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Properties

Label 20.8.27296637038...4832.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{34}\cdot 7^{9}\cdot 13^{14}$
Root discriminant $46.97$
Ramified primes $2, 7, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T633

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1262, -8216, -13836, 14828, 63383, 33940, -59876, -68044, 5410, 31926, 6324, -7898, -2029, 1958, 323, -330, -42, 30, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 9*x^18 + 30*x^17 - 42*x^16 - 330*x^15 + 323*x^14 + 1958*x^13 - 2029*x^12 - 7898*x^11 + 6324*x^10 + 31926*x^9 + 5410*x^8 - 68044*x^7 - 59876*x^6 + 33940*x^5 + 63383*x^4 + 14828*x^3 - 13836*x^2 - 8216*x - 1262)
 
gp: K = bnfinit(x^20 - 6*x^19 + 9*x^18 + 30*x^17 - 42*x^16 - 330*x^15 + 323*x^14 + 1958*x^13 - 2029*x^12 - 7898*x^11 + 6324*x^10 + 31926*x^9 + 5410*x^8 - 68044*x^7 - 59876*x^6 + 33940*x^5 + 63383*x^4 + 14828*x^3 - 13836*x^2 - 8216*x - 1262, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 9 x^{18} + 30 x^{17} - 42 x^{16} - 330 x^{15} + 323 x^{14} + 1958 x^{13} - 2029 x^{12} - 7898 x^{11} + 6324 x^{10} + 31926 x^{9} + 5410 x^{8} - 68044 x^{7} - 59876 x^{6} + 33940 x^{5} + 63383 x^{4} + 14828 x^{3} - 13836 x^{2} - 8216 x - 1262 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2729663703817172900595808790904832=2^{34}\cdot 7^{9}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.97$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3259652335899721879790256279318613051964303} a^{19} - \frac{1013869815666348703719606120212170754023179}{3259652335899721879790256279318613051964303} a^{18} - \frac{267469277262058297563114223777115404813796}{3259652335899721879790256279318613051964303} a^{17} + \frac{512751893516520645899867873363492109886845}{3259652335899721879790256279318613051964303} a^{16} - \frac{63122361398530606531841219281658741251082}{3259652335899721879790256279318613051964303} a^{15} - \frac{549651637714287086632067867112612088638247}{3259652335899721879790256279318613051964303} a^{14} + \frac{1159984427364142888154622428989291865037959}{3259652335899721879790256279318613051964303} a^{13} - \frac{653388323400998797794379897367873803073684}{3259652335899721879790256279318613051964303} a^{12} + \frac{419721087197977139404196111968570922096727}{3259652335899721879790256279318613051964303} a^{11} + \frac{145608347413070120390907744453359967668076}{3259652335899721879790256279318613051964303} a^{10} + \frac{1014130385365425883473390466996692248665032}{3259652335899721879790256279318613051964303} a^{9} + \frac{610408794840724903313911441815442522366355}{3259652335899721879790256279318613051964303} a^{8} + \frac{1080983215034231946703827234147657037798584}{3259652335899721879790256279318613051964303} a^{7} - \frac{931072891128115787848861208697372588755328}{3259652335899721879790256279318613051964303} a^{6} + \frac{509328240668865905471882240950153665394246}{3259652335899721879790256279318613051964303} a^{5} + \frac{897058942053801477582761343009306080347153}{3259652335899721879790256279318613051964303} a^{4} + \frac{1033003535733595869046475866956015944131403}{3259652335899721879790256279318613051964303} a^{3} + \frac{329331772011198212063684864245457985219379}{3259652335899721879790256279318613051964303} a^{2} + \frac{242763745018591311545672958155810672655510}{3259652335899721879790256279318613051964303} a - \frac{1476513416685907703395450179967639530288978}{3259652335899721879790256279318613051964303}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 1181773051.36 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.6889792.1, 10.10.379753870426112.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$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}$