↑

Properties

Label 20.8.58343029654...2368.4
Degree $20$
Signature $[8, 6]$
Discriminant $2^{10}\cdot 19^{10}\cdot 43^{11}$
Root discriminant $48.79$
Ramified primes $2, 19, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T423

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-800, -3280, -3048, 16488, 24522, -20027, -32434, 6575, 18913, 6255, -8682, -4478, 3416, 249, -277, 123, -159, 44, 14, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 14*x^18 + 44*x^17 - 159*x^16 + 123*x^15 - 277*x^14 + 249*x^13 + 3416*x^12 - 4478*x^11 - 8682*x^10 + 6255*x^9 + 18913*x^8 + 6575*x^7 - 32434*x^6 - 20027*x^5 + 24522*x^4 + 16488*x^3 - 3048*x^2 - 3280*x - 800)
 
gp: K = bnfinit(x^20 - 8*x^19 + 14*x^18 + 44*x^17 - 159*x^16 + 123*x^15 - 277*x^14 + 249*x^13 + 3416*x^12 - 4478*x^11 - 8682*x^10 + 6255*x^9 + 18913*x^8 + 6575*x^7 - 32434*x^6 - 20027*x^5 + 24522*x^4 + 16488*x^3 - 3048*x^2 - 3280*x - 800, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 14 x^{18} + 44 x^{17} - 159 x^{16} + 123 x^{15} - 277 x^{14} + 249 x^{13} + 3416 x^{12} - 4478 x^{11} - 8682 x^{10} + 6255 x^{9} + 18913 x^{8} + 6575 x^{7} - 32434 x^{6} - 20027 x^{5} + 24522 x^{4} + 16488 x^{3} - 3048 x^{2} - 3280 x - 800 \)

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:  \(5834302965409512291058019417402368=2^{10}\cdot 19^{10}\cdot 43^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.79$
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:  $2, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{10} a^{16} + \frac{2}{5} a^{15} + \frac{1}{5} a^{14} - \frac{1}{10} a^{12} - \frac{1}{2} a^{11} + \frac{3}{10} a^{10} + \frac{3}{10} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} - \frac{3}{10} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} - \frac{1}{10} a$, $\frac{1}{20} a^{17} + \frac{3}{10} a^{15} - \frac{2}{5} a^{14} + \frac{9}{20} a^{13} - \frac{1}{20} a^{12} + \frac{3}{20} a^{11} + \frac{1}{20} a^{10} - \frac{3}{10} a^{8} - \frac{1}{2} a^{7} + \frac{3}{20} a^{6} - \frac{3}{20} a^{5} + \frac{7}{20} a^{4} + \frac{3}{10} a^{3} + \frac{1}{4} a^{2} - \frac{3}{10} a$, $\frac{1}{40} a^{18} - \frac{1}{20} a^{16} - \frac{1}{2} a^{15} - \frac{7}{40} a^{14} + \frac{19}{40} a^{13} + \frac{11}{40} a^{12} + \frac{1}{40} a^{11} + \frac{2}{5} a^{10} + \frac{1}{4} a^{9} + \frac{7}{20} a^{8} + \frac{7}{40} a^{7} - \frac{7}{40} a^{6} + \frac{7}{40} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{7}{20} a^{2} + \frac{1}{5} a$, $\frac{1}{3514678188793145526662466044102234480} a^{19} + \frac{9534637986377337306944061178688383}{878669547198286381665616511025558620} a^{18} - \frac{1088575970498232537541248472517163}{351467818879314552666246604410223448} a^{17} + \frac{3434131809865228943688292953550349}{878669547198286381665616511025558620} a^{16} + \frac{74889630236409815646062019589884725}{702935637758629105332493208820446896} a^{15} - \frac{19320227162651863784946205379609679}{44489597326495512995727418279775120} a^{14} - \frac{1005587170937361860085474489143737893}{3514678188793145526662466044102234480} a^{13} + \frac{1219126261667958385757540123620921761}{3514678188793145526662466044102234480} a^{12} + \frac{55197696641731107635885459083866823}{439334773599143190832808255512779310} a^{11} + \frac{791905922246564474375295721646898947}{1757339094396572763331233022051117240} a^{10} + \frac{174825171155078466283573346882374167}{1757339094396572763331233022051117240} a^{9} - \frac{1738758685151603413651654364501683457}{3514678188793145526662466044102234480} a^{8} - \frac{10582586482994098399351996843325797}{44489597326495512995727418279775120} a^{7} - \frac{1238168324063203524151679496078476161}{3514678188793145526662466044102234480} a^{6} - \frac{21241877158925248235962709203831361}{1757339094396572763331233022051117240} a^{5} - \frac{870970226116766435334853276504485111}{3514678188793145526662466044102234480} a^{4} - \frac{427193334202130208417760422493687261}{1757339094396572763331233022051117240} a^{3} - \frac{77715581002063365356417596707899803}{175733909439657276333123302205111724} a^{2} - \frac{77679525085447433538006325576891986}{219667386799571595416404127756389655} a + \frac{15338742338145080672802150005300530}{43933477359914319083280825551277931}$

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:  \( 2558852049.72 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T423:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 160 conjugacy class representatives for t20n423 are not computed
Character table for t20n423 is not computed

Intermediate fields

\(\Q(\sqrt{817}) \), 5.5.667489.1 x5, 10.10.364007458703857.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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}$
2.10.10.14$x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$