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Properties

Label 20.12.1703171284...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{32}\cdot 5^{10}\cdot 67^{8}$
Root discriminant $36.44$
Ramified primes $2, 5, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T226

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, -240, 0, 736, 0, 720, 0, -2468, 0, -132, 0, 1424, 0, -156, 0, -139, 0, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^18 - 139*x^16 - 156*x^14 + 1424*x^12 - 132*x^10 - 2468*x^8 + 720*x^6 + 736*x^4 - 240*x^2 + 16)
 
gp: K = bnfinit(x^20 - 3*x^18 - 139*x^16 - 156*x^14 + 1424*x^12 - 132*x^10 - 2468*x^8 + 720*x^6 + 736*x^4 - 240*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{18} - 139 x^{16} - 156 x^{14} + 1424 x^{12} - 132 x^{10} - 2468 x^{8} + 720 x^{6} + 736 x^{4} - 240 x^{2} + 16 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17031712842465295728640000000000=2^{32}\cdot 5^{10}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.44$
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:  $2, 5, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{16} a^{12} - \frac{1}{4} a^{11} - \frac{3}{16} a^{10} - \frac{1}{4} a^{9} + \frac{3}{16} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{16} a^{13} - \frac{3}{16} a^{11} + \frac{3}{16} a^{9} + \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{14} - \frac{1}{4} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{3}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{13} - \frac{3}{32} a^{11} - \frac{1}{4} a^{10} - \frac{3}{16} a^{7} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a$, $\frac{1}{160} a^{16} + \frac{1}{160} a^{14} + \frac{1}{160} a^{12} + \frac{3}{20} a^{10} - \frac{1}{4} a^{9} - \frac{1}{40} a^{8} - \frac{1}{4} a^{7} + \frac{1}{10} a^{6} - \frac{1}{4} a^{5} - \frac{3}{10} a^{4} + \frac{3}{20} a^{2} - \frac{1}{2} a - \frac{7}{20}$, $\frac{1}{160} a^{17} + \frac{1}{160} a^{15} + \frac{1}{160} a^{13} + \frac{3}{20} a^{11} - \frac{1}{4} a^{10} - \frac{1}{40} a^{9} - \frac{1}{4} a^{8} + \frac{1}{10} a^{7} - \frac{1}{4} a^{6} - \frac{3}{10} a^{5} + \frac{3}{20} a^{3} - \frac{1}{2} a^{2} - \frac{7}{20} a$, $\frac{1}{298831020320} a^{18} - \frac{404761153}{298831020320} a^{16} + \frac{524512007}{298831020320} a^{14} + \frac{218372671}{29883102032} a^{12} - \frac{1}{4} a^{11} - \frac{1166532869}{14941551016} a^{10} - \frac{1}{4} a^{9} + \frac{225278303}{1334067055} a^{8} - \frac{1}{4} a^{7} - \frac{2225608771}{9338469385} a^{6} - \frac{11904948793}{37353877540} a^{4} - \frac{1}{2} a^{3} - \frac{15264633409}{37353877540} a^{2} + \frac{4202088119}{18676938770}$, $\frac{1}{597662040640} a^{19} - \frac{404761153}{597662040640} a^{17} + \frac{524512007}{597662040640} a^{15} + \frac{218372671}{59766204064} a^{13} + \frac{6304242639}{29883102032} a^{11} + \frac{225278303}{2668134110} a^{9} + \frac{4887251843}{37353877540} a^{7} + \frac{6771989977}{74707755080} a^{5} - \frac{15264633409}{74707755080} a^{3} - \frac{14474850651}{37353877540} a - \frac{1}{2}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 769493539.25 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T226:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 18 conjugacy class representatives for t20n226
Character table for t20n226

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.5745920.1, 10.6.4126949580800000.1, 10.6.33015596646400.1, 10.10.4126949580800000.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 sibling: data not computed
Degree 40 siblings: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.16.8$x^{8} + 8 x^{5} + 12$$4$$2$$16$$S_4$$[8/3, 8/3]_{3}^{2}$
2.8.16.8$x^{8} + 8 x^{5} + 12$$4$$2$$16$$S_4$$[8/3, 8/3]_{3}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.6.4.1$x^{6} + 2345 x^{3} + 7756992$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
67.6.4.1$x^{6} + 2345 x^{3} + 7756992$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$