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Properties

Label 20.0.64950391969...5921.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 1483^{2}\cdot 2236369^{2}$
Root discriminant $15.51$
Ramified primes $3, 1483, 2236369$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1021

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 4, -1, 11, -9, 3, -18, 7, -7, 6, -1, 7, 3, 1, -3, -1, -2, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^17 - x^16 - 3*x^15 + x^14 + 3*x^13 + 7*x^12 - x^11 + 6*x^10 - 7*x^9 + 7*x^8 - 18*x^7 + 3*x^6 - 9*x^5 + 11*x^4 - x^3 + 4*x^2 - x + 1)
 
gp: K = bnfinit(x^20 - 2*x^17 - x^16 - 3*x^15 + x^14 + 3*x^13 + 7*x^12 - x^11 + 6*x^10 - 7*x^9 + 7*x^8 - 18*x^7 + 3*x^6 - 9*x^5 + 11*x^4 - x^3 + 4*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{17} - x^{16} - 3 x^{15} + x^{14} + 3 x^{13} + 7 x^{12} - x^{11} + 6 x^{10} - 7 x^{9} + 7 x^{8} - 18 x^{7} + 3 x^{6} - 9 x^{5} + 11 x^{4} - x^{3} + 4 x^{2} - x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(649503919693669215345921=3^{10}\cdot 1483^{2}\cdot 2236369^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.51$
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:  $3, 1483, 2236369$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{74467} a^{19} - \frac{3394}{74467} a^{18} - \frac{23149}{74467} a^{17} + \frac{5019}{74467} a^{16} + \frac{18456}{74467} a^{15} - \frac{12920}{74467} a^{14} - \frac{10582}{74467} a^{13} + \frac{22217}{74467} a^{12} + \frac{30580}{74467} a^{11} + \frac{18477}{74467} a^{10} - \frac{86}{659} a^{9} - \frac{5996}{74467} a^{8} + \frac{20940}{74467} a^{7} - \frac{28860}{74467} a^{6} + \frac{26738}{74467} a^{5} + \frac{26492}{74467} a^{4} - \frac{32168}{74467} a^{3} + \frac{9569}{74467} a^{2} - \frac{9570}{74467} a + \frac{12967}{74467}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{10766}{74467} a^{19} - \frac{23493}{74467} a^{18} - \frac{18915}{74467} a^{17} + \frac{28488}{74467} a^{16} + \frac{55127}{74467} a^{15} + \frac{66831}{74467} a^{14} + \frac{65769}{74467} a^{13} - \frac{218}{74467} a^{12} - \frac{154607}{74467} a^{11} - \frac{170959}{74467} a^{10} - \frac{678}{659} a^{9} - \frac{9953}{74467} a^{8} - \frac{102898}{74467} a^{7} + \frac{104903}{74467} a^{6} + \frac{251515}{74467} a^{5} + \frac{219139}{74467} a^{4} - \frac{25329}{74467} a^{3} - \frac{180927}{74467} a^{2} - \frac{31708}{74467} a + \frac{22903}{74467} \) (order $6$)
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:  \( 8617.1364564 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1021:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7257600
The 84 conjugacy class representatives for t20n1021 are not computed
Character table for t20n1021 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.4.3316535227.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ $18{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
3Data not computed
1483Data not computed
2236369Data not computed