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)

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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)
 

Normalized defining 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]~
 
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

10.2.11284439629824.1

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type 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$$[8/3, 8/3]_{3}^{2}$
2.4.8.7$x^{4} + 4 x^{2} + 4 x + 2$$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