↑

Properties

Label 20.12.1737920484...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{10}\cdot 19^{4}\cdot 149^{2}\cdot 765899^{2}$
Root discriminant $51.52$
Ramified primes $2, 5, 19, 149, 765899$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1030

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![130321, 0, -41154, 0, -100719, 0, 42199, 0, 10370, 0, -3649, 0, -460, 0, -287, 0, 182, 0, -25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 25*x^18 + 182*x^16 - 287*x^14 - 460*x^12 - 3649*x^10 + 10370*x^8 + 42199*x^6 - 100719*x^4 - 41154*x^2 + 130321)
 
gp: K = bnfinit(x^20 - 25*x^18 + 182*x^16 - 287*x^14 - 460*x^12 - 3649*x^10 + 10370*x^8 + 42199*x^6 - 100719*x^4 - 41154*x^2 + 130321, 1)
 

Normalized defining polynomial

\( x^{20} - 25 x^{18} + 182 x^{16} - 287 x^{14} - 460 x^{12} - 3649 x^{10} + 10370 x^{8} + 42199 x^{6} - 100719 x^{4} - 41154 x^{2} + 130321 \)

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:  \(17379204847431458802903040000000000=2^{20}\cdot 5^{10}\cdot 19^{4}\cdot 149^{2}\cdot 765899^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.52$
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:  $2, 5, 19, 149, 765899$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{19} a^{12} + \frac{8}{19} a^{10} + \frac{9}{19} a^{8} - \frac{9}{19} a^{6} + \frac{3}{19} a^{4} + \frac{3}{19} a^{2}$, $\frac{1}{19} a^{13} + \frac{8}{19} a^{11} + \frac{9}{19} a^{9} - \frac{9}{19} a^{7} + \frac{3}{19} a^{5} + \frac{3}{19} a^{3}$, $\frac{1}{19} a^{14} + \frac{2}{19} a^{10} - \frac{5}{19} a^{8} - \frac{1}{19} a^{6} - \frac{2}{19} a^{4} - \frac{5}{19} a^{2}$, $\frac{1}{19} a^{15} + \frac{2}{19} a^{11} - \frac{5}{19} a^{9} - \frac{1}{19} a^{7} - \frac{2}{19} a^{5} - \frac{5}{19} a^{3}$, $\frac{1}{1805} a^{16} - \frac{44}{1805} a^{14} - \frac{8}{1805} a^{12} - \frac{401}{1805} a^{10} + \frac{452}{1805} a^{8} - \frac{476}{1805} a^{6} + \frac{452}{1805} a^{4} - \frac{27}{95} a^{2} + \frac{1}{5}$, $\frac{1}{1805} a^{17} - \frac{44}{1805} a^{15} - \frac{8}{1805} a^{13} - \frac{401}{1805} a^{11} + \frac{452}{1805} a^{9} - \frac{476}{1805} a^{7} + \frac{452}{1805} a^{5} - \frac{27}{95} a^{3} + \frac{1}{5} a$, $\frac{1}{7478786974073586355} a^{18} + \frac{1540767091099807}{7478786974073586355} a^{16} + \frac{98164419599267238}{7478786974073586355} a^{14} + \frac{4212666584075146}{7478786974073586355} a^{12} + \frac{456702705299239441}{7478786974073586355} a^{10} + \frac{2287045430699759646}{7478786974073586355} a^{8} + \frac{2638999045767954281}{7478786974073586355} a^{6} - \frac{5754248717293329}{393620367056504545} a^{4} + \frac{9026538474048658}{20716861424026555} a^{2} - \frac{544103661294526}{1090361127580345}$, $\frac{1}{7478786974073586355} a^{19} + \frac{1540767091099807}{7478786974073586355} a^{17} + \frac{98164419599267238}{7478786974073586355} a^{15} + \frac{4212666584075146}{7478786974073586355} a^{13} + \frac{456702705299239441}{7478786974073586355} a^{11} + \frac{2287045430699759646}{7478786974073586355} a^{9} + \frac{2638999045767954281}{7478786974073586355} a^{7} - \frac{5754248717293329}{393620367056504545} a^{5} + \frac{9026538474048658}{20716861424026555} a^{3} - \frac{544103661294526}{1090361127580345} a$

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:  \( 7238646903.12 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1030:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1030 are not computed
Character table for t20n1030 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.8.356621721875.1

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 32 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $16{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.8.4.2$x^{8} - 13718 x^{2} + 1303210$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$149$149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
149.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
149.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
149.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
765899Data not computed