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Properties

Label 20.12.6249008059...3125.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{11}\cdot 61^{6}\cdot 397^{4}$
Root discriminant $27.53$
Ramified primes $5, 61, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T802

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, 25, 265, 585, 459, -674, -2097, -1797, 364, 2279, 1942, -69, -1087, -550, 34, 113, 48, 7, -9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 9*x^18 + 7*x^17 + 48*x^16 + 113*x^15 + 34*x^14 - 550*x^13 - 1087*x^12 - 69*x^11 + 1942*x^10 + 2279*x^9 + 364*x^8 - 1797*x^7 - 2097*x^6 - 674*x^5 + 459*x^4 + 585*x^3 + 265*x^2 + 25*x - 5)
 
gp: K = bnfinit(x^20 - 3*x^19 - 9*x^18 + 7*x^17 + 48*x^16 + 113*x^15 + 34*x^14 - 550*x^13 - 1087*x^12 - 69*x^11 + 1942*x^10 + 2279*x^9 + 364*x^8 - 1797*x^7 - 2097*x^6 - 674*x^5 + 459*x^4 + 585*x^3 + 265*x^2 + 25*x - 5, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 9 x^{18} + 7 x^{17} + 48 x^{16} + 113 x^{15} + 34 x^{14} - 550 x^{13} - 1087 x^{12} - 69 x^{11} + 1942 x^{10} + 2279 x^{9} + 364 x^{8} - 1797 x^{7} - 2097 x^{6} - 674 x^{5} + 459 x^{4} + 585 x^{3} + 265 x^{2} + 25 x - 5 \)

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:  \(62490080598623484767626953125=5^{11}\cdot 61^{6}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.53$
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:  $5, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1343313825852576776693963} a^{19} + \frac{641281613074594449891984}{1343313825852576776693963} a^{18} + \frac{86578467742505486023005}{1343313825852576776693963} a^{17} - \frac{179210371142536692097312}{1343313825852576776693963} a^{16} - \frac{493105945784087017217532}{1343313825852576776693963} a^{15} + \frac{288429602305672535211271}{1343313825852576776693963} a^{14} + \frac{156074225428340341792094}{1343313825852576776693963} a^{13} - \frac{101004377933923894485388}{1343313825852576776693963} a^{12} - \frac{399810321828904664034594}{1343313825852576776693963} a^{11} - \frac{575006260714641270741103}{1343313825852576776693963} a^{10} - \frac{442971527025691621258602}{1343313825852576776693963} a^{9} - \frac{446479449981778279234673}{1343313825852576776693963} a^{8} - \frac{8477855092358877639984}{1343313825852576776693963} a^{7} - \frac{143392461936735522928145}{1343313825852576776693963} a^{6} + \frac{127515811837573807836794}{1343313825852576776693963} a^{5} - \frac{8459300481597336775849}{1343313825852576776693963} a^{4} + \frac{345100719660253966344261}{1343313825852576776693963} a^{3} - \frac{622689684955961418272866}{1343313825852576776693963} a^{2} + \frac{91192386422966712314788}{1343313825852576776693963} a + \frac{643074638335472799384479}{1343313825852576776693963}$

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:  \( 15818728.1514 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T802:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n802 are not computed
Character table for t20n802 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.24217.1, 10.10.1832697153125.1

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 $20$ $20$ R $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
61Data not computed
397Data not computed