↑

Properties

Label 20.12.2662310085...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{4}\cdot 5^{22}\cdot 13^{10}$
Root discriminant $26.38$
Ramified primes $3, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5\wr C_2:C_2$ (as 20T96)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 15, 75, -120, -475, 990, 1155, -4310, 865, 3810, -1299, -1405, 575, 255, -225, -20, 80, 0, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 + 80*x^16 - 20*x^15 - 225*x^14 + 255*x^13 + 575*x^12 - 1405*x^11 - 1299*x^10 + 3810*x^9 + 865*x^8 - 4310*x^7 + 1155*x^6 + 990*x^5 - 475*x^4 - 120*x^3 + 75*x^2 + 15*x - 1)
 
gp: K = bnfinit(x^20 - 15*x^18 + 80*x^16 - 20*x^15 - 225*x^14 + 255*x^13 + 575*x^12 - 1405*x^11 - 1299*x^10 + 3810*x^9 + 865*x^8 - 4310*x^7 + 1155*x^6 + 990*x^5 - 475*x^4 - 120*x^3 + 75*x^2 + 15*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 15 x^{18} + 80 x^{16} - 20 x^{15} - 225 x^{14} + 255 x^{13} + 575 x^{12} - 1405 x^{11} - 1299 x^{10} + 3810 x^{9} + 865 x^{8} - 4310 x^{7} + 1155 x^{6} + 990 x^{5} - 475 x^{4} - 120 x^{3} + 75 x^{2} + 15 x - 1 \)

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:  \(26623100852415561676025390625=3^{4}\cdot 5^{22}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.38$
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:  $3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{1131} a^{18} + \frac{21}{377} a^{17} + \frac{40}{1131} a^{16} - \frac{11}{87} a^{15} + \frac{391}{1131} a^{14} + \frac{114}{377} a^{13} + \frac{2}{29} a^{12} + \frac{407}{1131} a^{11} - \frac{32}{377} a^{10} - \frac{86}{1131} a^{9} - \frac{433}{1131} a^{8} + \frac{75}{377} a^{7} + \frac{484}{1131} a^{6} - \frac{563}{1131} a^{5} + \frac{14}{377} a^{4} + \frac{20}{87} a^{3} + \frac{55}{1131} a^{2} - \frac{55}{377} a - \frac{48}{377}$, $\frac{1}{537285005698656571629} a^{19} - \frac{158220925663597460}{537285005698656571629} a^{18} + \frac{4966798439795962820}{76755000814093795947} a^{17} + \frac{1542660246097567485}{25585000271364598649} a^{16} + \frac{58502410670911664752}{179095001899552190543} a^{15} + \frac{44081640665303574612}{179095001899552190543} a^{14} + \frac{114335243745224110262}{537285005698656571629} a^{13} - \frac{27128356971655734317}{179095001899552190543} a^{12} - \frac{2224769671142761069}{76755000814093795947} a^{11} + \frac{5619100868274393035}{13776538607657860811} a^{10} - \frac{121079562419341890142}{537285005698656571629} a^{9} - \frac{278701315247839305}{1968076943951122973} a^{8} + \frac{139817433975035591548}{537285005698656571629} a^{7} + \frac{211096884347044768708}{537285005698656571629} a^{6} + \frac{148964070851357815598}{537285005698656571629} a^{5} - \frac{254244920401444278017}{537285005698656571629} a^{4} + \frac{89554182291776740177}{537285005698656571629} a^{3} - \frac{72823540150464530676}{179095001899552190543} a^{2} - \frac{12585336782935668061}{25585000271364598649} a - \frac{260776637210251423379}{537285005698656571629}$

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:  \( 12133358.0731 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2:C_2$ (as 20T96):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 16 conjugacy class representatives for $D_5\wr C_2:C_2$
Character table for $D_5\wr C_2:C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}, \sqrt{13})\), 10.6.32633173828125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$