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Properties

Label 20.0.19901427351...4809.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 71^{2}\cdot 401^{8}$
Root discriminant $29.17$
Ramified primes $3, 71, 401$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 10, -6, -10, 15, -3, -7, 7, 0, -5, 0, 7, -7, -3, 15, -10, -6, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 - 6*x^17 - 10*x^16 + 15*x^15 - 3*x^14 - 7*x^13 + 7*x^12 - 5*x^10 + 7*x^8 - 7*x^7 - 3*x^6 + 15*x^5 - 10*x^4 - 6*x^3 + 10*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 10*x^18 - 6*x^17 - 10*x^16 + 15*x^15 - 3*x^14 - 7*x^13 + 7*x^12 - 5*x^10 + 7*x^8 - 7*x^7 - 3*x^6 + 15*x^5 - 10*x^4 - 6*x^3 + 10*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 10 x^{18} - 6 x^{17} - 10 x^{16} + 15 x^{15} - 3 x^{14} - 7 x^{13} + 7 x^{12} - 5 x^{10} + 7 x^{8} - 7 x^{7} - 3 x^{6} + 15 x^{5} - 10 x^{4} - 6 x^{3} + 10 x^{2} - 4 x + 1 \)

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:  \(199014273518726850560805214809=3^{10}\cdot 71^{2}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.17$
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:  $3, 71, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{29607} a^{18} + \frac{1081}{9869} a^{17} - \frac{3354}{9869} a^{16} + \frac{3855}{9869} a^{15} - \frac{9676}{29607} a^{14} + \frac{13112}{29607} a^{13} + \frac{3157}{9869} a^{12} + \frac{2384}{9869} a^{11} + \frac{1192}{29607} a^{10} + \frac{14362}{29607} a^{9} + \frac{1192}{29607} a^{8} + \frac{2384}{9869} a^{7} + \frac{3157}{9869} a^{6} + \frac{13112}{29607} a^{5} - \frac{9676}{29607} a^{4} + \frac{3855}{9869} a^{3} - \frac{3354}{9869} a^{2} + \frac{1081}{9869} a + \frac{1}{29607}$, $\frac{1}{29607} a^{19} + \frac{4327}{9869} a^{17} - \frac{4630}{9869} a^{16} - \frac{2902}{29607} a^{15} + \frac{8960}{29607} a^{14} + \frac{969}{9869} a^{13} - \frac{1614}{9869} a^{12} - \frac{10463}{29607} a^{11} - \frac{2384}{29607} a^{10} - \frac{2963}{29607} a^{9} - \frac{3198}{9869} a^{8} - \frac{728}{9869} a^{7} + \frac{1118}{29607} a^{6} + \frac{13367}{29607} a^{5} + \frac{2471}{9869} a^{4} - \frac{1096}{9869} a^{3} + \frac{2465}{9869} a^{2} - \frac{6563}{29607} a - \frac{1081}{9869}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( \frac{8849}{29607} a^{19} - \frac{34882}{29607} a^{18} + \frac{29517}{9869} a^{17} - \frac{17287}{9869} a^{16} - \frac{84698}{29607} a^{15} + \frac{145961}{29607} a^{14} - \frac{37895}{29607} a^{13} - \frac{25753}{9869} a^{12} + \frac{45734}{29607} a^{11} + \frac{2659}{29607} a^{10} - \frac{42236}{29607} a^{9} + \frac{4454}{29607} a^{8} + \frac{19629}{9869} a^{7} - \frac{67286}{29607} a^{6} - \frac{28837}{29607} a^{5} + \frac{134992}{29607} a^{4} - \frac{31869}{9869} a^{3} - \frac{20210}{9869} a^{2} + \frac{107789}{29607} a - \frac{13399}{29607} \) (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:  \( 1197823.58007 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

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 t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.160801.1, 10.0.6283241669043.1, 10.8.148703386167351.1, 10.2.5507532821013.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed