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Properties

Label 20.4.21799845600...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 60662149^{3}$
Root discriminant $32.88$
Ramified primes $5, 60662149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2309, -1074, 3725, -3379, -16045, -9571, 1372, 12546, 2666, -1109, 2296, -1268, 152, 263, -181, 88, 7, -28, 16, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 16*x^18 - 28*x^17 + 7*x^16 + 88*x^15 - 181*x^14 + 263*x^13 + 152*x^12 - 1268*x^11 + 2296*x^10 - 1109*x^9 + 2666*x^8 + 12546*x^7 + 1372*x^6 - 9571*x^5 - 16045*x^4 - 3379*x^3 + 3725*x^2 - 1074*x + 2309)
 
gp: K = bnfinit(x^20 - 5*x^19 + 16*x^18 - 28*x^17 + 7*x^16 + 88*x^15 - 181*x^14 + 263*x^13 + 152*x^12 - 1268*x^11 + 2296*x^10 - 1109*x^9 + 2666*x^8 + 12546*x^7 + 1372*x^6 - 9571*x^5 - 16045*x^4 - 3379*x^3 + 3725*x^2 - 1074*x + 2309, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 16 x^{18} - 28 x^{17} + 7 x^{16} + 88 x^{15} - 181 x^{14} + 263 x^{13} + 152 x^{12} - 1268 x^{11} + 2296 x^{10} - 1109 x^{9} + 2666 x^{8} + 12546 x^{7} + 1372 x^{6} - 9571 x^{5} - 16045 x^{4} - 3379 x^{3} + 3725 x^{2} - 1074 x + 2309 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2179984560030696704042470703125=5^{10}\cdot 60662149^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.88$
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:  $5, 60662149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{645416958862996559628301428187688685872404426} a^{19} - \frac{4685394649685930173939684981971484092902007}{645416958862996559628301428187688685872404426} a^{18} + \frac{4718738557830355162337099325991535172690849}{645416958862996559628301428187688685872404426} a^{17} - \frac{70595947040820805235137474750069633288227902}{322708479431498279814150714093844342936202213} a^{16} + \frac{107276929392208182612715944640626580941830019}{645416958862996559628301428187688685872404426} a^{15} + \frac{52764077789620075795827750301181202311247696}{322708479431498279814150714093844342936202213} a^{14} - \frac{44988115489676269788183734205551993072299745}{645416958862996559628301428187688685872404426} a^{13} + \frac{56499621602106790294859927238466957258244200}{322708479431498279814150714093844342936202213} a^{12} - \frac{223294863438945554260877740888416413990737699}{645416958862996559628301428187688685872404426} a^{11} + \frac{58782628087490099998781312304230851606900724}{322708479431498279814150714093844342936202213} a^{10} - \frac{70604542550995017642639099454165707014223539}{645416958862996559628301428187688685872404426} a^{9} - \frac{246460114814817396122565600489838920724109465}{645416958862996559628301428187688685872404426} a^{8} - \frac{275443981977239641371666634268203377678354985}{645416958862996559628301428187688685872404426} a^{7} - \frac{30739761507928954554848104831853196133620015}{322708479431498279814150714093844342936202213} a^{6} + \frac{107855606274331933911755154164068936578276777}{645416958862996559628301428187688685872404426} a^{5} + \frac{125371818407537626550283946396813777811809117}{645416958862996559628301428187688685872404426} a^{4} - \frac{94262514377722884283500779517041615486642659}{322708479431498279814150714093844342936202213} a^{3} + \frac{68419982754624381175693391979621669411222729}{322708479431498279814150714093844342936202213} a^{2} + \frac{160616792537186345714850807797731409717127936}{322708479431498279814150714093844342936202213} a - \frac{223473372781338632069051090540927117449678353}{645416958862996559628301428187688685872404426}$

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:  $11$
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:  \( 22934659.8404 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 10.6.189569215625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ $16{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
5Data not computed
60662149Data not computed