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Properties

Label 20.12.1529841644...5625.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{4}\cdot 5^{16}\cdot 23^{4}\cdot 89^{7}$
Root discriminant $40.67$
Ramified primes $3, 5, 23, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8059, -9768, 23592, 12707, -24733, 18740, 24753, -15652, 3673, 15328, -3517, -4272, 3353, 753, -1867, -315, 387, 57, -33, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 33*x^18 + 57*x^17 + 387*x^16 - 315*x^15 - 1867*x^14 + 753*x^13 + 3353*x^12 - 4272*x^11 - 3517*x^10 + 15328*x^9 + 3673*x^8 - 15652*x^7 + 24753*x^6 + 18740*x^5 - 24733*x^4 + 12707*x^3 + 23592*x^2 - 9768*x - 8059)
 
gp: K = bnfinit(x^20 - 3*x^19 - 33*x^18 + 57*x^17 + 387*x^16 - 315*x^15 - 1867*x^14 + 753*x^13 + 3353*x^12 - 4272*x^11 - 3517*x^10 + 15328*x^9 + 3673*x^8 - 15652*x^7 + 24753*x^6 + 18740*x^5 - 24733*x^4 + 12707*x^3 + 23592*x^2 - 9768*x - 8059, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 33 x^{18} + 57 x^{17} + 387 x^{16} - 315 x^{15} - 1867 x^{14} + 753 x^{13} + 3353 x^{12} - 4272 x^{11} - 3517 x^{10} + 15328 x^{9} + 3673 x^{8} - 15652 x^{7} + 24753 x^{6} + 18740 x^{5} - 24733 x^{4} + 12707 x^{3} + 23592 x^{2} - 9768 x - 8059 \)

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:  \(152984164439159881898345947265625=3^{4}\cdot 5^{16}\cdot 23^{4}\cdot 89^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{25} a^{16} + \frac{1}{25} a^{15} - \frac{1}{25} a^{11} - \frac{1}{25} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{6}{25} a^{6} - \frac{6}{25} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{25} a + \frac{1}{25}$, $\frac{1}{25} a^{17} - \frac{1}{25} a^{15} - \frac{1}{25} a^{12} + \frac{1}{25} a^{10} + \frac{4}{25} a^{7} + \frac{2}{5} a^{6} + \frac{6}{25} a^{5} - \frac{4}{25} a^{2} - \frac{2}{5} a - \frac{6}{25}$, $\frac{1}{25} a^{18} + \frac{1}{25} a^{15} - \frac{1}{25} a^{13} - \frac{1}{25} a^{10} + \frac{9}{25} a^{8} - \frac{6}{25} a^{5} - \frac{1}{5} a^{4} + \frac{11}{25} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{25}$, $\frac{1}{84458642583168814566333676958646728225} a^{19} + \frac{1059847099001138536281789256023726228}{84458642583168814566333676958646728225} a^{18} + \frac{907013215908174972465632055876701726}{84458642583168814566333676958646728225} a^{17} + \frac{1009465363121078204650573159246044551}{84458642583168814566333676958646728225} a^{16} - \frac{223693608968040167592852623910375118}{84458642583168814566333676958646728225} a^{15} + \frac{5962853118322868904214730971442560164}{84458642583168814566333676958646728225} a^{14} - \frac{950351686268105573865340168556435683}{84458642583168814566333676958646728225} a^{13} + \frac{7124151074347149622133749279639582284}{84458642583168814566333676958646728225} a^{12} - \frac{4221577073739097398843107331845804471}{84458642583168814566333676958646728225} a^{11} - \frac{3290513350430474340104892482977531662}{84458642583168814566333676958646728225} a^{10} - \frac{867445979358173408679788527345474326}{84458642583168814566333676958646728225} a^{9} + \frac{39536298251200677020082357406332291552}{84458642583168814566333676958646728225} a^{8} + \frac{10815511699952512829837139194270051764}{84458642583168814566333676958646728225} a^{7} + \frac{27430372565472928488521414245656287209}{84458642583168814566333676958646728225} a^{6} + \frac{5301712974264756249895636761105886583}{84458642583168814566333676958646728225} a^{5} + \frac{2805843321419116379087339796988456466}{84458642583168814566333676958646728225} a^{4} + \frac{7298987605129270592266126154621804208}{84458642583168814566333676958646728225} a^{3} - \frac{16622572507421919203986714017782683369}{84458642583168814566333676958646728225} a^{2} + \frac{23948470869734793023698483635216330691}{84458642583168814566333676958646728225} a + \frac{30283789864388054930808331682330561227}{84458642583168814566333676958646728225}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 455499895.436 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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}$
5.8.6.2$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$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$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}$
89Data not computed