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Properties

Label 20.12.2549038515...0000.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{15}\cdot 6029^{5}$
Root discriminant $58.93$
Ramified primes $2, 5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![30145, 0, -60290, 0, -28755, 0, 86200, 0, -2990, 0, -27260, 0, 1746, 0, 2392, 0, -147, 0, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 18*x^18 - 147*x^16 + 2392*x^14 + 1746*x^12 - 27260*x^10 - 2990*x^8 + 86200*x^6 - 28755*x^4 - 60290*x^2 + 30145)
 
gp: K = bnfinit(x^20 - 18*x^18 - 147*x^16 + 2392*x^14 + 1746*x^12 - 27260*x^10 - 2990*x^8 + 86200*x^6 - 28755*x^4 - 60290*x^2 + 30145, 1)
 

Normalized defining polynomial

\( x^{20} - 18 x^{18} - 147 x^{16} + 2392 x^{14} + 1746 x^{12} - 27260 x^{10} - 2990 x^{8} + 86200 x^{6} - 28755 x^{4} - 60290 x^{2} + 30145 \)

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:  \(254903851560926116768000000000000000=2^{20}\cdot 5^{15}\cdot 6029^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.93$
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:  $2, 5, 6029$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} + \frac{1}{16}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{5} + \frac{1}{16} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4} + \frac{1}{32} a^{2} - \frac{1}{32}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} - \frac{1}{16} a^{7} + \frac{1}{16} a^{5} + \frac{1}{32} a^{3} - \frac{1}{32} a$, $\frac{1}{64} a^{12} + \frac{1}{64} a^{8} - \frac{5}{64} a^{4} + \frac{3}{64}$, $\frac{1}{64} a^{13} + \frac{1}{64} a^{9} - \frac{5}{64} a^{5} + \frac{3}{64} a$, $\frac{1}{128} a^{14} - \frac{1}{128} a^{12} + \frac{1}{128} a^{10} - \frac{1}{128} a^{8} - \frac{5}{128} a^{6} + \frac{5}{128} a^{4} + \frac{3}{128} a^{2} - \frac{3}{128}$, $\frac{1}{128} a^{15} - \frac{1}{128} a^{13} + \frac{1}{128} a^{11} - \frac{1}{128} a^{9} - \frac{5}{128} a^{7} + \frac{5}{128} a^{5} + \frac{3}{128} a^{3} - \frac{3}{128} a$, $\frac{1}{4352} a^{16} + \frac{3}{2176} a^{14} + \frac{1}{2176} a^{12} + \frac{11}{2176} a^{10} + \frac{27}{1088} a^{8} + \frac{97}{2176} a^{6} + \frac{15}{2176} a^{4} + \frac{1}{128} a^{2} - \frac{1677}{4352}$, $\frac{1}{4352} a^{17} + \frac{3}{2176} a^{15} + \frac{1}{2176} a^{13} + \frac{11}{2176} a^{11} + \frac{27}{1088} a^{9} + \frac{97}{2176} a^{7} + \frac{15}{2176} a^{5} + \frac{1}{128} a^{3} - \frac{1677}{4352} a$, $\frac{1}{444155903882752} a^{18} - \frac{941118043}{444155903882752} a^{16} + \frac{23815307253}{13879871996336} a^{14} + \frac{7615722741}{3469967999084} a^{12} + \frac{3163310442613}{222077951941376} a^{10} + \frac{149866951545}{13063408937728} a^{8} + \frac{862294017513}{55519487985344} a^{6} - \frac{4448858622967}{55519487985344} a^{4} - \frac{3472654269395}{444155903882752} a^{2} - \frac{221903500180351}{444155903882752}$, $\frac{1}{444155903882752} a^{19} - \frac{941118043}{444155903882752} a^{17} + \frac{23815307253}{13879871996336} a^{15} + \frac{7615722741}{3469967999084} a^{13} + \frac{3163310442613}{222077951941376} a^{11} + \frac{149866951545}{13063408937728} a^{9} + \frac{862294017513}{55519487985344} a^{7} - \frac{4448858622967}{55519487985344} a^{5} - \frac{3472654269395}{444155903882752} a^{3} - \frac{221903500180351}{444155903882752} a$

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:  \( 33616823933.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
6029Data not computed