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Properties

Label 20.20.1719105272...6576.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 11^{18}\cdot 7369^{4}$
Root discriminant $102.75$
Ramified primes $2, 11, 7369$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T331

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6570561481, 0, -10085441839, 0, 6447181169, 0, -2248682392, 0, 472121067, 0, -62146183, 0, 5181616, 0, -270083, 0, 8437, 0, -143, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 143*x^18 + 8437*x^16 - 270083*x^14 + 5181616*x^12 - 62146183*x^10 + 472121067*x^8 - 2248682392*x^6 + 6447181169*x^4 - 10085441839*x^2 + 6570561481)
 
gp: K = bnfinit(x^20 - 143*x^18 + 8437*x^16 - 270083*x^14 + 5181616*x^12 - 62146183*x^10 + 472121067*x^8 - 2248682392*x^6 + 6447181169*x^4 - 10085441839*x^2 + 6570561481, 1)
 

Normalized defining polynomial

\( x^{20} - 143 x^{18} + 8437 x^{16} - 270083 x^{14} + 5181616 x^{12} - 62146183 x^{10} + 472121067 x^{8} - 2248682392 x^{6} + 6447181169 x^{4} - 10085441839 x^{2} + 6570561481 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17191052721619032309920296885375992856576=2^{20}\cdot 11^{18}\cdot 7369^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $102.75$
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:  $2, 11, 7369$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10}$, $\frac{1}{11} a^{11}$, $\frac{1}{11} a^{12}$, $\frac{1}{11} a^{13}$, $\frac{1}{11} a^{14}$, $\frac{1}{11} a^{15}$, $\frac{1}{11} a^{16}$, $\frac{1}{11} a^{17}$, $\frac{1}{787530434310336039837686596562083} a^{18} - \frac{2761554414850983865797833327967}{71593675846394185439789690596553} a^{16} + \frac{400162151252212313765089263921}{71593675846394185439789690596553} a^{14} + \frac{355301300874578013624289139447}{71593675846394185439789690596553} a^{12} - \frac{3048454525185599356229548153453}{71593675846394185439789690596553} a^{10} + \frac{21640835895880296804537344691781}{71593675846394185439789690596553} a^{8} + \frac{2111562728526631421393383611334}{6508515986035835039980880963323} a^{6} + \frac{719463196249867770957189957996}{6508515986035835039980880963323} a^{4} - \frac{98587785208640431829893632757}{6508515986035835039980880963323} a^{2} - \frac{79445388556442580841430239}{883229201524743525577538467}$, $\frac{1}{787530434310336039837686596562083} a^{19} - \frac{2761554414850983865797833327967}{71593675846394185439789690596553} a^{17} + \frac{400162151252212313765089263921}{71593675846394185439789690596553} a^{15} + \frac{355301300874578013624289139447}{71593675846394185439789690596553} a^{13} - \frac{3048454525185599356229548153453}{71593675846394185439789690596553} a^{11} + \frac{21640835895880296804537344691781}{71593675846394185439789690596553} a^{9} + \frac{2111562728526631421393383611334}{6508515986035835039980880963323} a^{7} + \frac{719463196249867770957189957996}{6508515986035835039980880963323} a^{5} - \frac{98587785208640431829893632757}{6508515986035835039980880963323} a^{3} - \frac{79445388556442580841430239}{883229201524743525577538467} a$

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:  $19$
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:  \( 47784305792100 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T331:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 56 conjugacy class representatives for t20n331 are not computed
Character table for t20n331 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.1579610594089.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.12$x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.10.12$x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
11Data not computed
7369Data not computed