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Properties

Label 20.4.84403903732...8257.1
Degree $20$
Signature $[4, 8]$
Discriminant $23\cdot 5003\cdot 322537\cdot 2274179039448269$
Root discriminant $19.78$
Ramified primes $23, 5003, 322537, 2274179039448269$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1117

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -2, 6, 1, -6, -3, 4, 7, -5, -6, 2, 4, 1, -6, 0, 3, 1, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 + x^17 + 3*x^16 - 6*x^14 + x^13 + 4*x^12 + 2*x^11 - 6*x^10 - 5*x^9 + 7*x^8 + 4*x^7 - 3*x^6 - 6*x^5 + x^4 + 6*x^3 - 2*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - x^19 - x^18 + x^17 + 3*x^16 - 6*x^14 + x^13 + 4*x^12 + 2*x^11 - 6*x^10 - 5*x^9 + 7*x^8 + 4*x^7 - 3*x^6 - 6*x^5 + x^4 + 6*x^3 - 2*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - x^{18} + x^{17} + 3 x^{16} - 6 x^{14} + x^{13} + 4 x^{12} + 2 x^{11} - 6 x^{10} - 5 x^{9} + 7 x^{8} + 4 x^{7} - 3 x^{6} - 6 x^{5} + x^{4} + 6 x^{3} - 2 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(84403903732404939239448257=23\cdot 5003\cdot 322537\cdot 2274179039448269\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.78$
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:  $23, 5003, 322537, 2274179039448269$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{53} a^{19} - \frac{17}{53} a^{18} + \frac{6}{53} a^{17} + \frac{11}{53} a^{16} - \frac{14}{53} a^{15} + \frac{12}{53} a^{14} + \frac{14}{53} a^{13} - \frac{11}{53} a^{12} + \frac{21}{53} a^{11} - \frac{16}{53} a^{10} - \frac{15}{53} a^{9} + \frac{23}{53} a^{8} + \frac{10}{53} a^{7} + \frac{3}{53} a^{6} + \frac{2}{53} a^{5} + \frac{15}{53} a^{4} + \frac{26}{53} a^{3} + \frac{14}{53} a^{2} - \frac{14}{53} a + \frac{10}{53}$

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:  $11$
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:  \( 105884.714067 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1117:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2432902008176640000
The 627 conjugacy class representatives for t20n1117 are not computed
Character table for t20n1117 is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ $16{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ $17{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $19{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $17{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ $17{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.11.0.1}{11} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
23.13.0.1$x^{13} - 5 x + 3$$1$$13$$0$$C_{13}$$[\ ]^{13}$
5003Data not computed
322537Data not computed
2274179039448269Data not computed