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Properties

Label 20.2.18256727419...6811.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,11^{16}\cdot 331^{5}$
Root discriminant $29.04$
Ramified primes $11, 331$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T303

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11659, 33365, -40635, 41746, -33613, 29415, -18446, 10655, -4485, 1297, 810, -1121, 920, -593, 371, -155, 72, -31, 9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 9*x^18 - 31*x^17 + 72*x^16 - 155*x^15 + 371*x^14 - 593*x^13 + 920*x^12 - 1121*x^11 + 810*x^10 + 1297*x^9 - 4485*x^8 + 10655*x^7 - 18446*x^6 + 29415*x^5 - 33613*x^4 + 41746*x^3 - 40635*x^2 + 33365*x - 11659)
 
gp: K = bnfinit(x^20 - 2*x^19 + 9*x^18 - 31*x^17 + 72*x^16 - 155*x^15 + 371*x^14 - 593*x^13 + 920*x^12 - 1121*x^11 + 810*x^10 + 1297*x^9 - 4485*x^8 + 10655*x^7 - 18446*x^6 + 29415*x^5 - 33613*x^4 + 41746*x^3 - 40635*x^2 + 33365*x - 11659, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 9 x^{18} - 31 x^{17} + 72 x^{16} - 155 x^{15} + 371 x^{14} - 593 x^{13} + 920 x^{12} - 1121 x^{11} + 810 x^{10} + 1297 x^{9} - 4485 x^{8} + 10655 x^{7} - 18446 x^{6} + 29415 x^{5} - 33613 x^{4} + 41746 x^{3} - 40635 x^{2} + 33365 x - 11659 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-182567274194490055859030886811=-\,11^{16}\cdot 331^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.04$
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:  $11, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} - \frac{9}{23} a^{15} - \frac{11}{23} a^{14} - \frac{2}{23} a^{13} + \frac{11}{23} a^{12} - \frac{6}{23} a^{11} - \frac{3}{23} a^{10} - \frac{8}{23} a^{9} + \frac{4}{23} a^{8} - \frac{3}{23} a^{7} + \frac{8}{23} a^{6} - \frac{3}{23} a^{5} - \frac{10}{23} a^{4} - \frac{6}{23} a^{3} + \frac{6}{23} a^{2} - \frac{11}{23} a + \frac{2}{23}$, $\frac{1}{23} a^{17} - \frac{9}{23} a^{14} - \frac{7}{23} a^{13} + \frac{1}{23} a^{12} - \frac{11}{23} a^{11} + \frac{11}{23} a^{10} + \frac{1}{23} a^{9} + \frac{10}{23} a^{8} + \frac{4}{23} a^{7} + \frac{9}{23} a^{5} - \frac{4}{23} a^{4} - \frac{2}{23} a^{3} - \frac{3}{23} a^{2} - \frac{5}{23} a - \frac{5}{23}$, $\frac{1}{2047} a^{18} + \frac{14}{2047} a^{17} + \frac{40}{2047} a^{16} + \frac{666}{2047} a^{15} + \frac{71}{2047} a^{14} - \frac{821}{2047} a^{13} - \frac{569}{2047} a^{12} + \frac{215}{2047} a^{11} + \frac{702}{2047} a^{10} + \frac{923}{2047} a^{9} - \frac{915}{2047} a^{8} + \frac{580}{2047} a^{7} + \frac{720}{2047} a^{6} - \frac{320}{2047} a^{5} + \frac{255}{2047} a^{4} - \frac{64}{2047} a^{3} - \frac{14}{2047} a^{2} + \frac{474}{2047} a - \frac{11}{23}$, $\frac{1}{62591921438950698491219423203510153} a^{19} - \frac{5360433574495800647955508340737}{62591921438950698491219423203510153} a^{18} - \frac{1195477527100738670909559346682754}{62591921438950698491219423203510153} a^{17} + \frac{137523898030121258425285797120755}{62591921438950698491219423203510153} a^{16} + \frac{24801249083363446452896345070334306}{62591921438950698491219423203510153} a^{15} - \frac{8449527375730576914644110623638585}{62591921438950698491219423203510153} a^{14} - \frac{15834639155761011278106511197575879}{62591921438950698491219423203510153} a^{13} + \frac{25433252149034482602162213714702606}{62591921438950698491219423203510153} a^{12} - \frac{10913793663167414867519151152210183}{62591921438950698491219423203510153} a^{11} + \frac{371385420889209524095939703626898}{62591921438950698491219423203510153} a^{10} + \frac{13814550854638867949062746558133501}{62591921438950698491219423203510153} a^{9} + \frac{27923538208627383418116775791603845}{62591921438950698491219423203510153} a^{8} + \frac{15039166089472773570495939430184480}{62591921438950698491219423203510153} a^{7} + \frac{25652565781704122692426207023769849}{62591921438950698491219423203510153} a^{6} + \frac{25374293718409376135973774580381716}{62591921438950698491219423203510153} a^{5} + \frac{25606461673933290718296930773097055}{62591921438950698491219423203510153} a^{4} + \frac{4647702300221364077222281041075404}{62591921438950698491219423203510153} a^{3} - \frac{17644631440991150352628576703468329}{62591921438950698491219423203510153} a^{2} + \frac{13147307676495353041521917757060824}{62591921438950698491219423203510153} a - \frac{218707530713420633870789334964721}{703280016167985376305836215769777}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $10$
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:  \( 1032943.16606 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T303:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 50 conjugacy class representatives for t20n303 are not computed
Character table for t20n303 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.70952789611.1

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
Arithmetically equvalently 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 $20$ $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
11Data not computed
331Data not computed