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Properties

Label 20.12.3702706290...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{16}\cdot 5^{10}\cdot 3541^{4}\cdot 60662149^{2}$
Root discriminant $119.79$
Ramified primes $2, 5, 3541, 60662149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1030

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![157218521219761, 0, -69041174949655, 0, 13305245801616, 0, -1478902722150, 0, 104758408713, 0, -4929882768, 0, 155690232, 0, -3247364, 0, 42619, 0, -316, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 316*x^18 + 42619*x^16 - 3247364*x^14 + 155690232*x^12 - 4929882768*x^10 + 104758408713*x^8 - 1478902722150*x^6 + 13305245801616*x^4 - 69041174949655*x^2 + 157218521219761)
 
gp: K = bnfinit(x^20 - 316*x^18 + 42619*x^16 - 3247364*x^14 + 155690232*x^12 - 4929882768*x^10 + 104758408713*x^8 - 1478902722150*x^6 + 13305245801616*x^4 - 69041174949655*x^2 + 157218521219761, 1)
 

Normalized defining polynomial

\( x^{20} - 316 x^{18} + 42619 x^{16} - 3247364 x^{14} + 155690232 x^{12} - 4929882768 x^{10} + 104758408713 x^{8} - 1478902722150 x^{6} + 13305245801616 x^{4} - 69041174949655 x^{2} + 157218521219761 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(370270629040986660253142367975040000000000=2^{16}\cdot 5^{10}\cdot 3541^{4}\cdot 60662149^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $119.79$
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:  $2, 5, 3541, 60662149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3541} a^{14} - \frac{316}{3541} a^{12} + \frac{127}{3541} a^{10} - \frac{267}{3541} a^{8} - \frac{456}{3541} a^{6} + \frac{121}{3541} a^{4} - \frac{1261}{3541} a^{2}$, $\frac{1}{3541} a^{15} - \frac{316}{3541} a^{13} + \frac{127}{3541} a^{11} - \frac{267}{3541} a^{9} - \frac{456}{3541} a^{7} + \frac{121}{3541} a^{5} - \frac{1261}{3541} a^{3}$, $\frac{1}{12538681} a^{16} - \frac{316}{12538681} a^{14} + \frac{42619}{12538681} a^{12} - \frac{3247364}{12538681} a^{10} + \frac{5226060}{12538681} a^{8} - \frac{2181135}{12538681} a^{6} - \frac{2271042}{12538681} a^{4} - \frac{823}{3541} a^{2}$, $\frac{1}{12538681} a^{17} - \frac{316}{12538681} a^{15} + \frac{42619}{12538681} a^{13} - \frac{3247364}{12538681} a^{11} + \frac{5226060}{12538681} a^{9} - \frac{2181135}{12538681} a^{7} - \frac{2271042}{12538681} a^{5} - \frac{823}{3541} a^{3}$, $\frac{1}{3627078336199582601125044195522952540273138} a^{18} - \frac{1}{25077362} a^{17} + \frac{27731378738084715653562286243425621}{1813539168099791300562522097761476270136569} a^{16} - \frac{3225}{25077362} a^{15} + \frac{30837948582168236817333755552155015632}{1813539168099791300562522097761476270136569} a^{14} + \frac{1076337}{25077362} a^{13} - \frac{1000525806887862878552263718072850987615699}{3627078336199582601125044195522952540273138} a^{12} - \frac{4870512}{12538681} a^{11} + \frac{776396790973009836205176410030666804698883}{3627078336199582601125044195522952540273138} a^{10} - \frac{4280613}{25077362} a^{9} + \frac{372283625715387977288990376921634369931120}{1813539168099791300562522097761476270136569} a^{8} + \frac{3795831}{25077362} a^{7} - \frac{720947423417945347262051194492689357093130}{1813539168099791300562522097761476270136569} a^{6} + \frac{1842581}{25077362} a^{5} + \frac{15064164368914359530340618100722590445}{44535175966007914751728745202447755366} a^{4} + \frac{1042}{3541} a^{3} - \frac{115374315495554121949566648669886065}{289271123190675526486800660733210498} a^{2} - \frac{1}{2} a + \frac{24253813573063698584006373282711}{81691929734729038827111172192378}$, $\frac{1}{3627078336199582601125044195522952540273138} a^{19} - \frac{89172804119168331936275757879754007}{3627078336199582601125044195522952540273138} a^{17} - \frac{1}{25077362} a^{16} - \frac{404773788980627812825298554327991896761}{3627078336199582601125044195522952540273138} a^{15} - \frac{3225}{25077362} a^{14} - \frac{422424600213510908238075968343525096861393}{1813539168099791300562522097761476270136569} a^{13} + \frac{1076337}{25077362} a^{12} - \frac{632501685780653603655104049678363724336093}{3627078336199582601125044195522952540273138} a^{11} - \frac{4870512}{12538681} a^{10} + \frac{125438386203472385847359135471683545124603}{3627078336199582601125044195522952540273138} a^{9} - \frac{4280613}{25077362} a^{8} - \frac{892882698429898157334142869569577145269341}{3627078336199582601125044195522952540273138} a^{7} + \frac{3795831}{25077362} a^{6} + \frac{9168212592963122885725714819593947914}{22267587983003957375864372601223877683} a^{5} + \frac{1842581}{25077362} a^{4} - \frac{30251324711966463491716807245428189}{289271123190675526486800660733210498} a^{3} + \frac{1042}{3541} a^{2} - \frac{8296075647150410414774606406739}{40845964867364519413555586096189} a - \frac{1}{2}$

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:  $15$
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:  \( 50916740385800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1030:

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 t20n1030 are not computed
Character table for t20n1030 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3541Data not computed
60662149Data not computed