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Properties

Label 20.12.3935781606...2368.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{24}\cdot 17\cdot 53^{14}$
Root discriminant $42.63$
Ramified primes $2, 17, 53$
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![17, 0, -585, 0, 588, 0, 3529, 0, -797, 0, -4700, 0, -273, 0, 1045, 0, -112, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^18 - 112*x^16 + 1045*x^14 - 273*x^12 - 4700*x^10 - 797*x^8 + 3529*x^6 + 588*x^4 - 585*x^2 + 17)
 
gp: K = bnfinit(x^20 - 9*x^18 - 112*x^16 + 1045*x^14 - 273*x^12 - 4700*x^10 - 797*x^8 + 3529*x^6 + 588*x^4 - 585*x^2 + 17, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{18} - 112 x^{16} + 1045 x^{14} - 273 x^{12} - 4700 x^{10} - 797 x^{8} + 3529 x^{6} + 588 x^{4} - 585 x^{2} + 17 \)

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:  \(393578160617730190869682473402368=2^{24}\cdot 17\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.63$
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:  $2, 17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{12} a^{6} - \frac{1}{2} a^{5} + \frac{1}{12} a^{4} - \frac{1}{4} a^{3} + \frac{1}{6} a^{2} + \frac{1}{4} a - \frac{1}{6}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{6} a^{7} - \frac{1}{4} a^{6} - \frac{5}{12} a^{5} - \frac{1}{2} a^{4} - \frac{1}{12} a^{3} - \frac{1}{4} a^{2} - \frac{5}{12} a + \frac{1}{4}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{10} - \frac{1}{4} a^{9} + \frac{1}{6} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{12}$, $\frac{1}{24} a^{15} - \frac{1}{24} a^{12} + \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{6} a^{6} + \frac{1}{4} a^{5} - \frac{5}{12} a^{4} + \frac{1}{8} a^{3} - \frac{1}{12} a^{2} + \frac{5}{12} a + \frac{5}{24}$, $\frac{1}{72} a^{16} + \frac{1}{36} a^{14} - \frac{1}{24} a^{13} + \frac{1}{36} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} + \frac{5}{36} a^{8} - \frac{1}{12} a^{7} + \frac{1}{12} a^{5} + \frac{7}{24} a^{4} + \frac{1}{6} a^{3} + \frac{1}{18} a^{2} + \frac{11}{24} a + \frac{1}{36}$, $\frac{1}{72} a^{17} - \frac{1}{72} a^{15} - \frac{1}{24} a^{14} + \frac{1}{36} a^{13} - \frac{1}{24} a^{12} + \frac{1}{12} a^{11} + \frac{1}{12} a^{10} + \frac{1}{18} a^{9} - \frac{1}{12} a^{8} - \frac{1}{4} a^{7} + \frac{1}{6} a^{6} + \frac{7}{24} a^{5} + \frac{1}{12} a^{4} + \frac{13}{72} a^{3} + \frac{7}{24} a^{2} - \frac{7}{18} a - \frac{11}{24}$, $\frac{1}{43130870592552} a^{18} + \frac{15036575027}{2537110034856} a^{16} - \frac{222392065651}{7188478432092} a^{14} - \frac{1}{24} a^{13} - \frac{1244154112133}{43130870592552} a^{12} - \frac{1}{12} a^{11} - \frac{391870979752}{5391358824069} a^{10} - \frac{1}{4} a^{9} - \frac{2599777914563}{21565435296276} a^{8} + \frac{1}{6} a^{7} - \frac{772785967333}{14376956864184} a^{6} - \frac{5}{12} a^{5} - \frac{8197417230563}{43130870592552} a^{4} - \frac{1}{12} a^{3} + \frac{3434909114503}{10782717648138} a^{2} + \frac{5}{24} a + \frac{503563168739}{2537110034856}$, $\frac{1}{86261741185104} a^{19} - \frac{1}{86261741185104} a^{18} - \frac{10100532173}{2537110034856} a^{17} + \frac{10100532173}{2537110034856} a^{16} + \frac{176967847243}{14376956864184} a^{15} - \frac{176967847243}{14376956864184} a^{14} + \frac{1152005365231}{86261741185104} a^{13} - \frac{1152005365231}{86261741185104} a^{12} + \frac{1013377648519}{21565435296276} a^{11} - \frac{1013377648519}{21565435296276} a^{10} - \frac{1000369022611}{21565435296276} a^{9} - \frac{2195494900729}{10782717648138} a^{8} - \frac{1970865706015}{28753913728368} a^{7} - \frac{5217612726077}{28753913728368} a^{6} + \frac{2191210012799}{43130870592552} a^{5} + \frac{19374225283477}{43130870592552} a^{4} - \frac{17690816413975}{43130870592552} a^{3} + \frac{6908098765837}{43130870592552} a^{2} + \frac{855939562469}{5074220069712} a + \frac{412615454959}{5074220069712}$

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:  \( 4206577043.91 \) (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

\(\Q(\sqrt{53}) \), 5.5.2382032.1, 10.10.300726051798272.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.12.19$x^{8} + 12 x^{4} + 80$$4$$2$$12$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.12.19$x^{8} + 12 x^{4} + 80$$4$$2$$12$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
$53$53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$