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Properties

Label 20.12.1147425991...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{28}\cdot 5^{22}\cdot 7^{8}\cdot 311$
Root discriminant $44.98$
Ramified primes $2, 5, 7, 311$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T513

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![311, 0, 3020, 0, 2085, 0, -7440, 0, 1670, 0, 2472, 0, -1110, 0, -80, 0, 115, 0, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^18 + 115*x^16 - 80*x^14 - 1110*x^12 + 2472*x^10 + 1670*x^8 - 7440*x^6 + 2085*x^4 + 3020*x^2 + 311)
 
gp: K = bnfinit(x^20 - 20*x^18 + 115*x^16 - 80*x^14 - 1110*x^12 + 2472*x^10 + 1670*x^8 - 7440*x^6 + 2085*x^4 + 3020*x^2 + 311, 1)
 

Normalized defining polynomial

\( x^{20} - 20 x^{18} + 115 x^{16} - 80 x^{14} - 1110 x^{12} + 2472 x^{10} + 1670 x^{8} - 7440 x^{6} + 2085 x^{4} + 3020 x^{2} + 311 \)

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:  \(1147425991040000000000000000000000=2^{28}\cdot 5^{22}\cdot 7^{8}\cdot 311\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.98$
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:  $2, 5, 7, 311$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} + \frac{1}{16}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{5} + \frac{1}{16} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4} + \frac{1}{32} a^{2} - \frac{1}{32}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} - \frac{1}{16} a^{7} + \frac{1}{16} a^{5} + \frac{1}{32} a^{3} - \frac{1}{32} a$, $\frac{1}{192} a^{12} - \frac{1}{96} a^{10} + \frac{1}{64} a^{8} - \frac{1}{48} a^{6} - \frac{17}{192} a^{4} + \frac{1}{32} a^{2} + \frac{77}{192}$, $\frac{1}{384} a^{13} - \frac{1}{384} a^{12} + \frac{1}{96} a^{11} - \frac{1}{96} a^{10} - \frac{1}{128} a^{9} + \frac{1}{128} a^{8} + \frac{1}{48} a^{7} - \frac{1}{48} a^{6} - \frac{29}{384} a^{5} + \frac{29}{384} a^{4} - \frac{1}{32} a^{3} + \frac{1}{32} a^{2} + \frac{95}{384} a - \frac{95}{384}$, $\frac{1}{384} a^{14} - \frac{1}{384} a^{12} + \frac{1}{384} a^{10} - \frac{1}{384} a^{8} - \frac{7}{128} a^{6} - \frac{11}{384} a^{4} + \frac{83}{384} a^{2} + \frac{77}{384}$, $\frac{1}{768} a^{15} - \frac{1}{768} a^{14} - \frac{1}{768} a^{13} + \frac{1}{768} a^{12} + \frac{1}{768} a^{11} - \frac{1}{768} a^{10} - \frac{1}{768} a^{9} + \frac{1}{768} a^{8} - \frac{7}{256} a^{7} + \frac{7}{256} a^{6} + \frac{85}{768} a^{5} - \frac{85}{768} a^{4} + \frac{83}{768} a^{3} - \frac{83}{768} a^{2} + \frac{365}{768} a - \frac{365}{768}$, $\frac{1}{768} a^{16} - \frac{11}{384} a^{8} - \frac{1}{24} a^{6} + \frac{3}{32} a^{4} + \frac{5}{24} a^{2} + \frac{77}{768}$, $\frac{1}{1536} a^{17} - \frac{1}{1536} a^{16} - \frac{11}{768} a^{9} + \frac{11}{768} a^{8} + \frac{1}{24} a^{7} - \frac{1}{24} a^{6} + \frac{7}{64} a^{5} - \frac{7}{64} a^{4} - \frac{5}{24} a^{3} + \frac{5}{24} a^{2} - \frac{403}{1536} a + \frac{403}{1536}$, $\frac{1}{1536} a^{18} - \frac{1}{1536} a^{16} - \frac{11}{768} a^{10} - \frac{5}{768} a^{8} - \frac{11}{192} a^{6} - \frac{13}{192} a^{4} + \frac{109}{1536} a^{2} + \frac{115}{1536}$, $\frac{1}{3072} a^{19} - \frac{1}{3072} a^{18} - \frac{1}{3072} a^{17} + \frac{1}{3072} a^{16} - \frac{11}{1536} a^{11} + \frac{11}{1536} a^{10} + \frac{43}{1536} a^{9} - \frac{43}{1536} a^{8} + \frac{13}{384} a^{7} - \frac{13}{384} a^{6} + \frac{35}{384} a^{5} - \frac{35}{384} a^{4} - \frac{83}{3072} a^{3} + \frac{83}{3072} a^{2} - \frac{365}{3072} a + \frac{365}{3072}$

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:  \( 3114981738.87 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T513:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 128 conjugacy class representatives for t20n513 are not computed
Character table for t20n513 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.2450000.1, 10.10.30012500000000.1

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 R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$5$5.10.11.1$x^{10} + 20 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.1$x^{10} + 20 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
311Data not computed