↑

Properties

Label 20.0.69761895802...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{6}\cdot 67^{8}$
Root discriminant $34.85$
Ramified primes $2, 5, 67$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 20T288

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![676, 0, 42280, 0, 44361, 0, 6288, 0, -5176, 0, -160, 0, 762, 0, 96, 0, 4, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 8*x^18 + 4*x^16 + 96*x^14 + 762*x^12 - 160*x^10 - 5176*x^8 + 6288*x^6 + 44361*x^4 + 42280*x^2 + 676)
 
gp: K = bnfinit(x^20 + 8*x^18 + 4*x^16 + 96*x^14 + 762*x^12 - 160*x^10 - 5176*x^8 + 6288*x^6 + 44361*x^4 + 42280*x^2 + 676, 1)
 

Normalized defining polynomial

\( x^{20} + 8 x^{18} + 4 x^{16} + 96 x^{14} + 762 x^{12} - 160 x^{10} - 5176 x^{8} + 6288 x^{6} + 44361 x^{4} + 42280 x^{2} + 676 \)

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:  \(6976189580273785130450944000000=2^{40}\cdot 5^{6}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.85$
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:  $2, 5, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2}$, $\frac{1}{8} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{3}{8} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{3}{8} a^{6}$, $\frac{1}{8} a^{15} - \frac{1}{2} a^{8} - \frac{3}{8} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{16} - \frac{1}{2} a^{9} - \frac{3}{8} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{17} - \frac{3}{8} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{1239468443664816} a^{18} - \frac{56760752267087}{1239468443664816} a^{16} + \frac{15170341227001}{413156147888272} a^{14} + \frac{13291590738321}{413156147888272} a^{12} - \frac{1370051888677}{413156147888272} a^{10} - \frac{1}{2} a^{9} - \frac{238651137966775}{1239468443664816} a^{8} - \frac{126267527522667}{413156147888272} a^{6} - \frac{153315448412931}{413156147888272} a^{4} + \frac{2945494082151}{103289036972068} a^{2} + \frac{21769298364391}{309867110916204}$, $\frac{1}{16113089767642608} a^{19} - \frac{676494974099495}{16113089767642608} a^{17} + \frac{221748415171137}{5371029922547536} a^{15} - \frac{38352927747713}{5371029922547536} a^{13} + \frac{205208022055459}{5371029922547536} a^{11} - \frac{3957056468961223}{16113089767642608} a^{9} - \frac{1}{2} a^{8} - \frac{746001749355075}{5371029922547536} a^{7} - \frac{1}{2} a^{6} + \frac{1654242698598259}{5371029922547536} a^{5} - \frac{461855172292155}{1342757480636884} a^{3} - \frac{1}{2} a^{2} - \frac{133164257093711}{4028272441910652} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}$, which has order $4$ (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:  \( -\frac{5586974899}{33709392819336} a^{19} - \frac{45113426155}{33709392819336} a^{17} - \frac{8638218397}{11236464273112} a^{15} - \frac{179969387501}{11236464273112} a^{13} - \frac{1434380170671}{11236464273112} a^{11} + \frac{512289509689}{33709392819336} a^{9} + \frac{9542111880855}{11236464273112} a^{7} - \frac{11162326717929}{11236464273112} a^{5} - \frac{41792532686317}{5618232136556} a^{3} - \frac{33547646840519}{4213674102417} a \) (order $4$)
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:  \( 22547882.1224 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T288:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n288
Character table for t20n288 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.5745920.1, 10.0.528249546342400.1, 10.4.2641247731712000.1, 10.6.165077983232000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.6.4.1$x^{6} + 2345 x^{3} + 7756992$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
67.6.4.1$x^{6} + 2345 x^{3} + 7756992$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$