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Properties

Label 20.8.14609322487...8784.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 11^{18}\cdot 1583^{2}$
Root discriminant $36.16$
Ramified primes $2, 11, 1583$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T340

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-131, 546, 703, -6748, 10982, -3330, -4601, 5692, -10408, 9438, -3002, 484, 2535, -3166, 2151, -1180, 493, -128, 29, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 29*x^18 - 128*x^17 + 493*x^16 - 1180*x^15 + 2151*x^14 - 3166*x^13 + 2535*x^12 + 484*x^11 - 3002*x^10 + 9438*x^9 - 10408*x^8 + 5692*x^7 - 4601*x^6 - 3330*x^5 + 10982*x^4 - 6748*x^3 + 703*x^2 + 546*x - 131)
 
gp: K = bnfinit(x^20 - 8*x^19 + 29*x^18 - 128*x^17 + 493*x^16 - 1180*x^15 + 2151*x^14 - 3166*x^13 + 2535*x^12 + 484*x^11 - 3002*x^10 + 9438*x^9 - 10408*x^8 + 5692*x^7 - 4601*x^6 - 3330*x^5 + 10982*x^4 - 6748*x^3 + 703*x^2 + 546*x - 131, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 29 x^{18} - 128 x^{17} + 493 x^{16} - 1180 x^{15} + 2151 x^{14} - 3166 x^{13} + 2535 x^{12} + 484 x^{11} - 3002 x^{10} + 9438 x^{9} - 10408 x^{8} + 5692 x^{7} - 4601 x^{6} - 3330 x^{5} + 10982 x^{4} - 6748 x^{3} + 703 x^{2} + 546 x - 131 \)

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:  \(14609322487882432594514132598784=2^{20}\cdot 11^{18}\cdot 1583^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.16$
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:  $2, 11, 1583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{9} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} + \frac{3}{8} a^{5} + \frac{7}{16} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a + \frac{1}{16}$, $\frac{1}{16} a^{15} - \frac{1}{8} a^{13} - \frac{1}{4} a^{10} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} + \frac{3}{8} a^{6} + \frac{7}{16} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} + \frac{1}{16} a$, $\frac{1}{64} a^{16} - \frac{1}{64} a^{14} - \frac{1}{32} a^{12} + \frac{3}{16} a^{11} + \frac{1}{4} a^{10} - \frac{11}{32} a^{9} + \frac{17}{64} a^{8} - \frac{7}{16} a^{7} - \frac{3}{8} a^{6} + \frac{3}{16} a^{5} + \frac{11}{64} a^{4} + \frac{3}{8} a^{3} + \frac{21}{64} a^{2} + \frac{9}{32} a - \frac{15}{64}$, $\frac{1}{64} a^{17} - \frac{1}{64} a^{15} - \frac{1}{32} a^{13} - \frac{1}{16} a^{12} + \frac{1}{4} a^{11} + \frac{13}{32} a^{10} + \frac{17}{64} a^{9} - \frac{3}{16} a^{8} - \frac{3}{8} a^{7} - \frac{1}{16} a^{6} - \frac{21}{64} a^{5} + \frac{3}{8} a^{4} + \frac{21}{64} a^{3} - \frac{15}{32} a^{2} + \frac{17}{64} a - \frac{1}{4}$, $\frac{1}{256} a^{18} - \frac{3}{256} a^{14} + \frac{3}{64} a^{13} + \frac{7}{128} a^{12} + \frac{27}{128} a^{11} - \frac{31}{256} a^{10} - \frac{57}{128} a^{9} + \frac{57}{256} a^{8} - \frac{5}{16} a^{7} - \frac{13}{256} a^{6} + \frac{25}{64} a^{5} - \frac{3}{8} a^{4} - \frac{43}{128} a^{3} + \frac{35}{128} a^{2} - \frac{55}{128} a + \frac{49}{256}$, $\frac{1}{16187720526712168864270826726144} a^{19} + \frac{1633615401660088557237804207}{1011732532919510554016926670384} a^{18} + \frac{6119883972843891231537490949}{4046930131678042216067706681536} a^{17} + \frac{1108243414854212292521161407}{4046930131678042216067706681536} a^{16} + \frac{257016680293493556140464816649}{16187720526712168864270826726144} a^{15} + \frac{4095566109630084157551833265}{505866266459755277008463335192} a^{14} - \frac{396178331938744822596773271533}{8093860263356084432135413363072} a^{13} - \frac{977146501227497852068160147033}{8093860263356084432135413363072} a^{12} - \frac{3344722422250188140185379766575}{16187720526712168864270826726144} a^{11} - \frac{532252434297268253656223563549}{8093860263356084432135413363072} a^{10} - \frac{2922806983183372329678062500027}{16187720526712168864270826726144} a^{9} + \frac{728795008299137832815019688619}{4046930131678042216067706681536} a^{8} + \frac{4525565776386287047896121614627}{16187720526712168864270826726144} a^{7} - \frac{1895978131723086957083246305}{15387567040600920973641470272} a^{6} + \frac{599182279165303422811893439979}{4046930131678042216067706681536} a^{5} - \frac{3787127759819283349763318698209}{8093860263356084432135413363072} a^{4} + \frac{3544888541149554001576243687349}{8093860263356084432135413363072} a^{3} + \frac{3966845672498453149251094171107}{8093860263356084432135413363072} a^{2} - \frac{5753805998655690656200962523587}{16187720526712168864270826726144} a - \frac{5366074243969986392773729967}{30892596425023223023417608256}$

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:  \( 95308373.4598 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T340:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n340 are not computed
Character table for t20n340 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1583Data not computed