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Properties

Label 20.12.1198207010...2437.1
Degree $20$
Signature $[12, 4]$
Discriminant $13^{11}\cdot 401^{8}$
Root discriminant $45.07$
Ramified primes $13, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, -306, 405, 2831, -6246, -407, 13978, -17153, 2821, 15105, -17839, 5889, 4433, -5209, 1904, 50, -343, 141, -14, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 14*x^18 + 141*x^17 - 343*x^16 + 50*x^15 + 1904*x^14 - 5209*x^13 + 4433*x^12 + 5889*x^11 - 17839*x^10 + 15105*x^9 + 2821*x^8 - 17153*x^7 + 13978*x^6 - 407*x^5 - 6246*x^4 + 2831*x^3 + 405*x^2 - 306*x - 27)
 
gp: K = bnfinit(x^20 - 5*x^19 - 14*x^18 + 141*x^17 - 343*x^16 + 50*x^15 + 1904*x^14 - 5209*x^13 + 4433*x^12 + 5889*x^11 - 17839*x^10 + 15105*x^9 + 2821*x^8 - 17153*x^7 + 13978*x^6 - 407*x^5 - 6246*x^4 + 2831*x^3 + 405*x^2 - 306*x - 27, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 14 x^{18} + 141 x^{17} - 343 x^{16} + 50 x^{15} + 1904 x^{14} - 5209 x^{13} + 4433 x^{12} + 5889 x^{11} - 17839 x^{10} + 15105 x^{9} + 2821 x^{8} - 17153 x^{7} + 13978 x^{6} - 407 x^{5} - 6246 x^{4} + 2831 x^{3} + 405 x^{2} - 306 x - 27 \)

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:  \(1198207010758520322015083315072437=13^{11}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.07$
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:  $13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} + \frac{1}{3} a^{12} - \frac{4}{9} a^{11} - \frac{4}{9} a^{10} + \frac{1}{3} a^{9} + \frac{2}{9} a^{8} - \frac{4}{9} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{2}{9} a^{3} + \frac{1}{9} a^{2}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{15} - \frac{1}{9} a^{12} - \frac{1}{3} a^{11} + \frac{4}{9} a^{10} - \frac{4}{9} a^{9} + \frac{1}{3} a^{8} - \frac{2}{9} a^{7} + \frac{1}{3} a^{6} - \frac{2}{9} a^{4} - \frac{1}{3} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{15} - \frac{1}{9} a^{13} + \frac{1}{9} a^{10} - \frac{1}{3} a^{9} - \frac{1}{9} a^{7} + \frac{1}{3} a^{6} + \frac{4}{9} a^{5} + \frac{4}{9} a^{2}$, $\frac{1}{1749870658874480384737306623393} a^{19} + \frac{15190325630673797161876787282}{583290219624826794912435541131} a^{18} - \frac{79762810298079589833225525797}{1749870658874480384737306623393} a^{17} + \frac{39878946592983486900433377836}{1749870658874480384737306623393} a^{16} + \frac{7621939864937396790328628722}{194430073208275598304145180377} a^{15} + \frac{16224091514154093388956981356}{1749870658874480384737306623393} a^{14} - \frac{675827797489010670934741032}{64810024402758532768048393459} a^{13} - \frac{648359873361713341040112773620}{1749870658874480384737306623393} a^{12} + \frac{230664817726187310459090193645}{583290219624826794912435541131} a^{11} + \frac{104940693286926919229479083479}{583290219624826794912435541131} a^{10} - \frac{530559831267052974041341208710}{1749870658874480384737306623393} a^{9} + \frac{617512112059967612515415011435}{1749870658874480384737306623393} a^{8} - \frac{159118320037178044920276364798}{583290219624826794912435541131} a^{7} - \frac{202884320481157821849825977120}{1749870658874480384737306623393} a^{6} - \frac{105288097428382215544472898628}{583290219624826794912435541131} a^{5} - \frac{527482978148136989578612776596}{1749870658874480384737306623393} a^{4} + \frac{69648508289717075880144692108}{1749870658874480384737306623393} a^{3} + \frac{17341191271347432465598639721}{583290219624826794912435541131} a^{2} + \frac{19144881103334788894920320149}{64810024402758532768048393459} a - \frac{3360523660661597717996074118}{64810024402758532768048393459}$

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:  \( 14714319324.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.160801.1, 10.10.9600508843720093.1

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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ $20$ $20$ $20$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed