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Properties

Label 20.2.31951141691...7779.2
Degree $20$
Signature $[2, 9]$
Discriminant $-\,11^{17}\cdot 43^{6}$
Root discriminant $23.73$
Ramified primes $11, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T326

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, -52, 22, -257, 104, -316, -38, 5, -329, 444, 32, 55, 323, -70, 44, 51, -31, 11, 4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 4*x^18 + 11*x^17 - 31*x^16 + 51*x^15 + 44*x^14 - 70*x^13 + 323*x^12 + 55*x^11 + 32*x^10 + 444*x^9 - 329*x^8 + 5*x^7 - 38*x^6 - 316*x^5 + 104*x^4 - 257*x^3 + 22*x^2 - 52*x - 23)
 
gp: K = bnfinit(x^20 - 3*x^19 + 4*x^18 + 11*x^17 - 31*x^16 + 51*x^15 + 44*x^14 - 70*x^13 + 323*x^12 + 55*x^11 + 32*x^10 + 444*x^9 - 329*x^8 + 5*x^7 - 38*x^6 - 316*x^5 + 104*x^4 - 257*x^3 + 22*x^2 - 52*x - 23, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 4 x^{18} + 11 x^{17} - 31 x^{16} + 51 x^{15} + 44 x^{14} - 70 x^{13} + 323 x^{12} + 55 x^{11} + 32 x^{10} + 444 x^{9} - 329 x^{8} + 5 x^{7} - 38 x^{6} - 316 x^{5} + 104 x^{4} - 257 x^{3} + 22 x^{2} - 52 x - 23 \)

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:  \(-3195114169182285566595267779=-\,11^{17}\cdot 43^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.73$
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:  $11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{46} a^{18} + \frac{3}{23} a^{17} + \frac{11}{46} a^{16} - \frac{11}{46} a^{15} - \frac{3}{46} a^{14} - \frac{11}{46} a^{13} + \frac{17}{46} a^{12} + \frac{1}{23} a^{11} + \frac{1}{23} a^{10} - \frac{21}{46} a^{9} + \frac{1}{23} a^{8} + \frac{7}{23} a^{6} - \frac{7}{46} a^{5} + \frac{13}{46} a^{3} + \frac{7}{23} a^{2} - \frac{3}{23} a$, $\frac{1}{52718313482781274346191226} a^{19} + \frac{42374846810808578972537}{26359156741390637173095613} a^{18} + \frac{3550885895242732066094208}{26359156741390637173095613} a^{17} - \frac{4735533559895478437373295}{26359156741390637173095613} a^{16} + \frac{3695149418606428295270094}{26359156741390637173095613} a^{15} - \frac{16654931004749072384767755}{52718313482781274346191226} a^{14} + \frac{10196149536399473864073943}{52718313482781274346191226} a^{13} + \frac{6358583165872633807907283}{26359156741390637173095613} a^{12} - \frac{1408887544707315663575389}{26359156741390637173095613} a^{11} - \frac{9587940725452627886715834}{26359156741390637173095613} a^{10} - \frac{10193517382482941536427551}{26359156741390637173095613} a^{9} + \frac{24903404875051014636790993}{52718313482781274346191226} a^{8} - \frac{25947942215321258173642579}{52718313482781274346191226} a^{7} - \frac{18597528995372703930044563}{52718313482781274346191226} a^{6} - \frac{6632923299941049634534657}{26359156741390637173095613} a^{5} - \frac{11047091825939340975980200}{26359156741390637173095613} a^{4} + \frac{564517023323458056140651}{52718313482781274346191226} a^{3} - \frac{3438247699207717174499185}{52718313482781274346191226} a^{2} - \frac{424178971322279553404329}{52718313482781274346191226} a - \frac{213588256990531468545140}{1146050293103940746656331}$

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:  $10$
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:  \( 556828.06097 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T326:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n326 are not computed
Character table for t20n326 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.396349570969.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
43Data not computed