↑

Properties

Label 20.8.64945430755...3125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 71^{4}\cdot 179\cdot 263^{4}\cdot 30559$
Root discriminant $34.72$
Ramified primes $5, 71, 179, 263, 30559$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1036

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 24, -105, -159, 486, 440, -1102, -417, 1503, -201, -1248, 805, 492, -743, 71, 288, -138, -21, 34, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 34*x^18 - 21*x^17 - 138*x^16 + 288*x^15 + 71*x^14 - 743*x^13 + 492*x^12 + 805*x^11 - 1248*x^10 - 201*x^9 + 1503*x^8 - 417*x^7 - 1102*x^6 + 440*x^5 + 486*x^4 - 159*x^3 - 105*x^2 + 24*x + 11)
 
gp: K = bnfinit(x^20 - 10*x^19 + 34*x^18 - 21*x^17 - 138*x^16 + 288*x^15 + 71*x^14 - 743*x^13 + 492*x^12 + 805*x^11 - 1248*x^10 - 201*x^9 + 1503*x^8 - 417*x^7 - 1102*x^6 + 440*x^5 + 486*x^4 - 159*x^3 - 105*x^2 + 24*x + 11, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 34 x^{18} - 21 x^{17} - 138 x^{16} + 288 x^{15} + 71 x^{14} - 743 x^{13} + 492 x^{12} + 805 x^{11} - 1248 x^{10} - 201 x^{9} + 1503 x^{8} - 417 x^{7} - 1102 x^{6} + 440 x^{5} + 486 x^{4} - 159 x^{3} - 105 x^{2} + 24 x + 11 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6494543075586159968315439453125=5^{10}\cdot 71^{4}\cdot 179\cdot 263^{4}\cdot 30559\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.72$
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:  $5, 71, 179, 263, 30559$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{29} a^{18} - \frac{9}{29} a^{17} + \frac{14}{29} a^{16} + \frac{5}{29} a^{15} + \frac{3}{29} a^{14} + \frac{4}{29} a^{13} + \frac{13}{29} a^{12} + \frac{9}{29} a^{11} + \frac{10}{29} a^{10} - \frac{9}{29} a^{9} - \frac{4}{29} a^{8} + \frac{10}{29} a^{7} - \frac{9}{29} a^{6} - \frac{14}{29} a^{5} - \frac{2}{29} a^{4} + \frac{12}{29} a^{3} - \frac{2}{29} a^{2} - \frac{3}{29} a + \frac{1}{29}$, $\frac{1}{29} a^{19} - \frac{9}{29} a^{17} - \frac{14}{29} a^{16} - \frac{10}{29} a^{15} + \frac{2}{29} a^{14} - \frac{9}{29} a^{13} + \frac{10}{29} a^{12} + \frac{4}{29} a^{11} - \frac{6}{29} a^{10} + \frac{2}{29} a^{9} + \frac{3}{29} a^{8} - \frac{6}{29} a^{7} - \frac{8}{29} a^{6} - \frac{12}{29} a^{5} - \frac{6}{29} a^{4} - \frac{10}{29} a^{3} + \frac{8}{29} a^{2} + \frac{3}{29} a + \frac{9}{29}$

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:  $13$
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:  \( 61728800.5029 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1036:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 396 conjugacy class representatives for t20n1036 are not computed
Character table for t20n1036 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.1089627903125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ $20$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
71.6.4.1$x^{6} + 2272 x^{3} + 6709571$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
71.6.0.1$x^{6} - 2 x + 13$$1$$6$$0$$C_6$$[\ ]^{6}$
179Data not computed
263Data not computed
30559Data not computed