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Properties

Label 20.12.2016049563...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{16}\cdot 6029^{4}$
Root discriminant $20.66$
Ramified primes $5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T279

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 15, -85, 235, -320, 75, 545, -1085, 880, 140, -1151, 1120, -185, -470, 380, -50, -85, 45, 0, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 45*x^17 - 85*x^16 - 50*x^15 + 380*x^14 - 470*x^13 - 185*x^12 + 1120*x^11 - 1151*x^10 + 140*x^9 + 880*x^8 - 1085*x^7 + 545*x^6 + 75*x^5 - 320*x^4 + 235*x^3 - 85*x^2 + 15*x - 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 45*x^17 - 85*x^16 - 50*x^15 + 380*x^14 - 470*x^13 - 185*x^12 + 1120*x^11 - 1151*x^10 + 140*x^9 + 880*x^8 - 1085*x^7 + 545*x^6 + 75*x^5 - 320*x^4 + 235*x^3 - 85*x^2 + 15*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 45 x^{17} - 85 x^{16} - 50 x^{15} + 380 x^{14} - 470 x^{13} - 185 x^{12} + 1120 x^{11} - 1151 x^{10} + 140 x^{9} + 880 x^{8} - 1085 x^{7} + 545 x^{6} + 75 x^{5} - 320 x^{4} + 235 x^{3} - 85 x^{2} + 15 x - 1 \)

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:  \(201604956366467437744140625=5^{16}\cdot 6029^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.66$
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:  $5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{10} + \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{10} + \frac{2}{5} a^{4} + \frac{2}{5}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{11} + \frac{2}{5} a^{5} + \frac{2}{5} a$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{10} + \frac{2}{5} a^{6} + \frac{1}{5}$, $\frac{1}{35} a^{17} + \frac{1}{35} a^{14} - \frac{3}{35} a^{13} - \frac{2}{35} a^{12} - \frac{3}{35} a^{11} - \frac{3}{35} a^{10} - \frac{3}{7} a^{8} + \frac{12}{35} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{8}{35} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{4}{35} a - \frac{11}{35}$, $\frac{1}{35} a^{18} + \frac{1}{35} a^{15} - \frac{3}{35} a^{14} - \frac{2}{35} a^{13} - \frac{3}{35} a^{12} - \frac{3}{35} a^{11} - \frac{3}{7} a^{9} + \frac{12}{35} a^{8} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} - \frac{8}{35} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{4}{35} a^{2} - \frac{11}{35} a$, $\frac{1}{2509366895} a^{19} + \frac{7098332}{501873379} a^{18} + \frac{539573}{71696197} a^{17} - \frac{49607289}{501873379} a^{16} + \frac{25242418}{2509366895} a^{15} + \frac{39830013}{501873379} a^{14} - \frac{110346682}{2509366895} a^{13} + \frac{226024114}{2509366895} a^{12} + \frac{371341583}{2509366895} a^{11} + \frac{163487673}{2509366895} a^{10} + \frac{70388392}{2509366895} a^{9} - \frac{83497674}{501873379} a^{8} - \frac{9973633}{501873379} a^{7} - \frac{4820316}{10678157} a^{6} - \frac{375000564}{2509366895} a^{5} + \frac{13586058}{71696197} a^{4} - \frac{901715629}{2509366895} a^{3} + \frac{119206679}{358480985} a^{2} + \frac{408013161}{2509366895} a - \frac{106012827}{358480985}$

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:  \( 684074.076253 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T279:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n279
Character table for t20n279 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1, 10.6.14198766015625.1, 10.6.2839753203125.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
6029Data not computed