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Properties

Label 15.9.31320212675...3727.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,3^{24}\cdot 223^{3}$
Root discriminant $17.10$
Ramified primes $3, 223$
Class number $1$
Class group Trivial
Galois group $S_5 \times C_3$ (as 15T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -18, -27, 0, -9, -36, 63, 99, 0, 3, 21, -9, -9, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^13 - 9*x^12 - 9*x^11 + 21*x^10 + 3*x^9 + 99*x^7 + 63*x^6 - 36*x^5 - 9*x^4 - 27*x^2 - 18*x - 3)
 
gp: K = bnfinit(x^15 - 3*x^13 - 9*x^12 - 9*x^11 + 21*x^10 + 3*x^9 + 99*x^7 + 63*x^6 - 36*x^5 - 9*x^4 - 27*x^2 - 18*x - 3, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{13} - 9 x^{12} - 9 x^{11} + 21 x^{10} + 3 x^{9} + 99 x^{7} + 63 x^{6} - 36 x^{5} - 9 x^{4} - 27 x^{2} - 18 x - 3 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3132021267584993727=-\,3^{24}\cdot 223^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.10$
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:  $3, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{1}{11} a^{12} - \frac{5}{11} a^{11} - \frac{1}{11} a^{10} - \frac{4}{11} a^{9} - \frac{5}{11} a^{8} - \frac{2}{11} a^{7} - \frac{5}{11} a^{6} + \frac{1}{11} a^{4} - \frac{4}{11} a^{3} + \frac{3}{11} a^{2} - \frac{2}{11} a - \frac{1}{11}$, $\frac{1}{4272739537} a^{14} - \frac{111219312}{4272739537} a^{13} + \frac{53046819}{388430867} a^{12} - \frac{1198426183}{4272739537} a^{11} + \frac{62893865}{4272739537} a^{10} + \frac{592929803}{4272739537} a^{9} - \frac{649613920}{4272739537} a^{8} + \frac{1775191445}{4272739537} a^{7} + \frac{1983240900}{4272739537} a^{6} - \frac{300940782}{4272739537} a^{5} - \frac{1526995886}{4272739537} a^{4} - \frac{699399293}{4272739537} a^{3} - \frac{2003697460}{4272739537} a^{2} - \frac{609592370}{4272739537} a + \frac{1659082947}{4272739537}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6161.99584614 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_5$ (as 15T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 21 conjugacy class representatives for $S_5 \times C_3$
Character table for $S_5 \times C_3$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 5.3.18063.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 45 sibling: 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 $15$ R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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
3Data not computed
223Data not computed