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Properties

Label 20.0.26696074417...6992.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{15}\cdot 36497^{4}$
Root discriminant $37.27$
Ramified primes $2, 3, 36497$
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![243, 0, -2916, 0, 14175, 0, -30132, 0, 34452, 0, -22572, 0, 8964, 0, -2187, 0, 324, 0, -27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 27*x^18 + 324*x^16 - 2187*x^14 + 8964*x^12 - 22572*x^10 + 34452*x^8 - 30132*x^6 + 14175*x^4 - 2916*x^2 + 243)
 
gp: K = bnfinit(x^20 - 27*x^18 + 324*x^16 - 2187*x^14 + 8964*x^12 - 22572*x^10 + 34452*x^8 - 30132*x^6 + 14175*x^4 - 2916*x^2 + 243, 1)
 

Normalized defining polynomial

\( x^{20} - 27 x^{18} + 324 x^{16} - 2187 x^{14} + 8964 x^{12} - 22572 x^{10} + 34452 x^{8} - 30132 x^{6} + 14175 x^{4} - 2916 x^{2} + 243 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26696074417804819521986853076992=2^{20}\cdot 3^{15}\cdot 36497^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.27$
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:  $2, 3, 36497$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4}$, $\frac{1}{3} a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{9} a^{8}$, $\frac{1}{9} a^{9}$, $\frac{1}{9} a^{10}$, $\frac{1}{9} a^{11}$, $\frac{1}{27} a^{12}$, $\frac{1}{27} a^{13}$, $\frac{1}{27} a^{14}$, $\frac{1}{27} a^{15}$, $\frac{1}{81} a^{16}$, $\frac{1}{81} a^{17}$, $\frac{1}{747596385} a^{18} + \frac{1815184}{747596385} a^{16} - \frac{473894}{249198795} a^{14} - \frac{1821193}{249198795} a^{12} + \frac{212771}{16613253} a^{10} - \frac{3232823}{83066265} a^{8} + \frac{146045}{1845917} a^{6} - \frac{1652186}{27688755} a^{4} + \frac{1459588}{9229585} a^{2} - \frac{2936238}{9229585}$, $\frac{1}{747596385} a^{19} + \frac{1815184}{747596385} a^{17} - \frac{473894}{249198795} a^{15} - \frac{1821193}{249198795} a^{13} + \frac{212771}{16613253} a^{11} - \frac{3232823}{83066265} a^{9} + \frac{146045}{1845917} a^{7} - \frac{1652186}{27688755} a^{5} + \frac{1459588}{9229585} a^{3} - \frac{2936238}{9229585} a$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{2481484}{747596385} a^{18} - \frac{21946643}{249198795} a^{16} + \frac{257933704}{249198795} a^{14} - \frac{564412934}{83066265} a^{12} + \frac{445058651}{16613253} a^{10} - \frac{1762907494}{27688755} a^{8} + \frac{493044266}{5537751} a^{6} - \frac{621182448}{9229585} a^{4} + \frac{236655422}{9229585} a^{2} - \frac{24034792}{9229585} \) (order $6$)
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:  \( 70827885.0617 \) (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

\(\Q(\sqrt{-3}) \), 5.5.36497.1, 10.0.323683535187.1

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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
3Data not computed
36497Data not computed