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Properties

Label 20.6.21800592438...0000.2
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{40}\cdot 5^{11}\cdot 67^{8}$
Root discriminant $52.11$
Ramified primes $2, 5, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4276, -28260, 72506, -108140, 104855, -110860, 181220, -275856, 296708, -210004, 92638, -20296, -3001, 4176, -1764, 600, -158, 12, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 6*x^18 + 12*x^17 - 158*x^16 + 600*x^15 - 1764*x^14 + 4176*x^13 - 3001*x^12 - 20296*x^11 + 92638*x^10 - 210004*x^9 + 296708*x^8 - 275856*x^7 + 181220*x^6 - 110860*x^5 + 104855*x^4 - 108140*x^3 + 72506*x^2 - 28260*x + 4276)
 
gp: K = bnfinit(x^20 - 4*x^19 + 6*x^18 + 12*x^17 - 158*x^16 + 600*x^15 - 1764*x^14 + 4176*x^13 - 3001*x^12 - 20296*x^11 + 92638*x^10 - 210004*x^9 + 296708*x^8 - 275856*x^7 + 181220*x^6 - 110860*x^5 + 104855*x^4 - 108140*x^3 + 72506*x^2 - 28260*x + 4276, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 6 x^{18} + 12 x^{17} - 158 x^{16} + 600 x^{15} - 1764 x^{14} + 4176 x^{13} - 3001 x^{12} - 20296 x^{11} + 92638 x^{10} - 210004 x^{9} + 296708 x^{8} - 275856 x^{7} + 181220 x^{6} - 110860 x^{5} + 104855 x^{4} - 108140 x^{3} + 72506 x^{2} - 28260 x + 4276 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-21800592438355578532659200000000000=-\,2^{40}\cdot 5^{11}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{21895218576504205483573911087320735770823744858} a^{19} + \frac{2063863760322833850380964011718509612535980748}{10947609288252102741786955543660367885411872429} a^{18} - \frac{4356804263816202958884754141266157189216692826}{10947609288252102741786955543660367885411872429} a^{17} - \frac{105114973717948496430591460217990529958846083}{10947609288252102741786955543660367885411872429} a^{16} - \frac{1378341329005523530628603965651059456342756144}{10947609288252102741786955543660367885411872429} a^{15} - \frac{3233967878114793494383243358027193940918249809}{10947609288252102741786955543660367885411872429} a^{14} + \frac{2767708425997458589665210513594387748740336961}{10947609288252102741786955543660367885411872429} a^{13} + \frac{4814480974872380794217723605614109145349946615}{10947609288252102741786955543660367885411872429} a^{12} + \frac{6475603827175620167375004160871175176918986781}{21895218576504205483573911087320735770823744858} a^{11} - \frac{5020919392491103178610207545215692006345511707}{10947609288252102741786955543660367885411872429} a^{10} + \frac{4516252966818517937628725455380353622007620579}{10947609288252102741786955543660367885411872429} a^{9} + \frac{2978736651211322897636600013927969458693365476}{10947609288252102741786955543660367885411872429} a^{8} + \frac{242176782530193203130820265773932674270055966}{10947609288252102741786955543660367885411872429} a^{7} - \frac{1594711631220731476794957190731160633437579338}{10947609288252102741786955543660367885411872429} a^{6} + \frac{5332649116494146143474715733964149095624124243}{10947609288252102741786955543660367885411872429} a^{5} + \frac{3095948951511844732470908751545153781125606714}{10947609288252102741786955543660367885411872429} a^{4} - \frac{10887205798826653170004917909253061334558261733}{21895218576504205483573911087320735770823744858} a^{3} + \frac{3081362336999567281347372429578323081023945843}{10947609288252102741786955543660367885411872429} a^{2} + \frac{477584028864466045322719144861781879714434696}{995237208022918431071541413060033444128352039} a - \frac{1807814010124231649396936673407716873802443013}{10947609288252102741786955543660367885411872429}$

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:  $12$
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:  \( 8200675031.25 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

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

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

Sibling fields

Degree 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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$67$67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
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}$