↑

Properties

Label 20.12.3178202722...0849.1
Degree $20$
Signature $[12, 4]$
Discriminant $13^{12}\cdot 97^{2}\cdot 347^{4}$
Root discriminant $23.72$
Ramified primes $13, 97, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T794

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -16, -125, 105, 871, -1147, -905, 1993, 92, -1456, 207, 483, -84, -38, 44, -36, -31, 25, 5, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 5*x^18 + 25*x^17 - 31*x^16 - 36*x^15 + 44*x^14 - 38*x^13 - 84*x^12 + 483*x^11 + 207*x^10 - 1456*x^9 + 92*x^8 + 1993*x^7 - 905*x^6 - 1147*x^5 + 871*x^4 + 105*x^3 - 125*x^2 - 16*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 5*x^18 + 25*x^17 - 31*x^16 - 36*x^15 + 44*x^14 - 38*x^13 - 84*x^12 + 483*x^11 + 207*x^10 - 1456*x^9 + 92*x^8 + 1993*x^7 - 905*x^6 - 1147*x^5 + 871*x^4 + 105*x^3 - 125*x^2 - 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 5 x^{18} + 25 x^{17} - 31 x^{16} - 36 x^{15} + 44 x^{14} - 38 x^{13} - 84 x^{12} + 483 x^{11} + 207 x^{10} - 1456 x^{9} + 92 x^{8} + 1993 x^{7} - 905 x^{6} - 1147 x^{5} + 871 x^{4} + 105 x^{3} - 125 x^{2} - 16 x + 1 \)

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:  \(3178202722755606120397450849=13^{12}\cdot 97^{2}\cdot 347^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.72$
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:  $13, 97, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} + \frac{5}{13} a^{17} - \frac{6}{13} a^{16} + \frac{6}{13} a^{15} + \frac{2}{13} a^{14} + \frac{6}{13} a^{13} + \frac{4}{13} a^{12} + \frac{3}{13} a^{10} - \frac{4}{13} a^{9} + \frac{4}{13} a^{8} - \frac{4}{13} a^{7} + \frac{5}{13} a^{6} - \frac{2}{13} a^{5} + \frac{4}{13} a^{4} + \frac{4}{13} a^{3} + \frac{1}{13} a^{2} - \frac{5}{13} a + \frac{1}{13}$, $\frac{1}{940601018697250919983} a^{19} - \frac{23698620602779705265}{940601018697250919983} a^{18} - \frac{56015609517899670536}{940601018697250919983} a^{17} + \frac{2465988742163528593}{72353924515173147691} a^{16} - \frac{101153219463439984131}{940601018697250919983} a^{15} - \frac{409676457734849904903}{940601018697250919983} a^{14} + \frac{319698984631604457422}{940601018697250919983} a^{13} + \frac{189893603871863466007}{940601018697250919983} a^{12} + \frac{344395359038061404349}{940601018697250919983} a^{11} - \frac{262321591837894638570}{940601018697250919983} a^{10} - \frac{15043329936740454283}{72353924515173147691} a^{9} + \frac{12650241534496926928}{72353924515173147691} a^{8} + \frac{140173751772410776603}{940601018697250919983} a^{7} - \frac{250151080587062898747}{940601018697250919983} a^{6} - \frac{139993252351863915249}{940601018697250919983} a^{5} + \frac{393113927354514773103}{940601018697250919983} a^{4} + \frac{274976361461134680495}{940601018697250919983} a^{3} - \frac{352948582180689078913}{940601018697250919983} a^{2} + \frac{274112682198716252602}{940601018697250919983} a + \frac{279132489154196268605}{940601018697250919983}$

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:  \( 3027268.40607 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T794:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n794 are not computed
Character table for t20n794 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$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}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$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}$
97.2.0.1$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$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
347Data not computed