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Properties

Label 20.8.30091033986...3776.3
Degree $20$
Signature $[8, 6]$
Discriminant $2^{10}\cdot 11^{18}\cdot 727^{2}$
Root discriminant $23.65$
Ramified primes $2, 11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43, 295, -856, 1617, -2259, 2054, -855, -583, 1589, -1473, 560, -42, 31, -22, -51, 58, -25, 11, -5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 5*x^18 + 11*x^17 - 25*x^16 + 58*x^15 - 51*x^14 - 22*x^13 + 31*x^12 - 42*x^11 + 560*x^10 - 1473*x^9 + 1589*x^8 - 583*x^7 - 855*x^6 + 2054*x^5 - 2259*x^4 + 1617*x^3 - 856*x^2 + 295*x - 43)
 
gp: K = bnfinit(x^20 - x^19 - 5*x^18 + 11*x^17 - 25*x^16 + 58*x^15 - 51*x^14 - 22*x^13 + 31*x^12 - 42*x^11 + 560*x^10 - 1473*x^9 + 1589*x^8 - 583*x^7 - 855*x^6 + 2054*x^5 - 2259*x^4 + 1617*x^3 - 856*x^2 + 295*x - 43, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 5 x^{18} + 11 x^{17} - 25 x^{16} + 58 x^{15} - 51 x^{14} - 22 x^{13} + 31 x^{12} - 42 x^{11} + 560 x^{10} - 1473 x^{9} + 1589 x^{8} - 583 x^{7} - 855 x^{6} + 2054 x^{5} - 2259 x^{4} + 1617 x^{3} - 856 x^{2} + 295 x - 43 \)

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:  \(3009103398689521267119563776=2^{10}\cdot 11^{18}\cdot 727^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.65$
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:  $2, 11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{697005940206193696755485051} a^{19} - \frac{279638348350135881791068682}{697005940206193696755485051} a^{18} + \frac{179909964867106329917761877}{697005940206193696755485051} a^{17} + \frac{98074976704181018411270697}{697005940206193696755485051} a^{16} + \frac{206053692536102876878514947}{697005940206193696755485051} a^{15} - \frac{215031991513007069431880917}{697005940206193696755485051} a^{14} + \frac{13104877351125319126798355}{30304606095921465076325437} a^{13} + \frac{210493954589163583579373577}{697005940206193696755485051} a^{12} - \frac{4092409923898661594059203}{30304606095921465076325437} a^{11} + \frac{174952818933559774739830903}{697005940206193696755485051} a^{10} - \frac{332384785174455240823549404}{697005940206193696755485051} a^{9} - \frac{94803352441254155312024409}{697005940206193696755485051} a^{8} + \frac{73054960740009193221790366}{697005940206193696755485051} a^{7} + \frac{345033151173535703243042127}{697005940206193696755485051} a^{6} - \frac{236388593739569083015600371}{697005940206193696755485051} a^{5} + \frac{199593994014715735417890026}{697005940206193696755485051} a^{4} - \frac{52587931756708815190502747}{697005940206193696755485051} a^{3} + \frac{325212227732993160630092740}{697005940206193696755485051} a^{2} + \frac{287623844431568941152871216}{697005940206193696755485051} a + \frac{105314658752364714981835931}{697005940206193696755485051}$

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:  \( 1358534.58752 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.155838906487.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.2$x^{10} - 5 x^{8} + 10 x^{6} - 2 x^{4} - 11 x^{2} + 39$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
11Data not computed
727Data not computed