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Properties

Label 20.12.6364883959...1904.3
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 13^{15}\cdot 17^{9}$
Root discriminant $49.00$
Ramified primes $2, 13, 17$
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![1439, 392, -12923, 5396, 36271, -28416, -38038, 43582, 9753, -29184, 8537, 7738, -6317, 448, 1376, -614, -39, 98, -17, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 17*x^18 + 98*x^17 - 39*x^16 - 614*x^15 + 1376*x^14 + 448*x^13 - 6317*x^12 + 7738*x^11 + 8537*x^10 - 29184*x^9 + 9753*x^8 + 43582*x^7 - 38038*x^6 - 28416*x^5 + 36271*x^4 + 5396*x^3 - 12923*x^2 + 392*x + 1439)
 
gp: K = bnfinit(x^20 - 4*x^19 - 17*x^18 + 98*x^17 - 39*x^16 - 614*x^15 + 1376*x^14 + 448*x^13 - 6317*x^12 + 7738*x^11 + 8537*x^10 - 29184*x^9 + 9753*x^8 + 43582*x^7 - 38038*x^6 - 28416*x^5 + 36271*x^4 + 5396*x^3 - 12923*x^2 + 392*x + 1439, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 17 x^{18} + 98 x^{17} - 39 x^{16} - 614 x^{15} + 1376 x^{14} + 448 x^{13} - 6317 x^{12} + 7738 x^{11} + 8537 x^{10} - 29184 x^{9} + 9753 x^{8} + 43582 x^{7} - 38038 x^{6} - 28416 x^{5} + 36271 x^{4} + 5396 x^{3} - 12923 x^{2} + 392 x + 1439 \)

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:  \(6364883959565411703183364041211904=2^{20}\cdot 13^{15}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.00$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{17578379456839225352263202641950499679} a^{19} - \frac{308665763119330640503038159612093720}{17578379456839225352263202641950499679} a^{18} - \frac{1514332949131844828500088895116405734}{17578379456839225352263202641950499679} a^{17} - \frac{6399271106793043268091311394867630922}{17578379456839225352263202641950499679} a^{16} + \frac{1987760355650090542287413876065410607}{17578379456839225352263202641950499679} a^{15} - \frac{7957305934208797296646938200780828152}{17578379456839225352263202641950499679} a^{14} + \frac{3732146448790639772382232514940160761}{17578379456839225352263202641950499679} a^{13} - \frac{7467637195447021041472402065003935264}{17578379456839225352263202641950499679} a^{12} - \frac{5737395535500465839883434459043380114}{17578379456839225352263202641950499679} a^{11} + \frac{7791435366857679400634802413567738912}{17578379456839225352263202641950499679} a^{10} - \frac{5802552525209724455453795428632735169}{17578379456839225352263202641950499679} a^{9} - \frac{7518495947816299316688022221320691372}{17578379456839225352263202641950499679} a^{8} - \frac{393269186021997986722755872204279768}{17578379456839225352263202641950499679} a^{7} - \frac{8728293982530842046631662806852728848}{17578379456839225352263202641950499679} a^{6} - \frac{3113630414155618714066820394129323279}{17578379456839225352263202641950499679} a^{5} - \frac{6150765547716637706466919887451858249}{17578379456839225352263202641950499679} a^{4} - \frac{7555348956190981408203756369580671388}{17578379456839225352263202641950499679} a^{3} - \frac{4562044163386629133813907965969576388}{17578379456839225352263202641950499679} a^{2} + \frac{108827219750496528930649104503984603}{297938634861681785631579705795771181} a + \frac{4772553736314074691922016290515817315}{17578379456839225352263202641950499679}$

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:  \( 7804560083.16 \) (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

\(\Q(\sqrt{13}) \), 5.5.10158928.1, 10.10.1341649635419392.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.12.11.6$x^{12} - 13312$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.6.5.2$x^{6} + 51$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$