↑

Properties

Label 20.8.97899079174...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{16}\cdot 3^{12}\cdot 5^{13}\cdot 101\cdot 691^{4}$
Root discriminant $44.62$
Ramified primes $2, 3, 5, 101, 691$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![505, 0, -3390, 0, 3085, 0, 1320, 0, 3874, 0, -676, 0, -846, 0, 200, 0, 37, 0, -14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 14*x^18 + 37*x^16 + 200*x^14 - 846*x^12 - 676*x^10 + 3874*x^8 + 1320*x^6 + 3085*x^4 - 3390*x^2 + 505)
 
gp: K = bnfinit(x^20 - 14*x^18 + 37*x^16 + 200*x^14 - 846*x^12 - 676*x^10 + 3874*x^8 + 1320*x^6 + 3085*x^4 - 3390*x^2 + 505, 1)
 

Normalized defining polynomial

\( x^{20} - 14 x^{18} + 37 x^{16} + 200 x^{14} - 846 x^{12} - 676 x^{10} + 3874 x^{8} + 1320 x^{6} + 3085 x^{4} - 3390 x^{2} + 505 \)

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:  \(978990791745334024080000000000000=2^{16}\cdot 3^{12}\cdot 5^{13}\cdot 101\cdot 691^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.62$
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:  $2, 3, 5, 101, 691$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} + \frac{1}{16}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{5} + \frac{1}{16} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4} + \frac{1}{32} a^{2} - \frac{1}{32}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} - \frac{1}{16} a^{7} + \frac{1}{16} a^{5} + \frac{1}{32} a^{3} - \frac{1}{32} a$, $\frac{1}{64} a^{12} + \frac{1}{64} a^{8} - \frac{5}{64} a^{4} + \frac{3}{64}$, $\frac{1}{64} a^{13} + \frac{1}{64} a^{9} - \frac{5}{64} a^{5} + \frac{3}{64} a$, $\frac{1}{384} a^{14} + \frac{1}{384} a^{12} + \frac{1}{384} a^{10} + \frac{1}{384} a^{8} + \frac{11}{384} a^{6} + \frac{43}{384} a^{4} + \frac{17}{128} a^{2} + \frac{19}{384}$, $\frac{1}{384} a^{15} + \frac{1}{384} a^{13} + \frac{1}{384} a^{11} + \frac{1}{384} a^{9} + \frac{11}{384} a^{7} + \frac{43}{384} a^{5} + \frac{17}{128} a^{3} + \frac{19}{384} a$, $\frac{1}{768} a^{16} + \frac{5}{384} a^{8} + \frac{1}{24} a^{6} + \frac{1}{96} a^{4} - \frac{1}{24} a^{2} - \frac{19}{768}$, $\frac{1}{1536} a^{17} - \frac{1}{1536} a^{16} - \frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{17}{768} a^{9} + \frac{7}{768} a^{8} + \frac{5}{96} a^{7} + \frac{1}{96} a^{6} - \frac{5}{192} a^{5} - \frac{7}{192} a^{4} - \frac{7}{192} a^{3} + \frac{1}{192} a^{2} - \frac{763}{1536} a - \frac{725}{1536}$, $\frac{1}{408576} a^{18} + \frac{29}{136192} a^{16} + \frac{13}{17024} a^{14} + \frac{5}{1064} a^{12} - \frac{193}{29184} a^{10} + \frac{1531}{204288} a^{8} - \frac{1177}{25536} a^{6} - \frac{6169}{51072} a^{4} - \frac{87067}{408576} a^{2} - \frac{45399}{136192}$, $\frac{1}{817152} a^{19} - \frac{1}{817152} a^{18} + \frac{29}{272384} a^{17} - \frac{29}{272384} a^{16} + \frac{13}{34048} a^{15} - \frac{13}{34048} a^{14} + \frac{5}{2128} a^{13} - \frac{5}{2128} a^{12} + \frac{719}{58368} a^{11} - \frac{719}{58368} a^{10} + \frac{7915}{408576} a^{9} - \frac{7915}{408576} a^{8} - \frac{2773}{51072} a^{7} + \frac{2773}{51072} a^{6} - \frac{9361}{102144} a^{5} + \frac{9361}{102144} a^{4} + \frac{129989}{817152} a^{3} - \frac{129989}{817152} a^{2} - \frac{109239}{272384} a + \frac{109239}{272384}$

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:  \( 1378368363.66 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.5438807015625.1

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.6.6.5$x^{6} + 6 x^{3} + 9 x^{2} + 9$$3$$2$$6$$S_3^2$$[3/2, 3/2]_{2}^{2}$
3.6.6.5$x^{6} + 6 x^{3} + 9 x^{2} + 9$$3$$2$$6$$S_3^2$$[3/2, 3/2]_{2}^{2}$
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$
101.8.0.1$x^{8} - x + 11$$1$$8$$0$$C_8$$[\ ]^{8}$
691Data not computed