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Properties

Label 20.0.10728332761...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{20}\cdot 7^{4}\cdot 59^{5}$
Root discriminant $35.61$
Ramified primes $2, 5, 7, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59, 0, 0, 0, 0, 0, 685, 0, -2560, 0, 2924, 0, -1155, 0, 60, 0, 65, 0, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 + 65*x^16 + 60*x^14 - 1155*x^12 + 2924*x^10 - 2560*x^8 + 685*x^6 + 59)
 
gp: K = bnfinit(x^20 - 15*x^18 + 65*x^16 + 60*x^14 - 1155*x^12 + 2924*x^10 - 2560*x^8 + 685*x^6 + 59, 1)
 

Normalized defining polynomial

\( x^{20} - 15 x^{18} + 65 x^{16} + 60 x^{14} - 1155 x^{12} + 2924 x^{10} - 2560 x^{8} + 685 x^{6} + 59 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10728332761868750000000000000000=2^{16}\cdot 5^{20}\cdot 7^{4}\cdot 59^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.61$
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:  $2, 5, 7, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} - \frac{2}{7} a^{14} + \frac{1}{7} a^{12} + \frac{2}{7} a^{10} + \frac{2}{7} a^{8} - \frac{3}{7} a^{6} - \frac{1}{7} a^{4} + \frac{2}{7} a^{2} + \frac{1}{7}$, $\frac{1}{14} a^{17} - \frac{1}{14} a^{16} + \frac{5}{14} a^{15} - \frac{5}{14} a^{14} - \frac{3}{7} a^{13} + \frac{3}{7} a^{12} - \frac{5}{14} a^{11} + \frac{5}{14} a^{10} - \frac{5}{14} a^{9} + \frac{5}{14} a^{8} - \frac{3}{14} a^{7} + \frac{3}{14} a^{6} - \frac{1}{14} a^{5} + \frac{1}{14} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{14} a - \frac{1}{14}$, $\frac{1}{6324980875696} a^{18} + \frac{158419937453}{3162490437848} a^{16} - \frac{114955774595}{903568696528} a^{14} - \frac{51776387377}{6324980875696} a^{12} - \frac{708199143471}{1581245218924} a^{10} + \frac{26220447833}{395311304731} a^{8} - \frac{62948334865}{395311304731} a^{6} - \frac{2237669812835}{6324980875696} a^{4} + \frac{1651678568213}{6324980875696} a^{2} - \frac{242503795635}{6324980875696}$, $\frac{1}{12649961751392} a^{19} - \frac{1}{12649961751392} a^{18} + \frac{158419937453}{6324980875696} a^{17} - \frac{158419937453}{6324980875696} a^{16} - \frac{114955774595}{1807137393056} a^{15} + \frac{114955774595}{1807137393056} a^{14} + \frac{6273204488319}{12649961751392} a^{13} - \frac{6273204488319}{12649961751392} a^{12} + \frac{873046075453}{3162490437848} a^{11} - \frac{873046075453}{3162490437848} a^{10} - \frac{184545428449}{395311304731} a^{9} + \frac{184545428449}{395311304731} a^{8} + \frac{166181484933}{395311304731} a^{7} - \frac{166181484933}{395311304731} a^{6} + \frac{4087311062861}{12649961751392} a^{5} - \frac{4087311062861}{12649961751392} a^{4} + \frac{1651678568213}{12649961751392} a^{3} - \frac{1651678568213}{12649961751392} a^{2} + \frac{6082477080061}{12649961751392} a - \frac{6082477080061}{12649961751392}$

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:  $9$
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:  \( 27878105.0757 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.1665712890625.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ R

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
5Data not computed
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.6.5.2$x^{6} + 177$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
59.10.0.1$x^{10} + x^{2} - x + 37$$1$$10$$0$$C_{10}$$[\ ]^{10}$