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Properties

Label 20.10.7482430020...7419.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{17}\cdot 23^{6}$
Root discriminant $19.67$
Ramified primes $11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T310

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -37, 152, -267, 322, -210, -518, 1401, -865, -819, 1385, -333, -538, 353, 62, -108, 3, 23, -4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 4*x^18 + 23*x^17 + 3*x^16 - 108*x^15 + 62*x^14 + 353*x^13 - 538*x^12 - 333*x^11 + 1385*x^10 - 819*x^9 - 865*x^8 + 1401*x^7 - 518*x^6 - 210*x^5 + 322*x^4 - 267*x^3 + 152*x^2 - 37*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 - 4*x^18 + 23*x^17 + 3*x^16 - 108*x^15 + 62*x^14 + 353*x^13 - 538*x^12 - 333*x^11 + 1385*x^10 - 819*x^9 - 865*x^8 + 1401*x^7 - 518*x^6 - 210*x^5 + 322*x^4 - 267*x^3 + 152*x^2 - 37*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 4 x^{18} + 23 x^{17} + 3 x^{16} - 108 x^{15} + 62 x^{14} + 353 x^{13} - 538 x^{12} - 333 x^{11} + 1385 x^{10} - 819 x^{9} - 865 x^{8} + 1401 x^{7} - 518 x^{6} - 210 x^{5} + 322 x^{4} - 267 x^{3} + 152 x^{2} - 37 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-74824300206301289262147419=-\,11^{17}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.67$
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:  $11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} - \frac{2}{23} a^{15} + \frac{1}{23} a^{14} - \frac{8}{23} a^{13} - \frac{10}{23} a^{12} - \frac{4}{23} a^{11} - \frac{8}{23} a^{10} + \frac{7}{23} a^{9} + \frac{11}{23} a^{8} - \frac{6}{23} a^{7} - \frac{11}{23} a^{6} - \frac{3}{23} a^{5} - \frac{8}{23} a^{4} - \frac{5}{23} a^{3} + \frac{3}{23} a^{2} - \frac{11}{23}$, $\frac{1}{23} a^{17} - \frac{3}{23} a^{15} - \frac{6}{23} a^{14} - \frac{3}{23} a^{13} - \frac{1}{23} a^{12} + \frac{7}{23} a^{11} - \frac{9}{23} a^{10} + \frac{2}{23} a^{9} - \frac{7}{23} a^{8} - \frac{2}{23} a^{6} + \frac{9}{23} a^{5} + \frac{2}{23} a^{4} - \frac{7}{23} a^{3} + \frac{6}{23} a^{2} - \frac{11}{23} a + \frac{1}{23}$, $\frac{1}{23} a^{18} + \frac{11}{23} a^{15} - \frac{2}{23} a^{13} + \frac{2}{23} a^{11} + \frac{1}{23} a^{10} - \frac{9}{23} a^{9} + \frac{10}{23} a^{8} + \frac{3}{23} a^{7} - \frac{1}{23} a^{6} - \frac{7}{23} a^{5} - \frac{8}{23} a^{4} - \frac{9}{23} a^{3} - \frac{2}{23} a^{2} + \frac{1}{23} a - \frac{10}{23}$, $\frac{1}{75998920962361} a^{19} + \frac{38686911819}{75998920962361} a^{18} - \frac{568716581779}{75998920962361} a^{17} + \frac{1457124330318}{75998920962361} a^{16} - \frac{19066484440724}{75998920962361} a^{15} + \frac{662517620575}{3304300911407} a^{14} - \frac{13010319558437}{75998920962361} a^{13} + \frac{26809512942862}{75998920962361} a^{12} - \frac{14963263152982}{75998920962361} a^{11} - \frac{521990037476}{3304300911407} a^{10} + \frac{16327088135880}{75998920962361} a^{9} + \frac{33981354350415}{75998920962361} a^{8} + \frac{1273661320487}{3304300911407} a^{7} + \frac{130723722901}{3304300911407} a^{6} + \frac{8539525035006}{75998920962361} a^{5} - \frac{1510246636969}{75998920962361} a^{4} + \frac{37850926631171}{75998920962361} a^{3} - \frac{7077231569056}{75998920962361} a^{2} + \frac{12829318967956}{75998920962361} a - \frac{19398842641614}{75998920962361}$

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:  \( 298913.61697 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T310:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 44 conjugacy class representatives for t20n310
Character table for t20n310 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.113395848049.1

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 ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$