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Properties

Label 20.4.31016763476...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{6}\cdot 19^{8}\cdot 43^{8}$
Root discriminant $23.69$
Ramified primes $5, 19, 43$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:D_5$ (as 20T85)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, -5, -124, 101, 581, -123, -1301, 1018, 246, -1437, 1136, -30, -240, 17, 10, 90, -92, 21, 12, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 12*x^18 + 21*x^17 - 92*x^16 + 90*x^15 + 10*x^14 + 17*x^13 - 240*x^12 - 30*x^11 + 1136*x^10 - 1437*x^9 + 246*x^8 + 1018*x^7 - 1301*x^6 - 123*x^5 + 581*x^4 + 101*x^3 - 124*x^2 - 5*x - 7)
 
gp: K = bnfinit(x^20 - 7*x^19 + 12*x^18 + 21*x^17 - 92*x^16 + 90*x^15 + 10*x^14 + 17*x^13 - 240*x^12 - 30*x^11 + 1136*x^10 - 1437*x^9 + 246*x^8 + 1018*x^7 - 1301*x^6 - 123*x^5 + 581*x^4 + 101*x^3 - 124*x^2 - 5*x - 7, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 12 x^{18} + 21 x^{17} - 92 x^{16} + 90 x^{15} + 10 x^{14} + 17 x^{13} - 240 x^{12} - 30 x^{11} + 1136 x^{10} - 1437 x^{9} + 246 x^{8} + 1018 x^{7} - 1301 x^{6} - 123 x^{5} + 581 x^{4} + 101 x^{3} - 124 x^{2} - 5 x - 7 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3101676347663598183510015625=5^{6}\cdot 19^{8}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.69$
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:  $5, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19} a^{15} + \frac{5}{19} a^{14} - \frac{2}{19} a^{13} - \frac{6}{19} a^{12} + \frac{2}{19} a^{11} + \frac{8}{19} a^{10} + \frac{7}{19} a^{9} - \frac{2}{19} a^{8} + \frac{1}{19} a^{7} - \frac{6}{19} a^{6} - \frac{4}{19} a^{5} - \frac{6}{19} a^{4} - \frac{9}{19} a^{3} - \frac{7}{19} a^{2} - \frac{5}{19} a + \frac{4}{19}$, $\frac{1}{19} a^{16} - \frac{8}{19} a^{14} + \frac{4}{19} a^{13} - \frac{6}{19} a^{12} - \frac{2}{19} a^{11} + \frac{5}{19} a^{10} + \frac{1}{19} a^{9} - \frac{8}{19} a^{8} + \frac{8}{19} a^{7} + \frac{7}{19} a^{6} - \frac{5}{19} a^{5} + \frac{2}{19} a^{4} - \frac{8}{19} a^{2} - \frac{9}{19} a - \frac{1}{19}$, $\frac{1}{19} a^{17} + \frac{6}{19} a^{14} - \frac{3}{19} a^{13} + \frac{7}{19} a^{12} + \frac{2}{19} a^{11} + \frac{8}{19} a^{10} - \frac{9}{19} a^{9} - \frac{8}{19} a^{8} - \frac{4}{19} a^{7} + \frac{4}{19} a^{6} + \frac{8}{19} a^{5} + \frac{9}{19} a^{4} - \frac{4}{19} a^{3} - \frac{8}{19} a^{2} - \frac{3}{19} a - \frac{6}{19}$, $\frac{1}{19} a^{18} + \frac{5}{19} a^{14} - \frac{4}{19} a^{11} + \frac{7}{19} a^{9} + \frac{8}{19} a^{8} - \frac{2}{19} a^{7} + \frac{6}{19} a^{6} - \frac{5}{19} a^{5} - \frac{6}{19} a^{4} + \frac{8}{19} a^{3} + \frac{1}{19} a^{2} + \frac{5}{19} a - \frac{5}{19}$, $\frac{1}{38946515115867669273853} a^{19} - \frac{578112276550659755637}{38946515115867669273853} a^{18} - \frac{9232413648506175529}{2049816585045666803887} a^{17} - \frac{478316906833354604082}{38946515115867669273853} a^{16} + \frac{627394305695338407685}{38946515115867669273853} a^{15} + \frac{19146014472641373927893}{38946515115867669273853} a^{14} + \frac{713986618716644416621}{2049816585045666803887} a^{13} + \frac{5462228418827798455486}{38946515115867669273853} a^{12} - \frac{8641142395314386128799}{38946515115867669273853} a^{11} + \frac{9900108537518337534135}{38946515115867669273853} a^{10} - \frac{11194417574157147429503}{38946515115867669273853} a^{9} - \frac{272359744848168775919}{38946515115867669273853} a^{8} + \frac{16614832211877311636128}{38946515115867669273853} a^{7} + \frac{15824030982811472685883}{38946515115867669273853} a^{6} - \frac{9019992564523620677313}{38946515115867669273853} a^{5} - \frac{105274409290410750399}{38946515115867669273853} a^{4} - \frac{8503919983940736175983}{38946515115867669273853} a^{3} - \frac{13677949618528110382228}{38946515115867669273853} a^{2} + \frac{7492410269095857301710}{38946515115867669273853} a - \frac{9649240174914212732765}{38946515115867669273853}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1285248.93015 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T85):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.667489.1, 10.2.2227707825605.1, 10.6.55692695640125.2, 10.6.11138539128025.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$43$43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$