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Properties

Label 20.2.20562460332...4256.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{48}\cdot 31^{7}\cdot 227^{4}$
Root discriminant $51.96$
Ramified primes $2, 31, 227$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6862, -84952, 321288, -546072, 607404, -438328, 390516, -280712, 163695, -102224, 59722, -29364, 15179, -7024, 2910, -1156, 423, -128, 34, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 34*x^18 - 128*x^17 + 423*x^16 - 1156*x^15 + 2910*x^14 - 7024*x^13 + 15179*x^12 - 29364*x^11 + 59722*x^10 - 102224*x^9 + 163695*x^8 - 280712*x^7 + 390516*x^6 - 438328*x^5 + 607404*x^4 - 546072*x^3 + 321288*x^2 - 84952*x + 6862)
 
gp: K = bnfinit(x^20 - 8*x^19 + 34*x^18 - 128*x^17 + 423*x^16 - 1156*x^15 + 2910*x^14 - 7024*x^13 + 15179*x^12 - 29364*x^11 + 59722*x^10 - 102224*x^9 + 163695*x^8 - 280712*x^7 + 390516*x^6 - 438328*x^5 + 607404*x^4 - 546072*x^3 + 321288*x^2 - 84952*x + 6862, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 34 x^{18} - 128 x^{17} + 423 x^{16} - 1156 x^{15} + 2910 x^{14} - 7024 x^{13} + 15179 x^{12} - 29364 x^{11} + 59722 x^{10} - 102224 x^{9} + 163695 x^{8} - 280712 x^{7} + 390516 x^{6} - 438328 x^{5} + 607404 x^{4} - 546072 x^{3} + 321288 x^{2} - 84952 x + 6862 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-20562460332300807986799747065184256=-\,2^{48}\cdot 31^{7}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.96$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{17} - \frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{13977944058139753250169761753251739909753015131905579} a^{19} + \frac{242726592143906223663364812268030396848465137819522}{4659314686046584416723253917750579969917671710635193} a^{18} + \frac{1925446413536268041874289333427989866562620546619867}{13977944058139753250169761753251739909753015131905579} a^{17} + \frac{1747303337881703259015592037943682364299343183768368}{4659314686046584416723253917750579969917671710635193} a^{16} - \frac{1600702447251439227353486604778269001224501896817217}{4659314686046584416723253917750579969917671710635193} a^{15} + \frac{2784053600398772469124237590624529514738457200183141}{13977944058139753250169761753251739909753015131905579} a^{14} + \frac{3347645791423348354433185605753402732948572298001384}{13977944058139753250169761753251739909753015131905579} a^{13} - \frac{2285958787625343326085502094202329104144167686050166}{13977944058139753250169761753251739909753015131905579} a^{12} + \frac{6163233112452087496626338824377460149141735651148138}{13977944058139753250169761753251739909753015131905579} a^{11} + \frac{2982590573090644194287873647496417632345020573733544}{13977944058139753250169761753251739909753015131905579} a^{10} + \frac{2488999899154834160661154604072948173511241241244417}{13977944058139753250169761753251739909753015131905579} a^{9} - \frac{5081273195518375155572482945994660737435027405762063}{13977944058139753250169761753251739909753015131905579} a^{8} - \frac{6449351964183724267456453820489006771655671605617491}{13977944058139753250169761753251739909753015131905579} a^{7} + \frac{188612028307016571652928447351972483748671995747340}{517701631782953824080361546416731107768630190070577} a^{6} - \frac{1348776831894856728189564924721652813999551663633471}{4659314686046584416723253917750579969917671710635193} a^{5} - \frac{2461177133300284178249026151066567848773342809391712}{13977944058139753250169761753251739909753015131905579} a^{4} - \frac{4258831690329533630792658499162178201042456796510932}{13977944058139753250169761753251739909753015131905579} a^{3} + \frac{3150852933187199922059738584524665133678328739725361}{13977944058139753250169761753251739909753015131905579} a^{2} - \frac{6868704382057012533582161869576302566245512471108305}{13977944058139753250169761753251739909753015131905579} a + \frac{5312768863627789097458273820311556533375289291543020}{13977944058139753250169761753251739909753015131905579}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  $10$
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:  \( 269909515.523 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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 $16{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.12.24.342$x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 4 x^{5} - 2 x^{4} + 4 x^{3} - 2 x^{2} + 4 x - 2$$12$$1$$24$$C_2 \times S_4$$[4/3, 4/3, 3]_{3}^{2}$
31Data not computed
227Data not computed