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Properties

Label 20.0.11720767680...9792.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 13^{13}$
Root discriminant $15.98$
Ramified primes $2, 3, 13$
Class number $1$
Class group Trivial
Galois group $C_2^2:F_5$ (as 20T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 45, -163, 464, -1081, 2121, -3545, 5100, -6329, 6805, -6329, 5100, -3545, 2121, -1081, 464, -163, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 45*x^18 - 163*x^17 + 464*x^16 - 1081*x^15 + 2121*x^14 - 3545*x^13 + 5100*x^12 - 6329*x^11 + 6805*x^10 - 6329*x^9 + 5100*x^8 - 3545*x^7 + 2121*x^6 - 1081*x^5 + 464*x^4 - 163*x^3 + 45*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^20 - 9*x^19 + 45*x^18 - 163*x^17 + 464*x^16 - 1081*x^15 + 2121*x^14 - 3545*x^13 + 5100*x^12 - 6329*x^11 + 6805*x^10 - 6329*x^9 + 5100*x^8 - 3545*x^7 + 2121*x^6 - 1081*x^5 + 464*x^4 - 163*x^3 + 45*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 45 x^{18} - 163 x^{17} + 464 x^{16} - 1081 x^{15} + 2121 x^{14} - 3545 x^{13} + 5100 x^{12} - 6329 x^{11} + 6805 x^{10} - 6329 x^{9} + 5100 x^{8} - 3545 x^{7} + 2121 x^{6} - 1081 x^{5} + 464 x^{4} - 163 x^{3} + 45 x^{2} - 9 x + 1 \)

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:  \(1172076768078459528609792=2^{16}\cdot 3^{10}\cdot 13^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.98$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{211} a^{17} - \frac{38}{211} a^{16} + \frac{63}{211} a^{15} - \frac{45}{211} a^{14} - \frac{20}{211} a^{13} - \frac{103}{211} a^{12} + \frac{36}{211} a^{11} - \frac{105}{211} a^{10} + \frac{30}{211} a^{9} + \frac{30}{211} a^{8} - \frac{105}{211} a^{7} + \frac{36}{211} a^{6} - \frac{103}{211} a^{5} - \frac{20}{211} a^{4} - \frac{45}{211} a^{3} + \frac{63}{211} a^{2} - \frac{38}{211} a + \frac{1}{211}$, $\frac{1}{211} a^{18} + \frac{96}{211} a^{16} + \frac{28}{211} a^{15} - \frac{42}{211} a^{14} - \frac{19}{211} a^{13} - \frac{80}{211} a^{12} - \frac{3}{211} a^{11} + \frac{49}{211} a^{10} - \frac{96}{211} a^{9} - \frac{20}{211} a^{8} + \frac{55}{211} a^{7} - \frac{1}{211} a^{6} + \frac{75}{211} a^{5} + \frac{39}{211} a^{4} + \frac{41}{211} a^{3} + \frac{35}{211} a^{2} + \frac{34}{211} a + \frac{38}{211}$, $\frac{1}{211} a^{19} + \frac{89}{211} a^{16} + \frac{29}{211} a^{15} + \frac{81}{211} a^{14} - \frac{59}{211} a^{13} - \frac{32}{211} a^{12} - \frac{31}{211} a^{11} + \frac{67}{211} a^{10} + \frac{54}{211} a^{9} - \frac{82}{211} a^{8} - \frac{49}{211} a^{7} - \frac{5}{211} a^{6} + \frac{10}{211} a^{5} + \frac{62}{211} a^{4} - \frac{76}{211} a^{3} + \frac{105}{211} a^{2} + \frac{99}{211} a - \frac{96}{211}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{292}{211} a^{19} + \frac{2131}{211} a^{18} - \frac{8807}{211} a^{17} + \frac{26688}{211} a^{16} - \frac{62440}{211} a^{15} + \frac{115627}{211} a^{14} - \frac{173327}{211} a^{13} + \frac{202027}{211} a^{12} - \frac{174078}{211} a^{11} + \frac{78447}{211} a^{10} + \frac{39781}{211} a^{9} - \frac{133287}{211} a^{8} + \frac{154012}{211} a^{7} - \frac{121493}{211} a^{6} + \frac{64941}{211} a^{5} - \frac{23660}{211} a^{4} + \frac{2432}{211} a^{3} + \frac{2025}{211} a^{2} - \frac{1799}{211} a + \frac{611}{211} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 19012.2439561 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:F_5$ (as 20T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_2^2:F_5$
Character table for $C_2^2:F_5$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.117.1, 5.1.35152.1, 10.0.300266134272.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
2Data not computed
3Data not computed
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$