Properties

Label 20.0.29103830456...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{35}$
Root discriminant $16.72$
Ramified prime $5$
Class number $1$
Class group Trivial
Galois group $C_{20}$ (as 20T1)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^15 + x^10 - x^5 + 1)
 
gp: K = bnfinit(x^20 - x^15 + x^10 - x^5 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1]);
 

Normalized defining polynomial

\( x^{20} - x^{15} + x^{10} - x^{5} + 1 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2910383045673370361328125=5^{35}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $16.72$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $20$
This field is Galois and abelian over $\Q$.
Conductor:  \(25=5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{25}(1,·)$, $\chi_{25}(2,·)$, $\chi_{25}(3,·)$, $\chi_{25}(4,·)$, $\chi_{25}(6,·)$, $\chi_{25}(7,·)$, $\chi_{25}(8,·)$, $\chi_{25}(9,·)$, $\chi_{25}(11,·)$, $\chi_{25}(12,·)$, $\chi_{25}(13,·)$, $\chi_{25}(14,·)$, $\chi_{25}(16,·)$, $\chi_{25}(17,·)$, $\chi_{25}(18,·)$, $\chi_{25}(19,·)$, $\chi_{25}(21,·)$, $\chi_{25}(22,·)$, $\chi_{25}(23,·)$, $\chi_{25}(24,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( a \) (order $50$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{19} + a^{9} \),  \( a^{19} + a^{16} - a^{14} + a^{9} - a^{4} \),  \( a^{18} - a^{13} + a^{8} - a^{3} - 1 \),  \( a^{10} + a^{6} \),  \( a^{19} - a^{15} + a^{10} - a^{5} + 1 \),  \( a^{16} + a^{4} \),  \( a^{7} + a \),  \( a^{16} - a^{7} \),  \( a^{8} - a \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 161406.837641 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_{20}$ (as 20T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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 $20$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.4t1.a.b$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.5.4t1.a.a$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 1.25.5t1.a.a$1$ $ 5^{2}$ $x^{5} - 10 x^{3} - 5 x^{2} + 10 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.25.20t1.a.c$1$ $ 5^{2}$ $x^{20} - x^{15} + x^{10} - x^{5} + 1$ $C_{20}$ (as 20T1) $0$ $-1$
* 1.25.10t1.a.d$1$ $ 5^{2}$ $x^{10} - 10 x^{8} + 35 x^{6} - x^{5} - 50 x^{4} + 5 x^{3} + 25 x^{2} - 5 x - 1$ $C_{10}$ (as 10T1) $0$ $1$
* 1.25.20t1.a.f$1$ $ 5^{2}$ $x^{20} - x^{15} + x^{10} - x^{5} + 1$ $C_{20}$ (as 20T1) $0$ $-1$
* 1.25.5t1.a.b$1$ $ 5^{2}$ $x^{5} - 10 x^{3} - 5 x^{2} + 10 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.25.20t1.a.h$1$ $ 5^{2}$ $x^{20} - x^{15} + x^{10} - x^{5} + 1$ $C_{20}$ (as 20T1) $0$ $-1$
* 1.25.10t1.a.c$1$ $ 5^{2}$ $x^{10} - 10 x^{8} + 35 x^{6} - x^{5} - 50 x^{4} + 5 x^{3} + 25 x^{2} - 5 x - 1$ $C_{10}$ (as 10T1) $0$ $1$
* 1.25.20t1.a.a$1$ $ 5^{2}$ $x^{20} - x^{15} + x^{10} - x^{5} + 1$ $C_{20}$ (as 20T1) $0$ $-1$
* 1.25.5t1.a.c$1$ $ 5^{2}$ $x^{5} - 10 x^{3} - 5 x^{2} + 10 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.25.20t1.a.e$1$ $ 5^{2}$ $x^{20} - x^{15} + x^{10} - x^{5} + 1$ $C_{20}$ (as 20T1) $0$ $-1$
* 1.25.10t1.a.b$1$ $ 5^{2}$ $x^{10} - 10 x^{8} + 35 x^{6} - x^{5} - 50 x^{4} + 5 x^{3} + 25 x^{2} - 5 x - 1$ $C_{10}$ (as 10T1) $0$ $1$
* 1.25.20t1.a.d$1$ $ 5^{2}$ $x^{20} - x^{15} + x^{10} - x^{5} + 1$ $C_{20}$ (as 20T1) $0$ $-1$
* 1.25.5t1.a.d$1$ $ 5^{2}$ $x^{5} - 10 x^{3} - 5 x^{2} + 10 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.25.20t1.a.b$1$ $ 5^{2}$ $x^{20} - x^{15} + x^{10} - x^{5} + 1$ $C_{20}$ (as 20T1) $0$ $-1$
* 1.25.10t1.a.a$1$ $ 5^{2}$ $x^{10} - 10 x^{8} + 35 x^{6} - x^{5} - 50 x^{4} + 5 x^{3} + 25 x^{2} - 5 x - 1$ $C_{10}$ (as 10T1) $0$ $1$
* 1.25.20t1.a.g$1$ $ 5^{2}$ $x^{20} - x^{15} + x^{10} - x^{5} + 1$ $C_{20}$ (as 20T1) $0$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.