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Properties

Label 20.2.14113234290...1603.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,13^{12}\cdot 347^{7}$
Root discriminant $36.10$
Ramified primes $13, 347$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T796

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-12143, -52719, -104799, 52247, -163122, 215582, -227470, 167166, -154411, 92168, -49524, 22404, -12106, 5613, -2261, 625, -73, -20, 18, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 18*x^18 - 20*x^17 - 73*x^16 + 625*x^15 - 2261*x^14 + 5613*x^13 - 12106*x^12 + 22404*x^11 - 49524*x^10 + 92168*x^9 - 154411*x^8 + 167166*x^7 - 227470*x^6 + 215582*x^5 - 163122*x^4 + 52247*x^3 - 104799*x^2 - 52719*x - 12143)
 
gp: K = bnfinit(x^20 - 5*x^19 + 18*x^18 - 20*x^17 - 73*x^16 + 625*x^15 - 2261*x^14 + 5613*x^13 - 12106*x^12 + 22404*x^11 - 49524*x^10 + 92168*x^9 - 154411*x^8 + 167166*x^7 - 227470*x^6 + 215582*x^5 - 163122*x^4 + 52247*x^3 - 104799*x^2 - 52719*x - 12143, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 18 x^{18} - 20 x^{17} - 73 x^{16} + 625 x^{15} - 2261 x^{14} + 5613 x^{13} - 12106 x^{12} + 22404 x^{11} - 49524 x^{10} + 92168 x^{9} - 154411 x^{8} + 167166 x^{7} - 227470 x^{6} + 215582 x^{5} - 163122 x^{4} + 52247 x^{3} - 104799 x^{2} - 52719 x - 12143 \)

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:  \(-14113234290632913459536084681603=-\,13^{12}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.10$
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:  $13, 347$
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}{10194419456160577581637546004515799139813242524482583403167} a^{19} - \frac{3772186180899943497165888495030650545454886871594424627123}{10194419456160577581637546004515799139813242524482583403167} a^{18} - \frac{1559352640867694368347000849376777002212966171307252269849}{10194419456160577581637546004515799139813242524482583403167} a^{17} - \frac{203446088833320029988794530747007242616841297317272415616}{10194419456160577581637546004515799139813242524482583403167} a^{16} - \frac{749944954351963597945576284840908744840430135249119287854}{10194419456160577581637546004515799139813242524482583403167} a^{15} + \frac{3593522226359458503236250098584983427251093629799104466190}{10194419456160577581637546004515799139813242524482583403167} a^{14} - \frac{440562935080056230218401119614127144424585414006086627619}{10194419456160577581637546004515799139813242524482583403167} a^{13} + \frac{623003066225939990712689863467983795497849518750394050378}{10194419456160577581637546004515799139813242524482583403167} a^{12} + \frac{2926495960840544332327373191389869488497400124405071984850}{10194419456160577581637546004515799139813242524482583403167} a^{11} - \frac{19488562826021819172459741116356708679999658155730008100}{42654474711968943856224041859898741170766705123358089553} a^{10} - \frac{3825838858943367878164447821144633732221081909744745917679}{10194419456160577581637546004515799139813242524482583403167} a^{9} + \frac{1014064126483744002036441793939834298313785344603830336384}{10194419456160577581637546004515799139813242524482583403167} a^{8} - \frac{239145238202104579697731016706848283640637595628265855824}{10194419456160577581637546004515799139813242524482583403167} a^{7} - \frac{2146419984971169524146140561036448915055306951324669743382}{10194419456160577581637546004515799139813242524482583403167} a^{6} + \frac{3002801937472923021184806351935586778561986993139948811108}{10194419456160577581637546004515799139813242524482583403167} a^{5} + \frac{829786397837991436046952919585133120169042026510300664812}{10194419456160577581637546004515799139813242524482583403167} a^{4} - \frac{826528772009027037868109893491816472538876658592647426662}{10194419456160577581637546004515799139813242524482583403167} a^{3} - \frac{3043721922609410685361798332353615566441904975972287359068}{10194419456160577581637546004515799139813242524482583403167} a^{2} - \frac{2071799092437709896853296138708527560487197380870310532107}{10194419456160577581637546004515799139813242524482583403167} a + \frac{571037381962736252784424690959924919505370760137915537535}{10194419456160577581637546004515799139813242524482583403167}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 5951038.00917 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

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 t20n796 are not computed
Character table for t20n796 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$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}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
347Data not computed