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Properties

Label 20.0.12846961306...0752.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{16}\cdot 23^{15}$
Root discriminant $143.03$
Ramified primes $2, 11, 23$
Class number $96$ (GRH)
Class group $[2, 2, 2, 12]$ (GRH)
Galois group 20T130

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1801152661463, 0, 0, 0, 48999879259, 0, 0, 0, 449242141, 0, 0, 0, 1662647, 0, 0, 0, 2346, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2346*x^16 + 1662647*x^12 + 449242141*x^8 + 48999879259*x^4 + 1801152661463)
 
gp: K = bnfinit(x^20 + 2346*x^16 + 1662647*x^12 + 449242141*x^8 + 48999879259*x^4 + 1801152661463, 1)
 

Normalized defining polynomial

\( x^{20} + 2346 x^{16} + 1662647 x^{12} + 449242141 x^{8} + 48999879259 x^{4} + 1801152661463 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12846961306316291046734327469906326168010752=2^{20}\cdot 11^{16}\cdot 23^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.03$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{23} a^{4}$, $\frac{1}{23} a^{5}$, $\frac{1}{23} a^{6}$, $\frac{1}{23} a^{7}$, $\frac{1}{529} a^{8}$, $\frac{1}{529} a^{9}$, $\frac{1}{24334} a^{10} + \frac{21}{1058} a^{6} - \frac{1}{46} a^{4} - \frac{7}{46} a^{2} - \frac{1}{2}$, $\frac{1}{559682} a^{11} - \frac{163}{24334} a^{7} - \frac{1}{46} a^{5} + \frac{315}{1058} a^{3} - \frac{1}{2} a$, $\frac{1}{12872686} a^{12} - \frac{163}{559682} a^{8} - \frac{1}{46} a^{6} + \frac{315}{24334} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{12872686} a^{13} - \frac{163}{559682} a^{9} - \frac{1}{46} a^{7} + \frac{315}{24334} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{12872686} a^{14} - \frac{1}{279841} a^{10} - \frac{1}{1058} a^{8} + \frac{261}{12167} a^{6} - \frac{3}{46} a^{2} - \frac{1}{2}$, $\frac{1}{12872686} a^{15} - \frac{1}{1058} a^{9} + \frac{98}{12167} a^{7} - \frac{497}{1058} a^{3} - \frac{1}{2} a$, $\frac{1}{15111799620898} a^{16} + \frac{7741}{657034766126} a^{12} - \frac{9819749}{28566728962} a^{8} + \frac{2493463}{1242031694} a^{4} - \frac{1}{2} a^{2} + \frac{50179}{102082}$, $\frac{1}{15111799620898} a^{17} + \frac{7741}{657034766126} a^{13} - \frac{9819749}{28566728962} a^{9} + \frac{2493463}{1242031694} a^{5} - \frac{1}{2} a^{3} + \frac{50179}{102082} a$, $\frac{1}{347571391280654} a^{18} + \frac{131473}{7555899810449} a^{14} + \frac{1291607}{328517383063} a^{10} - \frac{1}{1058} a^{8} - \frac{255566060}{14283364481} a^{6} - \frac{460231}{2347886} a^{2}$, $\frac{1}{7994141999455042} a^{19} - \frac{5606769}{347571391280654} a^{15} - \frac{12678045}{15111799620898} a^{11} - \frac{6626201207}{657034766126} a^{7} - \frac{1}{46} a^{5} + \frac{3010557}{54001378} a^{3}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{12}$, which has order $96$ (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:  $9$
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:  \( 131114379145 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T130:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n130
Character table for t20n130 is not computed

Intermediate fields

\(\Q(\sqrt{-23}) \), \(\Q(\zeta_{11})^+\), 10.0.1379687283212183.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$2$2.10.10.4$x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.10.14$x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$