $\chi_{100}(89,·)$, $\chi_{100}(91,·)$, $\chi_{100}(93,·)$, $\chi_{100}(97,·)$, $\chi_{100}(99,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$

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

Class group and class number

$C_{55}$, which has order $55$ (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:  \( a \) (order $100$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( a^{28} - a^{18} \),  \( a^{4} + 1 \),  \( a^{16} + 1 \),  \( a^{8} + 1 \),  \( a^{36} - a^{18} \),  \( a^{34} - a^{24} - a^{22} + a^{14} - a^{4} \),  \( a^{24} + 1 \),  \( a^{28} - a^{14} \),  \( a^{4} - a^{2} \),  \( a^{15} - 1 \),  \( a^{3} - 1 \),  \( a^{17} - 1 \),  \( a^{13} - 1 \),  \( a^{38} + a^{11} \),  \( a - 1 \),  \( a^{11} - 1 \),  \( a^{23} - a^{18} \),  \( a^{9} - 1 \),  \( a^{21} - 1 \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 144312257071955.8 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 5.5.390625.1, \(\Q(\zeta_{20})\), 10.0.156250000000000.1, \(\Q(\zeta_{25})^+\), 10.0.781250000000000.1, 20.0.610351562500000000000000000000.1, \(\Q(\zeta_{25})\), \(\Q(\zeta_{100})^+\)

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 $20^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
5Data not computed