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Properties

Label 20.0.67050942866...0032.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 2297^{5}$
Root discriminant $13.85$
Ramified primes $2, 2297$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 16, -12, -31, 88, -49, -88, 174, -48, -99, 134, -16, -6, 51, -38, -3, 18, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 4*x^18 + 18*x^17 - 3*x^16 - 38*x^15 + 51*x^14 - 6*x^13 - 16*x^12 + 134*x^11 - 99*x^10 - 48*x^9 + 174*x^8 - 88*x^7 - 49*x^6 + 88*x^5 - 31*x^4 - 12*x^3 + 16*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 4*x^18 + 18*x^17 - 3*x^16 - 38*x^15 + 51*x^14 - 6*x^13 - 16*x^12 + 134*x^11 - 99*x^10 - 48*x^9 + 174*x^8 - 88*x^7 - 49*x^6 + 88*x^5 - 31*x^4 - 12*x^3 + 16*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 4 x^{18} + 18 x^{17} - 3 x^{16} - 38 x^{15} + 51 x^{14} - 6 x^{13} - 16 x^{12} + 134 x^{11} - 99 x^{10} - 48 x^{9} + 174 x^{8} - 88 x^{7} - 49 x^{6} + 88 x^{5} - 31 x^{4} - 12 x^{3} + 16 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(67050942866528672940032=2^{20}\cdot 2297^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.85$
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:  $2, 2297$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{8596213028145} a^{19} + \frac{411868793275}{1719242605629} a^{18} - \frac{3964213145039}{8596213028145} a^{17} - \frac{710491935979}{1719242605629} a^{16} + \frac{1211669798032}{8596213028145} a^{15} + \frac{2069881583516}{8596213028145} a^{14} - \frac{975851317282}{8596213028145} a^{13} + \frac{15849810601}{1719242605629} a^{12} + \frac{2973746555434}{8596213028145} a^{11} + \frac{23148559438}{955134780905} a^{10} - \frac{15422372146}{191026956181} a^{9} - \frac{749854713436}{2865404342715} a^{8} + \frac{97711738432}{955134780905} a^{7} - \frac{4186727716777}{8596213028145} a^{6} + \frac{1818167251687}{8596213028145} a^{5} - \frac{2814844703323}{8596213028145} a^{4} - \frac{3078222854867}{8596213028145} a^{3} + \frac{2920405287184}{8596213028145} a^{2} + \frac{1387213454174}{8596213028145} a + \frac{2472326246492}{8596213028145}$

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:  \( \frac{100132199}{241432749} a^{19} - \frac{111907934}{241432749} a^{18} - \frac{548894866}{241432749} a^{17} + \frac{1433850899}{241432749} a^{16} + \frac{1067853653}{241432749} a^{15} - \frac{3667085162}{241432749} a^{14} + \frac{2476946866}{241432749} a^{13} + \frac{2282112685}{241432749} a^{12} - \frac{2022803362}{241432749} a^{11} + \frac{1604948974}{26825861} a^{10} + \frac{14578080}{26825861} a^{9} - \frac{3113086187}{80477583} a^{8} + \frac{1631544648}{26825861} a^{7} - \frac{1048890809}{241432749} a^{6} - \frac{7050280993}{241432749} a^{5} + \frac{7923720370}{241432749} a^{4} - \frac{905807953}{241432749} a^{3} - \frac{1779919063}{241432749} a^{2} + \frac{1856784145}{241432749} a - \frac{437415623}{241432749} \) (order $4$)
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:  \( 2467.78725248 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.1.2297.1, 10.0.5402838016.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2297Data not computed