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Properties

Label 20.0.33060197995...8961.1
Degree $20$
Signature $[0, 10]$
Discriminant $653^{8}$
Root discriminant $13.36$
Ramified prime $653$
Class number $1$
Class group Trivial
Galois group $C_2\times A_5$ (as 20T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 4, -6, 10, -15, 21, -27, 37, -34, 47, -34, 37, -27, 21, -15, 10, -6, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 4*x^18 - 6*x^17 + 10*x^16 - 15*x^15 + 21*x^14 - 27*x^13 + 37*x^12 - 34*x^11 + 47*x^10 - 34*x^9 + 37*x^8 - 27*x^7 + 21*x^6 - 15*x^5 + 10*x^4 - 6*x^3 + 4*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 4*x^18 - 6*x^17 + 10*x^16 - 15*x^15 + 21*x^14 - 27*x^13 + 37*x^12 - 34*x^11 + 47*x^10 - 34*x^9 + 37*x^8 - 27*x^7 + 21*x^6 - 15*x^5 + 10*x^4 - 6*x^3 + 4*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 4 x^{18} - 6 x^{17} + 10 x^{16} - 15 x^{15} + 21 x^{14} - 27 x^{13} + 37 x^{12} - 34 x^{11} + 47 x^{10} - 34 x^{9} + 37 x^{8} - 27 x^{7} + 21 x^{6} - 15 x^{5} + 10 x^{4} - 6 x^{3} + 4 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33060197995068669948961=653^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.36$
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:  $653$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{137} a^{18} + \frac{14}{137} a^{17} - \frac{47}{137} a^{16} + \frac{50}{137} a^{15} + \frac{35}{137} a^{14} - \frac{53}{137} a^{13} - \frac{40}{137} a^{12} - \frac{66}{137} a^{11} - \frac{20}{137} a^{10} - \frac{14}{137} a^{9} - \frac{20}{137} a^{8} - \frac{66}{137} a^{7} - \frac{40}{137} a^{6} - \frac{53}{137} a^{5} + \frac{35}{137} a^{4} + \frac{50}{137} a^{3} - \frac{47}{137} a^{2} + \frac{14}{137} a + \frac{1}{137}$, $\frac{1}{13015} a^{19} - \frac{41}{13015} a^{18} + \frac{3978}{13015} a^{17} - \frac{4763}{13015} a^{16} - \frac{3948}{13015} a^{15} - \frac{32}{685} a^{14} + \frac{2738}{13015} a^{13} + \frac{1586}{13015} a^{12} - \frac{637}{13015} a^{11} - \frac{1791}{13015} a^{10} - \frac{3634}{13015} a^{9} + \frac{4322}{13015} a^{8} + \frac{5234}{13015} a^{7} + \frac{3517}{13015} a^{6} + \frac{112}{685} a^{5} - \frac{642}{13015} a^{4} + \frac{4053}{13015} a^{3} - \frac{278}{13015} a^{2} + \frac{3341}{13015} a - \frac{5946}{13015}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 932.282807106 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_5$ (as 20T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

10.2.181824635281.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 12 siblings: data not computed
Degree 20 sibling: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
653Data not computed