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Properties

Label 20.0.29194440139...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{8}\cdot 5^{10}\cdot 97^{8}\cdot 149$
Root discriminant $23.62$
Ramified primes $2, 5, 97, 149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![149, 0, 505, 0, 713, 0, 668, 0, 667, 0, 697, 0, 542, 0, 272, 0, 83, 0, 14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 14*x^18 + 83*x^16 + 272*x^14 + 542*x^12 + 697*x^10 + 667*x^8 + 668*x^6 + 713*x^4 + 505*x^2 + 149)
 
gp: K = bnfinit(x^20 + 14*x^18 + 83*x^16 + 272*x^14 + 542*x^12 + 697*x^10 + 667*x^8 + 668*x^6 + 713*x^4 + 505*x^2 + 149, 1)
 

Normalized defining polynomial

\( x^{20} + 14 x^{18} + 83 x^{16} + 272 x^{14} + 542 x^{12} + 697 x^{10} + 667 x^{8} + 668 x^{6} + 713 x^{4} + 505 x^{2} + 149 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2919444013905417972500000000=2^{8}\cdot 5^{10}\cdot 97^{8}\cdot 149\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.62$
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:  $2, 5, 97, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{8} + \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{9} + \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{13} + \frac{1}{5} a^{9} - \frac{3}{10} a^{8} + \frac{1}{5} a^{7} - \frac{1}{2} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{3}{10} a^{3} + \frac{1}{10} a^{2} + \frac{2}{5} a - \frac{1}{10}$, $\frac{1}{10} a^{15} - \frac{1}{10} a^{13} - \frac{1}{10} a^{9} - \frac{1}{2} a^{8} - \frac{3}{10} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{2}{5} a^{3} - \frac{1}{2} a^{2} + \frac{3}{10} a - \frac{1}{2}$, $\frac{1}{10} a^{16} - \frac{1}{10} a^{13} - \frac{1}{10} a^{10} - \frac{3}{10} a^{9} + \frac{2}{5} a^{8} - \frac{3}{10} a^{7} - \frac{1}{2} a^{6} - \frac{1}{10} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{10} a - \frac{1}{10}$, $\frac{1}{10} a^{17} - \frac{1}{10} a^{13} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{3}{10} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{3}{10} a^{3} + \frac{3}{10} a + \frac{3}{10}$, $\frac{1}{270} a^{18} + \frac{1}{135} a^{14} - \frac{1}{10} a^{13} + \frac{1}{270} a^{12} - \frac{1}{10} a^{11} - \frac{2}{45} a^{10} + \frac{14}{135} a^{8} + \frac{59}{270} a^{6} + \frac{1}{5} a^{5} + \frac{31}{270} a^{4} + \frac{3}{10} a^{3} + \frac{1}{3} a^{2} - \frac{3}{10} a - \frac{107}{270}$, $\frac{1}{270} a^{19} + \frac{1}{135} a^{15} - \frac{13}{135} a^{13} - \frac{1}{10} a^{12} - \frac{2}{45} a^{11} + \frac{41}{135} a^{9} - \frac{3}{10} a^{8} + \frac{113}{270} a^{7} - \frac{3}{10} a^{6} - \frac{131}{270} a^{5} - \frac{1}{10} a^{4} - \frac{11}{30} a^{3} - \frac{1}{5} a^{2} + \frac{1}{270} a - \frac{1}{10}$

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:  $9$
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:  \( 611590.771783 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.47045.1, 10.2.276654003125.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$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$$[\ ]_{2}^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.6.4.3$x^{6} + 873 x^{3} + 235225$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
97.6.4.3$x^{6} + 873 x^{3} + 235225$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$149$$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.3.0.1$x^{3} - x + 18$$1$$3$$0$$C_3$$[\ ]^{3}$
149.3.0.1$x^{3} - x + 18$$1$$3$$0$$C_3$$[\ ]^{3}$
149.6.0.1$x^{6} - x + 14$$1$$6$$0$$C_6$$[\ ]^{6}$