# 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)

# Related objects

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]);

## Normalizeddefining 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

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $20$ $20$ 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 *.