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Properties

Label 20.0.71311236675...4176.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{42}\cdot 359\cdot 461^{4}$
Root discriminant $19.62$
Ramified primes $2, 359, 461$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T887

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![167, 528, 442, -492, -963, -216, 652, 376, -58, -200, 48, -48, 34, -40, 36, -24, 19, -8, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 6*x^18 - 8*x^17 + 19*x^16 - 24*x^15 + 36*x^14 - 40*x^13 + 34*x^12 - 48*x^11 + 48*x^10 - 200*x^9 - 58*x^8 + 376*x^7 + 652*x^6 - 216*x^5 - 963*x^4 - 492*x^3 + 442*x^2 + 528*x + 167)
 
gp: K = bnfinit(x^20 - 4*x^19 + 6*x^18 - 8*x^17 + 19*x^16 - 24*x^15 + 36*x^14 - 40*x^13 + 34*x^12 - 48*x^11 + 48*x^10 - 200*x^9 - 58*x^8 + 376*x^7 + 652*x^6 - 216*x^5 - 963*x^4 - 492*x^3 + 442*x^2 + 528*x + 167, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 6 x^{18} - 8 x^{17} + 19 x^{16} - 24 x^{15} + 36 x^{14} - 40 x^{13} + 34 x^{12} - 48 x^{11} + 48 x^{10} - 200 x^{9} - 58 x^{8} + 376 x^{7} + 652 x^{6} - 216 x^{5} - 963 x^{4} - 492 x^{3} + 442 x^{2} + 528 x + 167 \)

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:  \(71311236675536751140274176=2^{42}\cdot 359\cdot 461^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.62$
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:  $2, 359, 461$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{10} - \frac{1}{8} a^{6} + \frac{1}{8} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{11} - \frac{1}{8} a^{7} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{8} + \frac{1}{16}$, $\frac{1}{32} a^{17} - \frac{1}{32} a^{16} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{3}{16} a^{5} - \frac{3}{16} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{5}{32} a - \frac{7}{32}$, $\frac{1}{64} a^{18} - \frac{1}{64} a^{16} - \frac{1}{16} a^{15} + \frac{1}{32} a^{14} - \frac{1}{16} a^{13} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{16} a^{9} - \frac{3}{16} a^{7} + \frac{7}{32} a^{6} - \frac{3}{16} a^{5} + \frac{1}{32} a^{4} + \frac{5}{16} a^{3} - \frac{25}{64} a^{2} + \frac{5}{16} a + \frac{17}{64}$, $\frac{1}{171999663942272} a^{19} + \frac{368357158881}{171999663942272} a^{18} - \frac{1241065074529}{171999663942272} a^{17} + \frac{4540660108979}{171999663942272} a^{16} - \frac{492907409553}{85999831971136} a^{15} - \frac{1706885379625}{85999831971136} a^{14} - \frac{4575307327671}{85999831971136} a^{13} - \frac{3368807220471}{85999831971136} a^{12} + \frac{3535904634073}{42999915985568} a^{11} - \frac{2768693060603}{42999915985568} a^{10} - \frac{2628141595771}{42999915985568} a^{9} + \frac{5310090217239}{42999915985568} a^{8} + \frac{8636176424913}{85999831971136} a^{7} + \frac{19314539300121}{85999831971136} a^{6} + \frac{6152135642975}{85999831971136} a^{5} + \frac{16701983578143}{85999831971136} a^{4} + \frac{35691968971723}{171999663942272} a^{3} + \frac{18733169687643}{171999663942272} a^{2} - \frac{79285168317107}{171999663942272} a - \frac{32670992156943}{171999663942272}$

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:  $9$
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:  \( 57586.1354732 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T887:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 245760
The 201 conjugacy class representatives for t20n887 are not computed
Character table for t20n887 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.1.29504.1, 10.2.6963888128.1

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

Sibling fields

Degree 20 siblings: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
359Data not computed
461Data not computed