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Properties

Label 20.2.17297979487...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{20}\cdot 5^{10}\cdot 11^{4}\cdot 1630891^{5}$
Root discriminant $258.17$
Ramified primes $2, 5, 11, 1630891$
Class number Not computed
Class group Not computed
Galois group 20T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4337852776485437971, 0, -321799222643246666, 0, -1875817034507589, 0, 351613585853425, 0, 8603288846964, 0, 9530036138, 0, -1395371322, 0, -8411745, 0, 59526, 0, 538, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 538*x^18 + 59526*x^16 - 8411745*x^14 - 1395371322*x^12 + 9530036138*x^10 + 8603288846964*x^8 + 351613585853425*x^6 - 1875817034507589*x^4 - 321799222643246666*x^2 - 4337852776485437971)
 
gp: K = bnfinit(x^20 + 538*x^18 + 59526*x^16 - 8411745*x^14 - 1395371322*x^12 + 9530036138*x^10 + 8603288846964*x^8 + 351613585853425*x^6 - 1875817034507589*x^4 - 321799222643246666*x^2 - 4337852776485437971, 1)
 

Normalized defining polynomial

\( x^{20} + 538 x^{18} + 59526 x^{16} - 8411745 x^{14} - 1395371322 x^{12} + 9530036138 x^{10} + 8603288846964 x^{8} + 351613585853425 x^{6} - 1875817034507589 x^{4} - 321799222643246666 x^{2} - 4337852776485437971 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1729797948719255084072017529251001251840000000000=-\,2^{20}\cdot 5^{10}\cdot 11^{4}\cdot 1630891^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $258.17$
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:  $2, 5, 11, 1630891$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{1630891} a^{16} + \frac{538}{1630891} a^{14} + \frac{59526}{1630891} a^{12} - \frac{257290}{1630891} a^{10} + \frac{671374}{1630891} a^{8} + \frac{740025}{1630891} a^{6} - \frac{403364}{1630891} a^{4} + \frac{32079}{1630891} a^{2}$, $\frac{1}{1630891} a^{17} + \frac{538}{1630891} a^{15} + \frac{59526}{1630891} a^{13} - \frac{257290}{1630891} a^{11} + \frac{671374}{1630891} a^{9} + \frac{740025}{1630891} a^{7} - \frac{403364}{1630891} a^{5} + \frac{32079}{1630891} a^{3}$, $\frac{1}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{18} - \frac{3643811975250418133985758425357031188258167616233445240266524529155892}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{16} - \frac{12160458210776675268081215034952653105297758942266395654310150684580668076110}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{14} - \frac{8638715543121341663939801224738553527401556693443225746925623647174040988370}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{12} + \frac{1124700392817236115313047835187406848321546432676494209011942889979556272322}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{10} + \frac{12065044765535614742182431361962005521995974451197151930954287903920539426471}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{8} + \frac{5604802100262782378482059586206605641861633717805273828557563758409957293225}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{6} - \frac{8262128905484941846755173478675939431438188564586550607988062642371227748193}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{4} + \frac{1394869784457099778616814862397116221641208368747872337552836816060835}{15413750071051564964035543574729581040414860192109790288497837418774091} a^{2} + \frac{1545812095716968987968339949016105292756114914911314393480659248}{9451122160249559881092938507067352165420534046793924479623615201}$, $\frac{1}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{19} - \frac{3643811975250418133985758425357031188258167616233445240266524529155892}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{17} - \frac{12160458210776675268081215034952653105297758942266395654310150684580668076110}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{15} - \frac{8638715543121341663939801224738553527401556693443225746925623647174040988370}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{13} + \frac{1124700392817236115313047835187406848321546432676494209011942889979556272322}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{11} + \frac{12065044765535614742182431361962005521995974451197151930954287903920539426471}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{9} + \frac{5604802100262782378482059586206605641861633717805273828557563758409957293225}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{7} - \frac{8262128905484941846755173478675939431438188564586550607988062642371227748193}{25138146267127357835760891696134301152583231753570127993398526565741896045081} a^{5} + \frac{1394869784457099778616814862397116221641208368747872337552836816060835}{15413750071051564964035543574729581040414860192109790288497837418774091} a^{3} + \frac{1545812095716968987968339949016105292756114914911314393480659248}{9451122160249559881092938507067352165420534046793924479623615201} a$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1022 are not computed
Character table for t20n1022 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.8.616680659375.1

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 32 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ R $16{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.8.8.7$x^{8} + 2 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.12.12.18$x^{12} + 80 x^{10} + 81 x^{8} - 160 x^{6} - 117 x^{4} + 80 x^{2} + 227$$2$$6$$12$$D_4 \times C_3$$[2, 2]^{6}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
1630891Data not computed