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Properties

Label 20.10.2584296997...6848.7
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{38}\cdot 7^{9}\cdot 13^{12}$
Root discriminant $41.75$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T633

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![46, 680, 3193, 5374, 1007, -11212, -23401, -15958, 15617, 32860, 14363, -6006, -6153, -876, 85, -78, 53, 24, -13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 13*x^18 + 24*x^17 + 53*x^16 - 78*x^15 + 85*x^14 - 876*x^13 - 6153*x^12 - 6006*x^11 + 14363*x^10 + 32860*x^9 + 15617*x^8 - 15958*x^7 - 23401*x^6 - 11212*x^5 + 1007*x^4 + 5374*x^3 + 3193*x^2 + 680*x + 46)
 
gp: K = bnfinit(x^20 - 2*x^19 - 13*x^18 + 24*x^17 + 53*x^16 - 78*x^15 + 85*x^14 - 876*x^13 - 6153*x^12 - 6006*x^11 + 14363*x^10 + 32860*x^9 + 15617*x^8 - 15958*x^7 - 23401*x^6 - 11212*x^5 + 1007*x^4 + 5374*x^3 + 3193*x^2 + 680*x + 46, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 13 x^{18} + 24 x^{17} + 53 x^{16} - 78 x^{15} + 85 x^{14} - 876 x^{13} - 6153 x^{12} - 6006 x^{11} + 14363 x^{10} + 32860 x^{9} + 15617 x^{8} - 15958 x^{7} - 23401 x^{6} - 11212 x^{5} + 1007 x^{4} + 5374 x^{3} + 3193 x^{2} + 680 x + 46 \)

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:  \(-258429699769673174020904974286848=-\,2^{38}\cdot 7^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.75$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{3}{7} a^{14} - \frac{1}{7} a^{13} - \frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{1}{7} a^{10} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7}$, $\frac{1}{7} a^{16} - \frac{3}{7} a^{14} + \frac{2}{7} a^{13} + \frac{2}{7} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} + \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{49} a^{17} + \frac{1}{49} a^{16} + \frac{1}{49} a^{15} - \frac{24}{49} a^{14} - \frac{3}{7} a^{13} + \frac{23}{49} a^{12} - \frac{10}{49} a^{11} + \frac{12}{49} a^{10} - \frac{9}{49} a^{9} + \frac{11}{49} a^{8} - \frac{20}{49} a^{7} + \frac{12}{49} a^{6} + \frac{2}{49} a^{5} + \frac{22}{49} a^{4} + \frac{2}{49} a^{3} - \frac{2}{7} a^{2} + \frac{15}{49} a - \frac{4}{49}$, $\frac{1}{49} a^{18} + \frac{3}{49} a^{15} - \frac{11}{49} a^{14} + \frac{16}{49} a^{13} - \frac{12}{49} a^{12} - \frac{6}{49} a^{11} + \frac{1}{7} a^{10} + \frac{20}{49} a^{9} - \frac{17}{49} a^{8} + \frac{4}{49} a^{7} - \frac{17}{49} a^{6} + \frac{13}{49} a^{5} + \frac{15}{49} a^{4} + \frac{19}{49} a^{3} - \frac{6}{49} a^{2} - \frac{19}{49} a - \frac{10}{49}$, $\frac{1}{16756255177221744519226234} a^{19} + \frac{53423665010351995467527}{16756255177221744519226234} a^{18} - \frac{11915758117547229515533}{1196875369801553179944731} a^{17} - \frac{18206451305964145820668}{8378127588610872259613117} a^{16} - \frac{10036595818861579104845}{341964391371872337127066} a^{15} + \frac{1805609099390006902768811}{16756255177221744519226234} a^{14} + \frac{31989986324110638810481}{170982195685936168563533} a^{13} - \frac{2189691528990239945611868}{8378127588610872259613117} a^{12} + \frac{2936671752491510023077807}{16756255177221744519226234} a^{11} - \frac{3209149064813599361929399}{16756255177221744519226234} a^{10} - \frac{2156339752960639072344092}{8378127588610872259613117} a^{9} + \frac{578403159598293920426990}{1196875369801553179944731} a^{8} - \frac{982798840745998291216209}{2393750739603106359889462} a^{7} + \frac{6998927114202197039369009}{16756255177221744519226234} a^{6} + \frac{3780906965090701627188274}{8378127588610872259613117} a^{5} + \frac{2098585532074093498704123}{8378127588610872259613117} a^{4} + \frac{927998566545626500692883}{16756255177221744519226234} a^{3} - \frac{2770671468776557963775675}{16756255177221744519226234} a^{2} + \frac{3003749411038445547836380}{8378127588610872259613117} a + \frac{74139726400514211538036}{8378127588610872259613117}$

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:  \( 3991083011.2 \) (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 $20$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{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}$
13Data not computed