Properties

Label 20.0.32830755744...1569.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{18}$
Root discriminant $14.99$
Ramified primes $3, 11$
Class number $1$
Class group Trivial
Galois Group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1)
gp: K = bnfinit(x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut x^{19} \) \(\mathstrut +\mathstrut x^{17} \) \(\mathstrut -\mathstrut x^{16} \) \(\mathstrut +\mathstrut x^{14} \) \(\mathstrut -\mathstrut x^{13} \) \(\mathstrut +\mathstrut x^{11} \) \(\mathstrut -\mathstrut x^{10} \) \(\mathstrut +\mathstrut x^{9} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut +\mathstrut x^{6} \) \(\mathstrut -\mathstrut x^{4} \) \(\mathstrut +\mathstrut x^{3} \) \(\mathstrut -\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 10]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(328307557444402776721569=3^{10}\cdot 11^{18}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $14.99$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 11$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois and abelian over $\Q$.
Conductor:  \(33=3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{33}(1,·)$, $\chi_{33}(2,·)$, $\chi_{33}(4,·)$, $\chi_{33}(5,·)$, $\chi_{33}(7,·)$, $\chi_{33}(8,·)$, $\chi_{33}(10,·)$, $\chi_{33}(13,·)$, $\chi_{33}(14,·)$, $\chi_{33}(16,·)$, $\chi_{33}(17,·)$, $\chi_{33}(19,·)$, $\chi_{33}(20,·)$, $\chi_{33}(23,·)$, $\chi_{33}(25,·)$, $\chi_{33}(26,·)$, $\chi_{33}(28,·)$, $\chi_{33}(29,·)$, $\chi_{33}(31,·)$, $\chi_{33}(32,·)$$\rbrace$
This is a CM field.

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

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

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $9$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( -a \) (order $66$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( a^{17} + a^{8} \),  \( a^{19} - a^{18} + a^{8} \),  \( a^{19} - a^{9} + a^{8} - a^{6} - a^{3} - 1 \),  \( a^{19} + a^{13} + a^{8} - a^{3} + a^{2} \),  \( a^{14} - a^{13} \),  \( a^{12} + a^{6} + a \),  \( a^{19} - a^{17} - a^{15} + a^{13} + a^{10} - a^{9} - a^{6} + a^{2} - 1 \),  \( a^{16} - a^{9} \),  \( a^{18} - a^{16} + a^{12} - a^{8} + a^{7} + a^{6} - a^{5} - a^{2} + a + 1 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 62791.3897584 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 10.0.52089208083.1, \(\Q(\zeta_{11})\), \(\Q(\zeta_{33})^+\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed