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Properties

Label 20.8.28211931605...1264.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{40}\cdot 11^{16}\cdot 89^{5}$
Root discriminant $83.66$
Ramified primes $2, 11, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T310

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47081, 0, -32918, 0, -434830, 0, -447664, 0, 46130, 0, 156886, 0, 14737, 0, -3068, 0, -242, 0, 14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 14*x^18 - 242*x^16 - 3068*x^14 + 14737*x^12 + 156886*x^10 + 46130*x^8 - 447664*x^6 - 434830*x^4 - 32918*x^2 + 47081)
 
gp: K = bnfinit(x^20 + 14*x^18 - 242*x^16 - 3068*x^14 + 14737*x^12 + 156886*x^10 + 46130*x^8 - 447664*x^6 - 434830*x^4 - 32918*x^2 + 47081, 1)
 

Normalized defining polynomial

\( x^{20} + 14 x^{18} - 242 x^{16} - 3068 x^{14} + 14737 x^{12} + 156886 x^{10} + 46130 x^{8} - 447664 x^{6} - 434830 x^{4} - 32918 x^{2} + 47081 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(282119316059236304095781411736743051264=2^{40}\cdot 11^{16}\cdot 89^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.66$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} - \frac{11}{23} a^{15} + \frac{10}{23} a^{13} - \frac{6}{23} a^{11} + \frac{6}{23} a^{9} - \frac{9}{23} a^{7} + \frac{10}{23} a^{5} + \frac{11}{23} a^{3} + \frac{9}{23} a$, $\frac{1}{1645206289698698532709255351} a^{18} - \frac{587517460349581224137635793}{1645206289698698532709255351} a^{16} + \frac{3237183407704811269187378}{1645206289698698532709255351} a^{14} + \frac{295099147778181445709589033}{1645206289698698532709255351} a^{12} + \frac{296261645820863693705427798}{1645206289698698532709255351} a^{10} + \frac{793169373455979345073395464}{1645206289698698532709255351} a^{8} - \frac{68855643090753428007937708}{1645206289698698532709255351} a^{6} - \frac{346596448073911233068712042}{1645206289698698532709255351} a^{4} + \frac{529177573818746616225473903}{1645206289698698532709255351} a^{2} + \frac{34799629204553973261920847}{71530708247769501422141537}$, $\frac{1}{1645206289698698532709255351} a^{19} - \frac{15271794367425212760503497}{1645206289698698532709255351} a^{17} + \frac{289360016398782816957753526}{1645206289698698532709255351} a^{15} - \frac{563269351195052571356109411}{1645206289698698532709255351} a^{13} + \frac{153200229325324690861144724}{1645206289698698532709255351} a^{11} - \frac{708975499747180184791576813}{1645206289698698532709255351} a^{9} - \frac{283447767834061932274362319}{1645206289698698532709255351} a^{7} + \frac{440241342651553282574844865}{1645206289698698532709255351} a^{5} + \frac{243054740827668610536907755}{1645206289698698532709255351} a^{3} - \frac{630222693250648643418651259}{1645206289698698532709255351} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 263616821657 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T310:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 44 conjugacy class representatives for t20n310
Character table for t20n310 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.1738687177114624.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
2Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
89Data not computed