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Properties

Label 20.6.30423511290...1232.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{30}\cdot 4903^{5}$
Root discriminant $23.67$
Ramified primes $2, 4903$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, -48, 314, -1062, 1935, -2398, 2475, -1670, 762, 354, -882, 1086, -927, 612, -365, 146, -52, 6, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 5*x^18 + 6*x^17 - 52*x^16 + 146*x^15 - 365*x^14 + 612*x^13 - 927*x^12 + 1086*x^11 - 882*x^10 + 354*x^9 + 762*x^8 - 1670*x^7 + 2475*x^6 - 2398*x^5 + 1935*x^4 - 1062*x^3 + 314*x^2 - 48*x - 9)
 
gp: K = bnfinit(x^20 - 2*x^19 + 5*x^18 + 6*x^17 - 52*x^16 + 146*x^15 - 365*x^14 + 612*x^13 - 927*x^12 + 1086*x^11 - 882*x^10 + 354*x^9 + 762*x^8 - 1670*x^7 + 2475*x^6 - 2398*x^5 + 1935*x^4 - 1062*x^3 + 314*x^2 - 48*x - 9, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 5 x^{18} + 6 x^{17} - 52 x^{16} + 146 x^{15} - 365 x^{14} + 612 x^{13} - 927 x^{12} + 1086 x^{11} - 882 x^{10} + 354 x^{9} + 762 x^{8} - 1670 x^{7} + 2475 x^{6} - 2398 x^{5} + 1935 x^{4} - 1062 x^{3} + 314 x^{2} - 48 x - 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3042351129053791179853791232=-\,2^{30}\cdot 4903^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.67$
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:  $2, 4903$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{126927343393902737525906704161} a^{19} + \frac{40473364030418731102201561186}{126927343393902737525906704161} a^{18} + \frac{33110457230344935757801751774}{126927343393902737525906704161} a^{17} + \frac{4235926966560101673813345688}{42309114464634245841968901387} a^{16} + \frac{32045109368642957349044987123}{126927343393902737525906704161} a^{15} - \frac{21643351965834359398675509331}{126927343393902737525906704161} a^{14} + \frac{1408867430823311103799341463}{126927343393902737525906704161} a^{13} - \frac{6753801538901537616341176985}{42309114464634245841968901387} a^{12} - \frac{2029370995509086807717615233}{4701012718292693982440989043} a^{11} - \frac{10317539652626914146758208625}{42309114464634245841968901387} a^{10} - \frac{902131298601981052140271756}{4701012718292693982440989043} a^{9} - \frac{15486098546538348443224088102}{42309114464634245841968901387} a^{8} + \frac{13473166100768336856920194004}{42309114464634245841968901387} a^{7} + \frac{62995271380072784775979592350}{126927343393902737525906704161} a^{6} + \frac{6986997994792881289558056697}{42309114464634245841968901387} a^{5} + \frac{19334112410452561506472393373}{126927343393902737525906704161} a^{4} + \frac{19060900001458849308799694630}{42309114464634245841968901387} a^{3} - \frac{1810286834749360243662002174}{14103038154878081947322967129} a^{2} - \frac{39386614784194566455785329394}{126927343393902737525906704161} a + \frac{13556193297443823230684916949}{42309114464634245841968901387}$

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:  $12$
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:  \( 962313.007835 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.3.4903.1, 10.6.787723354112.2

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
4903Data not computed