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Properties

Label 20.8.44980292500...5625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 4079^{6}$
Root discriminant $27.08$
Ramified primes $5, 4079$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T199

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, -256, 13, 901, -544, -1335, 1072, 1126, -840, -448, 608, -8, -389, 2, 52, -12, 8, 16, -3, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 3*x^18 + 16*x^17 + 8*x^16 - 12*x^15 + 52*x^14 + 2*x^13 - 389*x^12 - 8*x^11 + 608*x^10 - 448*x^9 - 840*x^8 + 1126*x^7 + 1072*x^6 - 1335*x^5 - 544*x^4 + 901*x^3 + 13*x^2 - 256*x + 29)
 
gp: K = bnfinit(x^20 - 4*x^19 - 3*x^18 + 16*x^17 + 8*x^16 - 12*x^15 + 52*x^14 + 2*x^13 - 389*x^12 - 8*x^11 + 608*x^10 - 448*x^9 - 840*x^8 + 1126*x^7 + 1072*x^6 - 1335*x^5 - 544*x^4 + 901*x^3 + 13*x^2 - 256*x + 29, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 3 x^{18} + 16 x^{17} + 8 x^{16} - 12 x^{15} + 52 x^{14} + 2 x^{13} - 389 x^{12} - 8 x^{11} + 608 x^{10} - 448 x^{9} - 840 x^{8} + 1126 x^{7} + 1072 x^{6} - 1335 x^{5} - 544 x^{4} + 901 x^{3} + 13 x^{2} - 256 x + 29 \)

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:  \(44980292500311338198447265625=5^{10}\cdot 4079^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.08$
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:  $5, 4079$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{468621467448502238555392802405} a^{19} + \frac{235681372470788194207446321}{93724293489700447711078560481} a^{18} - \frac{3612156185507892288060193403}{468621467448502238555392802405} a^{17} + \frac{45649182844104077634949389319}{468621467448502238555392802405} a^{16} + \frac{74365409792896968353829436534}{468621467448502238555392802405} a^{15} + \frac{228020676507703045813098690349}{468621467448502238555392802405} a^{14} - \frac{13339203747056298618566863907}{468621467448502238555392802405} a^{13} - \frac{219504131596489208150220446891}{468621467448502238555392802405} a^{12} - \frac{93714819467543049542236416763}{468621467448502238555392802405} a^{11} - \frac{18236578720176120530527607998}{93724293489700447711078560481} a^{10} + \frac{24411457185696127459574638758}{468621467448502238555392802405} a^{9} - \frac{181593297303769463431048516086}{468621467448502238555392802405} a^{8} + \frac{214370599980898954697664151436}{468621467448502238555392802405} a^{7} + \frac{10453067905649826490045298836}{93724293489700447711078560481} a^{6} - \frac{122886746092865506670753657153}{468621467448502238555392802405} a^{5} + \frac{129304895020232830831446528368}{468621467448502238555392802405} a^{4} + \frac{124730894278237444141388881783}{468621467448502238555392802405} a^{3} + \frac{101078409626069004203043132033}{468621467448502238555392802405} a^{2} - \frac{17401555814782891192762317682}{93724293489700447711078560481} a + \frac{179672484306642369377250510744}{468621467448502238555392802405}$

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:  \( 9209951.62422 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T199:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1440
The 22 conjugacy class representatives for t20n199
Character table for t20n199 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.4.42417115649375.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
4079Data not computed