Properties

Label 20.0.90255872342...5129.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11119^{4}$
Root discriminant $11.16$
Ramified primes $3, 11119$
Class number $1$
Class group Trivial
Galois group 20T279

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1, -6, -1, -4, 15, 7, 10, -17, -11, -17, 10, 7, 15, -4, -1, -6, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 - 6*x^17 - x^16 - 4*x^15 + 15*x^14 + 7*x^13 + 10*x^12 - 17*x^11 - 11*x^10 - 17*x^9 + 10*x^8 + 7*x^7 + 15*x^6 - 4*x^5 - x^4 - 6*x^3 + x^2 + 1)
 
gp: K = bnfinit(x^20 + x^18 - 6*x^17 - x^16 - 4*x^15 + 15*x^14 + 7*x^13 + 10*x^12 - 17*x^11 - 11*x^10 - 17*x^9 + 10*x^8 + 7*x^7 + 15*x^6 - 4*x^5 - x^4 - 6*x^3 + x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + x^{18} - 6 x^{17} - x^{16} - 4 x^{15} + 15 x^{14} + 7 x^{13} + 10 x^{12} - 17 x^{11} - 11 x^{10} - 17 x^{9} + 10 x^{8} + 7 x^{7} + 15 x^{6} - 4 x^{5} - x^{4} - 6 x^{3} + x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(902558723428708305129=3^{10}\cdot 11119^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.16$
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:  $3, 11119$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{15} - \frac{1}{9} a^{14} - \frac{4}{9} a^{12} + \frac{2}{9} a^{11} + \frac{2}{9} a^{10} - \frac{1}{3} a^{8} - \frac{1}{9} a^{6} - \frac{4}{9} a^{5} + \frac{2}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2} - \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{15} - \frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{2}{9} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{9} a^{7} + \frac{4}{9} a^{6} + \frac{4}{9} a^{5} + \frac{2}{9} a^{4} - \frac{1}{9} a^{3} - \frac{2}{9} a^{2} - \frac{2}{9}$, $\frac{1}{27} a^{18} + \frac{1}{27} a^{17} + \frac{1}{27} a^{16} + \frac{1}{9} a^{15} + \frac{1}{27} a^{14} + \frac{1}{9} a^{13} + \frac{8}{27} a^{12} + \frac{1}{9} a^{11} - \frac{13}{27} a^{10} + \frac{1}{9} a^{9} - \frac{4}{27} a^{8} + \frac{1}{9} a^{7} - \frac{10}{27} a^{6} + \frac{4}{9} a^{5} - \frac{8}{27} a^{4} + \frac{4}{9} a^{3} + \frac{10}{27} a^{2} + \frac{10}{27} a + \frac{10}{27}$, $\frac{1}{27} a^{19} - \frac{1}{27} a^{16} + \frac{1}{27} a^{15} - \frac{4}{27} a^{14} + \frac{5}{27} a^{13} - \frac{2}{27} a^{12} - \frac{4}{27} a^{11} - \frac{8}{27} a^{10} - \frac{7}{27} a^{9} - \frac{11}{27} a^{8} - \frac{13}{27} a^{7} - \frac{2}{27} a^{6} - \frac{8}{27} a^{5} - \frac{4}{27} a^{4} - \frac{2}{27} a^{3} - \frac{2}{9} a^{2} + \frac{1}{9} a - \frac{13}{27}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{26}{27} a^{19} + \frac{34}{27} a^{18} + \frac{1}{27} a^{17} - \frac{121}{27} a^{16} - \frac{244}{27} a^{15} + \frac{2}{27} a^{14} + \frac{283}{27} a^{13} + \frac{712}{27} a^{12} + \frac{166}{27} a^{11} - \frac{290}{27} a^{10} - \frac{899}{27} a^{9} - \frac{422}{27} a^{8} - \frac{41}{27} a^{7} + \frac{658}{27} a^{6} + \frac{323}{27} a^{5} + \frac{164}{27} a^{4} - \frac{295}{27} a^{3} - \frac{35}{27} a^{2} - \frac{56}{27} a + \frac{74}{27} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 293.866842819 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T279:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n279
Character table for t20n279 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.3.11119.1, 10.0.30042615123.1, 10.2.1112689449.1, 10.4.3338068347.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$$[\ ]_{2}^{4}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
11119Data not computed