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Properties

Label 20.10.7663347760...0000.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 5^{10}\cdot 71^{4}\cdot 131^{4}$
Root discriminant $27.81$
Ramified primes $2, 5, 71, 131$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T299

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 24, -28, -200, 654, -205, -878, 504, 652, -339, -630, 336, 342, -239, -94, 75, 20, -13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 13*x^18 + 20*x^17 + 75*x^16 - 94*x^15 - 239*x^14 + 342*x^13 + 336*x^12 - 630*x^11 - 339*x^10 + 652*x^9 + 504*x^8 - 878*x^7 - 205*x^6 + 654*x^5 - 200*x^4 - 28*x^3 + 24*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 13*x^18 + 20*x^17 + 75*x^16 - 94*x^15 - 239*x^14 + 342*x^13 + 336*x^12 - 630*x^11 - 339*x^10 + 652*x^9 + 504*x^8 - 878*x^7 - 205*x^6 + 654*x^5 - 200*x^4 - 28*x^3 + 24*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 13 x^{18} + 20 x^{17} + 75 x^{16} - 94 x^{15} - 239 x^{14} + 342 x^{13} + 336 x^{12} - 630 x^{11} - 339 x^{10} + 652 x^{9} + 504 x^{8} - 878 x^{7} - 205 x^{6} + 654 x^{5} - 200 x^{4} - 28 x^{3} + 24 x^{2} - 10 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-76633477601046538240000000000=-\,2^{20}\cdot 5^{10}\cdot 71^{4}\cdot 131^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.81$
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:  $2, 5, 71, 131$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{25} a^{16} - \frac{1}{25} a^{15} - \frac{1}{25} a^{14} + \frac{2}{25} a^{12} + \frac{1}{5} a^{11} + \frac{8}{25} a^{10} + \frac{3}{25} a^{9} + \frac{6}{25} a^{8} + \frac{12}{25} a^{7} + \frac{1}{5} a^{6} + \frac{4}{25} a^{5} + \frac{1}{5} a^{4} + \frac{7}{25} a^{3} - \frac{12}{25} a^{2} - \frac{11}{25} a + \frac{11}{25}$, $\frac{1}{75} a^{17} + \frac{1}{75} a^{16} + \frac{2}{75} a^{15} - \frac{2}{75} a^{14} + \frac{2}{75} a^{13} - \frac{2}{25} a^{12} - \frac{4}{25} a^{11} - \frac{11}{75} a^{10} + \frac{9}{25} a^{9} - \frac{2}{25} a^{8} + \frac{29}{75} a^{7} + \frac{34}{75} a^{6} + \frac{1}{25} a^{5} + \frac{2}{75} a^{4} + \frac{22}{75} a^{3} - \frac{31}{75} a - \frac{23}{75}$, $\frac{1}{225} a^{18} - \frac{1}{225} a^{17} - \frac{1}{75} a^{16} - \frac{1}{75} a^{15} - \frac{2}{75} a^{14} + \frac{4}{45} a^{13} - \frac{2}{75} a^{12} + \frac{43}{225} a^{11} + \frac{14}{45} a^{10} + \frac{22}{75} a^{9} + \frac{38}{225} a^{8} - \frac{4}{15} a^{7} + \frac{1}{9} a^{6} - \frac{46}{225} a^{5} - \frac{4}{75} a^{4} + \frac{4}{9} a^{3} + \frac{4}{45} a^{2} - \frac{11}{75} a - \frac{32}{225}$, $\frac{1}{3729336539294475} a^{19} + \frac{2569684635074}{1243112179764825} a^{18} - \frac{23653100302156}{3729336539294475} a^{17} + \frac{4426539615001}{1243112179764825} a^{16} + \frac{4344850760971}{82874145317655} a^{15} - \frac{321314761787413}{3729336539294475} a^{14} - \frac{173729397966571}{3729336539294475} a^{13} + \frac{39377229904451}{745867307858895} a^{12} - \frac{289324099630321}{3729336539294475} a^{11} - \frac{1626405100534874}{3729336539294475} a^{10} + \frac{670801287877061}{3729336539294475} a^{9} + \frac{1861863261479099}{3729336539294475} a^{8} - \frac{54739921867853}{149173461571779} a^{7} - \frac{185127054017072}{1243112179764825} a^{6} - \frac{238422249668891}{745867307858895} a^{5} + \frac{1581540919587529}{3729336539294475} a^{4} - \frac{981316599838}{5524943021177} a^{3} - \frac{1400949223070788}{3729336539294475} a^{2} - \frac{110781761488301}{3729336539294475} a + \frac{148342719473434}{3729336539294475}$

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:  \( 10696061.9951 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T299:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5000
The 230 conjugacy class representatives for t20n299 are not computed
Character table for t20n299 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.400.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
71.5.4.5$x^{5} - 1136$$5$$1$$4$$C_5$$[\ ]_{5}$
$131$$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
131.5.4.1$x^{5} - 131$$5$$1$$4$$C_5$$[\ ]_{5}$
131.2.0.1$x^{2} - x + 14$$1$$2$$0$$C_2$$[\ ]^{2}$