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Properties

Label 20.20.2166421400...5121.1
Degree $20$
Signature $[20, 0]$
Discriminant $67^{4}\cdot 401^{10}$
Root discriminant $46.43$
Ramified primes $67, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4:D_5$ (as 20T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, -96, 92, 1396, -1527, -6729, 10593, 8084, -21452, 1845, 17083, -8292, -4663, 4427, -292, -652, 186, 11, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 11*x^18 + 186*x^17 - 652*x^16 - 292*x^15 + 4427*x^14 - 4663*x^13 - 8292*x^12 + 17083*x^11 + 1845*x^10 - 21452*x^9 + 8084*x^8 + 10593*x^7 - 6729*x^6 - 1527*x^5 + 1396*x^4 + 92*x^3 - 96*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 11*x^18 + 186*x^17 - 652*x^16 - 292*x^15 + 4427*x^14 - 4663*x^13 - 8292*x^12 + 17083*x^11 + 1845*x^10 - 21452*x^9 + 8084*x^8 + 10593*x^7 - 6729*x^6 - 1527*x^5 + 1396*x^4 + 92*x^3 - 96*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 11 x^{18} + 186 x^{17} - 652 x^{16} - 292 x^{15} + 4427 x^{14} - 4663 x^{13} - 8292 x^{12} + 17083 x^{11} + 1845 x^{10} - 21452 x^{9} + 8084 x^{8} + 10593 x^{7} - 6729 x^{6} - 1527 x^{5} + 1396 x^{4} + 92 x^{3} - 96 x^{2} - 5 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2166421400000369059410353975835121=67^{4}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.43$
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:  $67, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{22969179} a^{18} - \frac{1}{2552131} a^{17} - \frac{175876}{2552131} a^{16} - \frac{294390}{2552131} a^{15} - \frac{9367009}{22969179} a^{14} - \frac{2911121}{22969179} a^{13} - \frac{6798940}{22969179} a^{12} + \frac{4026518}{22969179} a^{11} + \frac{2455978}{22969179} a^{10} + \frac{1916605}{22969179} a^{9} + \frac{9928085}{22969179} a^{8} - \frac{11122256}{22969179} a^{7} - \frac{292048}{22969179} a^{6} - \frac{868788}{2552131} a^{5} - \frac{6831847}{22969179} a^{4} + \frac{4357385}{22969179} a^{3} - \frac{3271837}{22969179} a^{2} - \frac{2773328}{7656393} a - \frac{8347951}{22969179}$, $\frac{1}{6224647509} a^{19} + \frac{14}{691627501} a^{18} - \frac{174072941}{2074882503} a^{17} + \frac{24868028}{2074882503} a^{16} - \frac{680962972}{6224647509} a^{15} - \frac{2561387753}{6224647509} a^{14} + \frac{2701038890}{6224647509} a^{13} + \frac{2952648083}{6224647509} a^{12} + \frac{1150890955}{6224647509} a^{11} - \frac{1955787872}{6224647509} a^{10} + \frac{613207445}{6224647509} a^{9} - \frac{500708708}{6224647509} a^{8} + \frac{2892972974}{6224647509} a^{7} - \frac{145616713}{691627501} a^{6} - \frac{2287432147}{6224647509} a^{5} + \frac{973187111}{6224647509} a^{4} - \frac{1428656221}{6224647509} a^{3} + \frac{457401185}{2074882503} a^{2} + \frac{3033532001}{6224647509} a - \frac{783998755}{2074882503}$

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:  $19$
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:  \( 24330067237.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 20T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $C_2^4:D_5$
Character table for $C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.116071900626889.2, 10.10.46544832151382489.2, 10.10.10368641602001.1

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 16 sibling: data not computed
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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$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}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed