↑

Properties

Label 20.8.21742693620...8125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{15}\cdot 97^{2}\cdot 27517559^{2}$
Root discriminant $29.30$
Ramified primes $5, 97, 27517559$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1039

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-79, -46, 1307, -2041, 3745, -4624, 3209, -3465, 2525, -1763, 2059, -1448, 401, 252, -414, 338, -182, 51, -1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - x^18 + 51*x^17 - 182*x^16 + 338*x^15 - 414*x^14 + 252*x^13 + 401*x^12 - 1448*x^11 + 2059*x^10 - 1763*x^9 + 2525*x^8 - 3465*x^7 + 3209*x^6 - 4624*x^5 + 3745*x^4 - 2041*x^3 + 1307*x^2 - 46*x - 79)
 
gp: K = bnfinit(x^20 - 4*x^19 - x^18 + 51*x^17 - 182*x^16 + 338*x^15 - 414*x^14 + 252*x^13 + 401*x^12 - 1448*x^11 + 2059*x^10 - 1763*x^9 + 2525*x^8 - 3465*x^7 + 3209*x^6 - 4624*x^5 + 3745*x^4 - 2041*x^3 + 1307*x^2 - 46*x - 79, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - x^{18} + 51 x^{17} - 182 x^{16} + 338 x^{15} - 414 x^{14} + 252 x^{13} + 401 x^{12} - 1448 x^{11} + 2059 x^{10} - 1763 x^{9} + 2525 x^{8} - 3465 x^{7} + 3209 x^{6} - 4624 x^{5} + 3745 x^{4} - 2041 x^{3} + 1307 x^{2} - 46 x - 79 \)

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:  \(217426936208300406768798828125=5^{15}\cdot 97^{2}\cdot 27517559^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.30$
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:  $5, 97, 27517559$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4429172315267198829273453463093758479} a^{19} - \frac{1061095771678470925386244279529215457}{4429172315267198829273453463093758479} a^{18} - \frac{705415844980318950970093695887920077}{4429172315267198829273453463093758479} a^{17} - \frac{2080036638994654889318425822215918777}{4429172315267198829273453463093758479} a^{16} + \frac{466694165665223219870974495067789683}{4429172315267198829273453463093758479} a^{15} + \frac{1204650892665562102863012901589458578}{4429172315267198829273453463093758479} a^{14} + \frac{1868860959037032558827630485706958703}{4429172315267198829273453463093758479} a^{13} - \frac{2086164759383115727690157314257346359}{4429172315267198829273453463093758479} a^{12} - \frac{1873713187705835759026803364119471668}{4429172315267198829273453463093758479} a^{11} + \frac{1082125373772504244544309105810352233}{4429172315267198829273453463093758479} a^{10} - \frac{659455755142535101556879939874438554}{4429172315267198829273453463093758479} a^{9} + \frac{1945548943512386520233079381182990456}{4429172315267198829273453463093758479} a^{8} - \frac{563586091307819776038837612998745821}{4429172315267198829273453463093758479} a^{7} + \frac{1423074710778147086505578695398871185}{4429172315267198829273453463093758479} a^{6} - \frac{232446046521973036058502152285041159}{4429172315267198829273453463093758479} a^{5} + \frac{1363017422571611791229004454621527856}{4429172315267198829273453463093758479} a^{4} - \frac{950357619328074676443350447105110099}{4429172315267198829273453463093758479} a^{3} - \frac{895603879502801792778988675363954463}{4429172315267198829273453463093758479} a^{2} + \frac{1590823512764992650091704284228321453}{4429172315267198829273453463093758479} a - \frac{1698157129330560384475290873261066502}{4429172315267198829273453463093758479}$

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:  $13$
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:  \( 8839123.1462 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1039:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1039 are not computed
Character table for t20n1039 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.8.85992371875.1

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 $20$ $20$ R $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
97Data not computed
27517559Data not computed