Properties

 Label 20.4.12733857775...0976.1 Degree $20$ Signature $[4, 8]$ Discriminant $2^{36}\cdot 3^{32}$ Root discriminant $20.20$ Ramified primes $2, 3$ Class number $1$ Class group Trivial Galois group $A_6$ (as 20T89)

Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -14, 0, 45, 0, -60, 0, 30, 0, 0, 0, -6, 0, 12, 0, 9, 0, -2, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^18 + 9*x^16 + 12*x^14 - 6*x^12 + 30*x^8 - 60*x^6 + 45*x^4 - 14*x^2 + 1)

gp: K = bnfinit(x^20 - 2*x^18 + 9*x^16 + 12*x^14 - 6*x^12 + 30*x^8 - 60*x^6 + 45*x^4 - 14*x^2 + 1, 1)

Normalizeddefining polynomial

$$x^{20} - 2 x^{18} + 9 x^{16} + 12 x^{14} - 6 x^{12} + 30 x^{8} - 60 x^{6} + 45 x^{4} - 14 x^{2} + 1$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

Invariants

 Degree: $20$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[4, 8]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$127338577759142414150270976=2^{36}\cdot 3^{32}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $20.20$ 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: $2, 3$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{16} a^{12} + \frac{3}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{7}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{16}$, $\frac{1}{16} a^{13} + \frac{3}{16} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{7}{16} a^{5} + \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{16} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{16} a^{6} + \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{5}{16} a^{2} - \frac{1}{4}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{11} + \frac{1}{8} a^{9} + \frac{1}{16} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{5}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{32} a^{16} - \frac{1}{16} a^{8} - \frac{1}{8} a^{6} + \frac{1}{4} a^{4} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{15}{32}$, $\frac{1}{32} a^{17} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{4} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{15}{32} a$, $\frac{1}{96} a^{18} - \frac{1}{16} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{9}{32} a^{2} - \frac{1}{2} a - \frac{11}{24}$, $\frac{1}{96} a^{19} - \frac{1}{16} a^{11} - \frac{1}{4} a^{7} - \frac{3}{8} a^{5} - \frac{9}{32} a^{3} - \frac{11}{24} a - \frac{1}{2}$

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: $11$ magma: UnitRank(K);  sage: UK.rank()  gp: K.fu Torsion generator: $$-1$$ (order $2$) 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 magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$616938.09204$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

Galois group

$A_6$ (as 20T89):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A non-solvable group of order 360 The 7 conjugacy class representatives for $A_6$ Character table for $A_6$

Intermediate fields

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

Sibling fields

 Degree 6 siblings: 6.2.1679616.2, 6.2.6718464.1 Degree 10 sibling: 10.2.11284439629824.1 Degree 15 siblings: Deg 15, 15.3.18953525353286467584.1 Degree 30 siblings: data not computed Degree 36 sibling: data not computed Degree 40 sibling: data not computed Degree 45 sibling: data not computed

Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.4.8.7$x^{4} + 4 x^{2} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2} 2.4.8.7x^{4} + 4 x^{2} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
2.12.20.34$x^{12} + 14 x^{10} + 16 x^{8} - 8 x^{6} - 8 x^{4} + 16 x^{2} + 16$$6$$2$$20$$S_4$$[8/3, 8/3]_{3}^{2}$
3Data not computed