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Properties

Label 20.6.85587764130...8384.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{50}\cdot 31^{5}\cdot 227^{4}$
Root discriminant $39.50$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3881, -22704, -60976, -99528, -111125, -91508, -58856, -30456, -11586, -1580, 2252, 2344, 1272, 484, 96, -8, -33, -20, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^18 - 20*x^17 - 33*x^16 - 8*x^15 + 96*x^14 + 484*x^13 + 1272*x^12 + 2344*x^11 + 2252*x^10 - 1580*x^9 - 11586*x^8 - 30456*x^7 - 58856*x^6 - 91508*x^5 - 111125*x^4 - 99528*x^3 - 60976*x^2 - 22704*x - 3881)
 
gp: K = bnfinit(x^20 - 4*x^18 - 20*x^17 - 33*x^16 - 8*x^15 + 96*x^14 + 484*x^13 + 1272*x^12 + 2344*x^11 + 2252*x^10 - 1580*x^9 - 11586*x^8 - 30456*x^7 - 58856*x^6 - 91508*x^5 - 111125*x^4 - 99528*x^3 - 60976*x^2 - 22704*x - 3881, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{18} - 20 x^{17} - 33 x^{16} - 8 x^{15} + 96 x^{14} + 484 x^{13} + 1272 x^{12} + 2344 x^{11} + 2252 x^{10} - 1580 x^{9} - 11586 x^{8} - 30456 x^{7} - 58856 x^{6} - 91508 x^{5} - 111125 x^{4} - 99528 x^{3} - 60976 x^{2} - 22704 x - 3881 \)

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:  \(-85587764130284320444535887888384=-\,2^{50}\cdot 31^{5}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.50$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{8} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8} a$, $\frac{1}{8} a^{18} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4}$, $\frac{1}{33019761240038709160318432088} a^{19} + \frac{621192320687968184619267561}{16509880620019354580159216044} a^{18} + \frac{799099587216142977557818957}{16509880620019354580159216044} a^{17} + \frac{156866721478460946848851225}{4127470155004838645039804011} a^{16} - \frac{189864503141094298958622590}{4127470155004838645039804011} a^{15} - \frac{918829412804120910842902079}{4127470155004838645039804011} a^{14} - \frac{3636331641832368025580829559}{16509880620019354580159216044} a^{13} - \frac{1483102567005974079892007953}{8254940310009677290079608022} a^{12} - \frac{1357954407626937445422865047}{8254940310009677290079608022} a^{11} + \frac{1476819322484790317034254218}{4127470155004838645039804011} a^{10} - \frac{6409392179953227850223446213}{16509880620019354580159216044} a^{9} - \frac{1804864811543631973879852789}{4127470155004838645039804011} a^{8} + \frac{5115386914657442142562306075}{16509880620019354580159216044} a^{7} - \frac{1830109930975541120298655253}{8254940310009677290079608022} a^{6} - \frac{3068864729738237375014121923}{16509880620019354580159216044} a^{5} + \frac{2537815757765763548676487919}{8254940310009677290079608022} a^{4} - \frac{7973156890470784790747632323}{33019761240038709160318432088} a^{3} + \frac{6169620599966794982908878409}{16509880620019354580159216044} a^{2} - \frac{1172120830598428055525910410}{4127470155004838645039804011} a + \frac{1433430015610572587651616749}{4127470155004838645039804011}$

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:  \( 315279863.18 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
$2$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.12.28.26$x^{12} + 4 x^{11} - 2 x^{10} + 4 x^{8} + 4 x^{6} + 4 x^{5} + 2$$12$$1$$28$$C_2 \times S_4$$[8/3, 8/3, 3]_{3}^{2}$
31Data not computed
227Data not computed