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Properties

Label 20.12.7725497509...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{4}\cdot 5^{14}\cdot 23^{4}\cdot 89^{5}$
Root discriminant $22.10$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T887

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 16, 38, -68, -93, 158, -26, -151, 216, -126, -8, 121, -138, 93, -31, -12, 19, -9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 9*x^18 + 19*x^17 - 12*x^16 - 31*x^15 + 93*x^14 - 138*x^13 + 121*x^12 - 8*x^11 - 126*x^10 + 216*x^9 - 151*x^8 - 26*x^7 + 158*x^6 - 93*x^5 - 68*x^4 + 38*x^3 + 16*x^2 - 3*x - 1)
 
gp: K = bnfinit(x^20 - x^19 - 9*x^18 + 19*x^17 - 12*x^16 - 31*x^15 + 93*x^14 - 138*x^13 + 121*x^12 - 8*x^11 - 126*x^10 + 216*x^9 - 151*x^8 - 26*x^7 + 158*x^6 - 93*x^5 - 68*x^4 + 38*x^3 + 16*x^2 - 3*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 9 x^{18} + 19 x^{17} - 12 x^{16} - 31 x^{15} + 93 x^{14} - 138 x^{13} + 121 x^{12} - 8 x^{11} - 126 x^{10} + 216 x^{9} - 151 x^{8} - 26 x^{7} + 158 x^{6} - 93 x^{5} - 68 x^{4} + 38 x^{3} + 16 x^{2} - 3 x - 1 \)

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:  \(772549750986794000244140625=3^{4}\cdot 5^{14}\cdot 23^{4}\cdot 89^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.10$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} + \frac{5}{23} a^{16} - \frac{8}{23} a^{15} + \frac{9}{23} a^{13} + \frac{11}{23} a^{12} - \frac{10}{23} a^{11} - \frac{9}{23} a^{10} - \frac{6}{23} a^{9} - \frac{5}{23} a^{8} - \frac{7}{23} a^{7} + \frac{11}{23} a^{6} + \frac{7}{23} a^{5} - \frac{3}{23} a^{4} + \frac{11}{23} a^{3} - \frac{10}{23} a^{2} - \frac{3}{23} a - \frac{7}{23}$, $\frac{1}{23} a^{18} - \frac{10}{23} a^{16} - \frac{6}{23} a^{15} + \frac{9}{23} a^{14} - \frac{11}{23} a^{13} + \frac{4}{23} a^{12} - \frac{5}{23} a^{11} - \frac{7}{23} a^{10} + \frac{2}{23} a^{9} - \frac{5}{23} a^{8} - \frac{2}{23} a^{6} + \frac{8}{23} a^{5} + \frac{3}{23} a^{4} + \frac{4}{23} a^{3} + \frac{1}{23} a^{2} + \frac{8}{23} a - \frac{11}{23}$, $\frac{1}{28721788159267} a^{19} + \frac{623584550041}{28721788159267} a^{18} + \frac{66619519191}{4103112594181} a^{17} - \frac{543091775781}{28721788159267} a^{16} + \frac{9476971008124}{28721788159267} a^{15} - \frac{516678674890}{4103112594181} a^{14} + \frac{13849490005549}{28721788159267} a^{13} - \frac{27550877785}{4103112594181} a^{12} - \frac{6703167694907}{28721788159267} a^{11} + \frac{3570292347360}{28721788159267} a^{10} + \frac{4934893422077}{28721788159267} a^{9} + \frac{5216972470979}{28721788159267} a^{8} - \frac{1865617055254}{4103112594181} a^{7} + \frac{4529726523107}{28721788159267} a^{6} + \frac{79871298649}{28721788159267} a^{5} - \frac{4917081424392}{28721788159267} a^{4} + \frac{5025472169094}{28721788159267} a^{3} + \frac{11536632986708}{28721788159267} a^{2} + \frac{6965493238744}{28721788159267} a - \frac{14035122104284}{28721788159267}$

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:  \( 1857639.96504 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T887:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 245760
The 201 conjugacy class representatives for t20n887 are not computed
Character table for t20n887 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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89Data not computed