Properties

Label 20.4.24157770159...1376.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 3^{24}\cdot 13^{8}$
Root discriminant $20.85$
Ramified primes $2, 3, 13$
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, -2, 0, -3, -24, -36, -48, -30, 72, 156, 72, -30, -48, -36, -24, -3, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^18 - 3*x^16 - 24*x^15 - 36*x^14 - 48*x^13 - 30*x^12 + 72*x^11 + 156*x^10 + 72*x^9 - 30*x^8 - 48*x^7 - 36*x^6 - 24*x^5 - 3*x^4 - 2*x^2 + 1)
 
gp: K = bnfinit(x^20 - 2*x^18 - 3*x^16 - 24*x^15 - 36*x^14 - 48*x^13 - 30*x^12 + 72*x^11 + 156*x^10 + 72*x^9 - 30*x^8 - 48*x^7 - 36*x^6 - 24*x^5 - 3*x^4 - 2*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{18} - 3 x^{16} - 24 x^{15} - 36 x^{14} - 48 x^{13} - 30 x^{12} + 72 x^{11} + 156 x^{10} + 72 x^{9} - 30 x^{8} - 48 x^{7} - 36 x^{6} - 24 x^{5} - 3 x^{4} - 2 x^{2} + 1 \)

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:  \(241577701592627342528741376=2^{20}\cdot 3^{24}\cdot 13^{8}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.85$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{104} a^{16} - \frac{3}{104} a^{15} + \frac{3}{52} a^{14} - \frac{5}{104} a^{13} - \frac{3}{26} a^{12} - \frac{11}{104} a^{11} - \frac{1}{13} a^{10} + \frac{15}{104} a^{9} - \frac{11}{52} a^{8} + \frac{15}{104} a^{7} + \frac{9}{52} a^{6} + \frac{41}{104} a^{5} + \frac{5}{13} a^{4} + \frac{47}{104} a^{3} + \frac{4}{13} a^{2} - \frac{3}{104} a + \frac{1}{104}$, $\frac{1}{104} a^{17} - \frac{3}{104} a^{15} - \frac{1}{104} a^{13} - \frac{1}{13} a^{12} + \frac{11}{104} a^{11} + \frac{1}{26} a^{10} + \frac{23}{104} a^{9} + \frac{7}{52} a^{8} + \frac{11}{104} a^{7} + \frac{1}{26} a^{6} + \frac{33}{104} a^{5} + \frac{3}{13} a^{4} - \frac{35}{104} a^{3} - \frac{3}{13} a^{2} - \frac{1}{13} a + \frac{21}{52}$, $\frac{1}{624} a^{18} - \frac{1}{208} a^{16} + \frac{1}{52} a^{14} - \frac{5}{52} a^{13} + \frac{1}{26} a^{12} - \frac{1}{13} a^{11} - \frac{7}{104} a^{10} - \frac{1}{52} a^{9} - \frac{9}{104} a^{8} - \frac{1}{13} a^{7} + \frac{3}{26} a^{6} + \frac{15}{52} a^{5} + \frac{9}{52} a^{4} - \frac{1}{26} a^{3} + \frac{71}{208} a^{2} - \frac{3}{52} a - \frac{23}{48}$, $\frac{1}{624} a^{19} - \frac{1}{208} a^{17} + \frac{1}{52} a^{15} + \frac{3}{104} a^{14} + \frac{1}{26} a^{13} + \frac{5}{104} a^{12} - \frac{7}{104} a^{11} + \frac{11}{104} a^{10} + \frac{17}{104} a^{9} + \frac{5}{104} a^{8} + \frac{3}{26} a^{7} + \frac{17}{104} a^{6} + \frac{9}{52} a^{5} - \frac{17}{104} a^{4} + \frac{71}{208} a^{3} - \frac{19}{104} a^{2} + \frac{13}{48} a - \frac{1}{8}$

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:  \( 851623.989731 \)
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

10.2.15542770074624.1

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

Sibling fields

Degree 6 siblings: 6.2.7884864.1, 6.2.1971216.1
Degree 10 sibling: 10.2.15542770074624.1
Degree 15 siblings: Deg 15, 15.3.30638157055420022784.1
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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.4.5$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
2.4.4.5$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
2.12.12.28$x^{12} - x^{10} + 2 x^{8} - x^{6} - 2 x^{4} + 3 x^{2} + 1$$6$$2$$12$$S_4$$[4/3, 4/3]_{3}^{2}$
3Data not computed
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$