↑

Properties

Label 20.12.1147974208...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{4}\cdot 5^{16}\cdot 23^{6}\cdot 89^{4}$
Root discriminant $28.38$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T794

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25, -250, 500, 2230, -3770, -2250, 8471, -1676, -8419, 4394, 4146, -3358, -872, 1312, -62, -278, 71, 29, -14, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 14*x^18 + 29*x^17 + 71*x^16 - 278*x^15 - 62*x^14 + 1312*x^13 - 872*x^12 - 3358*x^11 + 4146*x^10 + 4394*x^9 - 8419*x^8 - 1676*x^7 + 8471*x^6 - 2250*x^5 - 3770*x^4 + 2230*x^3 + 500*x^2 - 250*x - 25)
 
gp: K = bnfinit(x^20 - x^19 - 14*x^18 + 29*x^17 + 71*x^16 - 278*x^15 - 62*x^14 + 1312*x^13 - 872*x^12 - 3358*x^11 + 4146*x^10 + 4394*x^9 - 8419*x^8 - 1676*x^7 + 8471*x^6 - 2250*x^5 - 3770*x^4 + 2230*x^3 + 500*x^2 - 250*x - 25, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 14 x^{18} + 29 x^{17} + 71 x^{16} - 278 x^{15} - 62 x^{14} + 1312 x^{13} - 872 x^{12} - 3358 x^{11} + 4146 x^{10} + 4394 x^{9} - 8419 x^{8} - 1676 x^{7} + 8471 x^{6} - 2250 x^{5} - 3770 x^{4} + 2230 x^{3} + 500 x^{2} - 250 x - 25 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(114797420862925288238525390625=3^{4}\cdot 5^{16}\cdot 23^{6}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.38$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{8} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{8} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{8} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3}$, $\frac{1}{75} a^{16} + \frac{1}{15} a^{15} + \frac{1}{15} a^{14} - \frac{1}{15} a^{13} - \frac{1}{25} a^{11} + \frac{1}{15} a^{10} + \frac{4}{15} a^{7} + \frac{1}{75} a^{6} - \frac{1}{15} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{75} a^{17} - \frac{1}{15} a^{15} - \frac{1}{15} a^{13} - \frac{1}{25} a^{12} + \frac{1}{15} a^{11} + \frac{1}{15} a^{10} + \frac{7}{15} a^{8} - \frac{8}{25} a^{7} - \frac{1}{3} a^{6} - \frac{7}{15} a^{5} - \frac{2}{5} a^{4} - \frac{1}{15} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{225} a^{18} - \frac{1}{225} a^{17} + \frac{1}{15} a^{15} - \frac{2}{45} a^{14} + \frac{22}{225} a^{13} - \frac{22}{225} a^{12} - \frac{2}{45} a^{10} - \frac{2}{45} a^{9} + \frac{91}{225} a^{8} + \frac{23}{75} a^{7} + \frac{2}{45} a^{6} - \frac{7}{45} a^{5} - \frac{13}{45} a^{4} - \frac{8}{45} a^{3} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{34036545825} a^{19} + \frac{1093631}{11345515275} a^{18} - \frac{42292211}{6807309165} a^{17} - \frac{7348717}{11345515275} a^{16} - \frac{412639793}{6807309165} a^{15} - \frac{353589751}{11345515275} a^{14} + \frac{41989172}{11345515275} a^{13} + \frac{354465979}{6807309165} a^{12} - \frac{1173983662}{34036545825} a^{11} - \frac{311618644}{6807309165} a^{10} - \frac{912576143}{11345515275} a^{9} - \frac{12842528432}{34036545825} a^{8} - \frac{1301025451}{6807309165} a^{7} + \frac{12577738319}{34036545825} a^{6} + \frac{1099005643}{6807309165} a^{5} + \frac{93775629}{756367685} a^{4} - \frac{482633462}{6807309165} a^{3} - \frac{548141191}{1361461833} a^{2} - \frac{88563710}{1361461833} a + \frac{334215905}{1361461833}$

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:  $15$
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:  \( 24768064.615 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T794:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n794 are not computed
Character table for t20n794 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$