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Properties

Label 20.8.67442553253...7456.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 11^{16}\cdot 241^{3}$
Root discriminant $31.01$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -26, -121, -206, 347, 1382, -321, -3398, -1888, 944, 1696, 72, 314, 116, -131, -44, -65, 10, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 4*x^18 + 10*x^17 - 65*x^16 - 44*x^15 - 131*x^14 + 116*x^13 + 314*x^12 + 72*x^11 + 1696*x^10 + 944*x^9 - 1888*x^8 - 3398*x^7 - 321*x^6 + 1382*x^5 + 347*x^4 - 206*x^3 - 121*x^2 - 26*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 4*x^18 + 10*x^17 - 65*x^16 - 44*x^15 - 131*x^14 + 116*x^13 + 314*x^12 + 72*x^11 + 1696*x^10 + 944*x^9 - 1888*x^8 - 3398*x^7 - 321*x^6 + 1382*x^5 + 347*x^4 - 206*x^3 - 121*x^2 - 26*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 4 x^{18} + 10 x^{17} - 65 x^{16} - 44 x^{15} - 131 x^{14} + 116 x^{13} + 314 x^{12} + 72 x^{11} + 1696 x^{10} + 944 x^{9} - 1888 x^{8} - 3398 x^{7} - 321 x^{6} + 1382 x^{5} + 347 x^{4} - 206 x^{3} - 121 x^{2} - 26 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(674425532537559799418460307456=2^{20}\cdot 11^{16}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.01$
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:  $2, 11, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} - \frac{9}{23} a^{15} - \frac{9}{23} a^{14} + \frac{2}{23} a^{13} + \frac{11}{23} a^{12} + \frac{6}{23} a^{11} - \frac{3}{23} a^{10} - \frac{8}{23} a^{9} + \frac{10}{23} a^{8} + \frac{9}{23} a^{7} + \frac{10}{23} a^{6} - \frac{4}{23} a^{5} - \frac{6}{23} a^{3} - \frac{5}{23} a^{2} + \frac{3}{23} a + \frac{1}{23}$, $\frac{1}{23} a^{17} + \frac{2}{23} a^{15} - \frac{10}{23} a^{14} + \frac{6}{23} a^{13} - \frac{10}{23} a^{12} + \frac{5}{23} a^{11} + \frac{11}{23} a^{10} + \frac{7}{23} a^{9} + \frac{7}{23} a^{8} - \frac{1}{23} a^{7} - \frac{6}{23} a^{6} + \frac{10}{23} a^{5} - \frac{6}{23} a^{4} + \frac{10}{23} a^{3} + \frac{4}{23} a^{2} + \frac{5}{23} a + \frac{9}{23}$, $\frac{1}{23} a^{18} + \frac{8}{23} a^{15} + \frac{1}{23} a^{14} + \frac{9}{23} a^{13} + \frac{6}{23} a^{12} - \frac{1}{23} a^{11} - \frac{10}{23} a^{10} + \frac{2}{23} a^{8} - \frac{1}{23} a^{7} - \frac{10}{23} a^{6} + \frac{2}{23} a^{5} + \frac{10}{23} a^{4} - \frac{7}{23} a^{3} - \frac{8}{23} a^{2} + \frac{3}{23} a - \frac{2}{23}$, $\frac{1}{2176241379499346418733327178821} a^{19} - \frac{45323240686429054789028483446}{2176241379499346418733327178821} a^{18} + \frac{30427867987770185359298242901}{2176241379499346418733327178821} a^{17} + \frac{44569699184423844669864380173}{2176241379499346418733327178821} a^{16} - \frac{118133176760644649699294984595}{2176241379499346418733327178821} a^{15} - \frac{738224061411650412413174590931}{2176241379499346418733327178821} a^{14} - \frac{944099010698420992014670613785}{2176241379499346418733327178821} a^{13} + \frac{506006630854297361771399921499}{2176241379499346418733327178821} a^{12} - \frac{976967670933082265058377068120}{2176241379499346418733327178821} a^{11} - \frac{353130211408223433471530340258}{2176241379499346418733327178821} a^{10} - \frac{851508563006740983352349984907}{2176241379499346418733327178821} a^{9} + \frac{542444732539285095362607733673}{2176241379499346418733327178821} a^{8} - \frac{676290079696641156005295442707}{2176241379499346418733327178821} a^{7} - \frac{38767433049236529706919497117}{2176241379499346418733327178821} a^{6} - \frac{677508417839217582484830249261}{2176241379499346418733327178821} a^{5} - \frac{43502557027072116629851727114}{94619190413015061684057703427} a^{4} + \frac{629907821007047124972379554898}{2176241379499346418733327178821} a^{3} + \frac{339433263728051302351779770163}{2176241379499346418733327178821} a^{2} + \frac{806122016126356921457945324298}{2176241379499346418733327178821} a + \frac{987297448226307588891895690298}{2176241379499346418733327178821}$

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:  \( 20282700.4586 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.52900342088704.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
11Data not computed
241Data not computed