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Properties

Label 20.10.4780633147...9375.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,5^{15}\cdot 11^{5}\cdot 9931^{4}$
Root discriminant $38.37$
Ramified primes $5, 11, 9931$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-31, 79, 3870, -804, -14062, 2854, 15301, -2908, -5131, -2284, -763, 2960, 1109, -965, -297, 68, 23, 27, -3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 3*x^18 + 27*x^17 + 23*x^16 + 68*x^15 - 297*x^14 - 965*x^13 + 1109*x^12 + 2960*x^11 - 763*x^10 - 2284*x^9 - 5131*x^8 - 2908*x^7 + 15301*x^6 + 2854*x^5 - 14062*x^4 - 804*x^3 + 3870*x^2 + 79*x - 31)
 
gp: K = bnfinit(x^20 - 5*x^19 - 3*x^18 + 27*x^17 + 23*x^16 + 68*x^15 - 297*x^14 - 965*x^13 + 1109*x^12 + 2960*x^11 - 763*x^10 - 2284*x^9 - 5131*x^8 - 2908*x^7 + 15301*x^6 + 2854*x^5 - 14062*x^4 - 804*x^3 + 3870*x^2 + 79*x - 31, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 3 x^{18} + 27 x^{17} + 23 x^{16} + 68 x^{15} - 297 x^{14} - 965 x^{13} + 1109 x^{12} + 2960 x^{11} - 763 x^{10} - 2284 x^{9} - 5131 x^{8} - 2908 x^{7} + 15301 x^{6} + 2854 x^{5} - 14062 x^{4} - 804 x^{3} + 3870 x^{2} + 79 x - 31 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-47806331471833622563812255859375=-\,5^{15}\cdot 11^{5}\cdot 9931^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.37$
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:  $5, 11, 9931$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{13030812211238124594312890262337341031} a^{19} + \frac{956257243796586503645779226970612390}{13030812211238124594312890262337341031} a^{18} - \frac{1364451088050318096685805550568211288}{13030812211238124594312890262337341031} a^{17} - \frac{3139911039461279894510257718681228938}{13030812211238124594312890262337341031} a^{16} + \frac{3488036730166101752334645696563215298}{13030812211238124594312890262337341031} a^{15} + \frac{4442576031411143049833456917368117860}{13030812211238124594312890262337341031} a^{14} - \frac{107812907872197175385594869278190352}{420348781007681438526222266527011001} a^{13} - \frac{5954430986372797325030611669100391892}{13030812211238124594312890262337341031} a^{12} + \frac{5033442829745084741970715178350381809}{13030812211238124594312890262337341031} a^{11} - \frac{3497159546521979316841556124838244599}{13030812211238124594312890262337341031} a^{10} + \frac{3933000113038527254537518668026349531}{13030812211238124594312890262337341031} a^{9} - \frac{1148471986576006164388675094683755572}{13030812211238124594312890262337341031} a^{8} + \frac{1903472295221870487097400844765388410}{13030812211238124594312890262337341031} a^{7} + \frac{2681141675250792366118435841811474861}{13030812211238124594312890262337341031} a^{6} - \frac{6421874679477255586372244740587868555}{13030812211238124594312890262337341031} a^{5} + \frac{1050015020207780072002661121296257346}{13030812211238124594312890262337341031} a^{4} - \frac{2998095889341052747059043945945671141}{13030812211238124594312890262337341031} a^{3} - \frac{1194400802562423567794425650415777464}{13030812211238124594312890262337341031} a^{2} - \frac{899823231732351777811936670769585214}{13030812211238124594312890262337341031} a - \frac{124202995186426982570768302915705139}{420348781007681438526222266527011001}$

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:  $14$
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:  \( 354389907.753 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

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

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
9931Data not computed