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Properties

Label 20.10.1825672741...6811.3
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{16}\cdot 331^{5}$
Root discriminant $29.04$
Ramified primes $11, 331$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T303

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![397, -3056, 4857, 4686, -11463, -1473, 8059, -2702, -1895, 2199, 211, -430, -718, 433, 161, -142, 32, -33, 28, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 28*x^18 - 33*x^17 + 32*x^16 - 142*x^15 + 161*x^14 + 433*x^13 - 718*x^12 - 430*x^11 + 211*x^10 + 2199*x^9 - 1895*x^8 - 2702*x^7 + 8059*x^6 - 1473*x^5 - 11463*x^4 + 4686*x^3 + 4857*x^2 - 3056*x + 397)
 
gp: K = bnfinit(x^20 - 9*x^19 + 28*x^18 - 33*x^17 + 32*x^16 - 142*x^15 + 161*x^14 + 433*x^13 - 718*x^12 - 430*x^11 + 211*x^10 + 2199*x^9 - 1895*x^8 - 2702*x^7 + 8059*x^6 - 1473*x^5 - 11463*x^4 + 4686*x^3 + 4857*x^2 - 3056*x + 397, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 28 x^{18} - 33 x^{17} + 32 x^{16} - 142 x^{15} + 161 x^{14} + 433 x^{13} - 718 x^{12} - 430 x^{11} + 211 x^{10} + 2199 x^{9} - 1895 x^{8} - 2702 x^{7} + 8059 x^{6} - 1473 x^{5} - 11463 x^{4} + 4686 x^{3} + 4857 x^{2} - 3056 x + 397 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{199} a^{18} + \frac{89}{199} a^{17} + \frac{95}{199} a^{16} - \frac{42}{199} a^{15} - \frac{61}{199} a^{14} - \frac{14}{199} a^{13} - \frac{9}{199} a^{12} - \frac{72}{199} a^{11} + \frac{73}{199} a^{10} + \frac{49}{199} a^{9} + \frac{48}{199} a^{8} - \frac{88}{199} a^{7} - \frac{99}{199} a^{6} + \frac{1}{199} a^{5} - \frac{51}{199} a^{4} - \frac{2}{199} a^{3} - \frac{94}{199} a^{2} + \frac{48}{199} a + \frac{67}{199}$, $\frac{1}{3562263681863476958825918477957357} a^{19} + \frac{2776954352100813812667719626748}{3562263681863476958825918477957357} a^{18} + \frac{1722046138424658963048746430684276}{3562263681863476958825918477957357} a^{17} + \frac{1336497915926636602396196551426820}{3562263681863476958825918477957357} a^{16} - \frac{144774485849074322672012639975317}{3562263681863476958825918477957357} a^{15} + \frac{116152320369765662220546172791243}{3562263681863476958825918477957357} a^{14} - \frac{498011257803941659964813563402431}{3562263681863476958825918477957357} a^{13} - \frac{877994127876250145052252137199459}{3562263681863476958825918477957357} a^{12} - \frac{1064114242010632140251078243113868}{3562263681863476958825918477957357} a^{11} - \frac{1292682102203189616963200254675413}{3562263681863476958825918477957357} a^{10} + \frac{751378386969169237533569843231657}{3562263681863476958825918477957357} a^{9} + \frac{1088990937619347585813848576783101}{3562263681863476958825918477957357} a^{8} + \frac{553871387044300809217077921635673}{3562263681863476958825918477957357} a^{7} + \frac{209385210448642783394093662490960}{3562263681863476958825918477957357} a^{6} - \frac{1307659153467883295921833428801382}{3562263681863476958825918477957357} a^{5} - \frac{579101658212656038637486488697707}{3562263681863476958825918477957357} a^{4} - \frac{148412186665202875849070781704434}{3562263681863476958825918477957357} a^{3} - \frac{769959742257387167608901804812502}{3562263681863476958825918477957357} a^{2} - \frac{769370975736089782416487358192437}{3562263681863476958825918477957357} a - \frac{36459254843006447460251842149934}{3562263681863476958825918477957357}$

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:  $14$
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:  \( 15014513.7978 \) (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 $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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