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Properties

Label 20.10.6566907333...4784.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{30}\cdot 11^{19}$
Root discriminant $27.60$
Ramified primes $2, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T130

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 20, 36, -180, -226, 742, 260, -1306, 31, 1244, -242, -758, 218, 322, -114, -94, 38, 18, -8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 8*x^18 + 18*x^17 + 38*x^16 - 94*x^15 - 114*x^14 + 322*x^13 + 218*x^12 - 758*x^11 - 242*x^10 + 1244*x^9 + 31*x^8 - 1306*x^7 + 260*x^6 + 742*x^5 - 226*x^4 - 180*x^3 + 36*x^2 + 20*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 8*x^18 + 18*x^17 + 38*x^16 - 94*x^15 - 114*x^14 + 322*x^13 + 218*x^12 - 758*x^11 - 242*x^10 + 1244*x^9 + 31*x^8 - 1306*x^7 + 260*x^6 + 742*x^5 - 226*x^4 - 180*x^3 + 36*x^2 + 20*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 8 x^{18} + 18 x^{17} + 38 x^{16} - 94 x^{15} - 114 x^{14} + 322 x^{13} + 218 x^{12} - 758 x^{11} - 242 x^{10} + 1244 x^{9} + 31 x^{8} - 1306 x^{7} + 260 x^{6} + 742 x^{5} - 226 x^{4} - 180 x^{3} + 36 x^{2} + 20 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-65669073332261612842630774784=-\,2^{30}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.60$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{8839354039} a^{19} + \frac{1277504808}{8839354039} a^{18} + \frac{106076525}{8839354039} a^{17} - \frac{1905615096}{8839354039} a^{16} - \frac{1001189693}{8839354039} a^{15} + \frac{3896724756}{8839354039} a^{14} - \frac{2716204737}{8839354039} a^{13} - \frac{2060372507}{8839354039} a^{12} - \frac{3071742088}{8839354039} a^{11} + \frac{372829061}{8839354039} a^{10} - \frac{3787507871}{8839354039} a^{9} - \frac{3408418187}{8839354039} a^{8} - \frac{2485274145}{8839354039} a^{7} - \frac{3746038245}{8839354039} a^{6} + \frac{207803913}{8839354039} a^{5} - \frac{552032694}{8839354039} a^{4} - \frac{1827554794}{8839354039} a^{3} + \frac{173318749}{8839354039} a^{2} - \frac{2997519917}{8839354039} a - \frac{4223722155}{8839354039}$

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:  $14$
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:  \( 8885050.66778 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T130:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n130
Character table for t20n130 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
11Data not computed