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Properties

Label 15.3.41809707512...6609.1
Degree $15$
Signature $[3, 6]$
Discriminant $3^{24}\cdot 23^{6}$
Root discriminant $20.33$
Ramified primes $3, 23$
Class number $1$
Class group Trivial
Galois group $\GL(2,4)$ (as 15T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, -18, -18, -24, -72, -17, -78, -27, -21, -21, 9, -6, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 6*x^13 - 6*x^12 + 9*x^11 - 21*x^10 - 21*x^9 - 27*x^8 - 78*x^7 - 17*x^6 - 72*x^5 - 24*x^4 - 18*x^3 - 18*x^2 - 3)
 
gp: K = bnfinit(x^15 + 6*x^13 - 6*x^12 + 9*x^11 - 21*x^10 - 21*x^9 - 27*x^8 - 78*x^7 - 17*x^6 - 72*x^5 - 24*x^4 - 18*x^3 - 18*x^2 - 3, 1)
 

Normalized defining polynomial

\( x^{15} + 6 x^{13} - 6 x^{12} + 9 x^{11} - 21 x^{10} - 21 x^{9} - 27 x^{8} - 78 x^{7} - 17 x^{6} - 72 x^{5} - 24 x^{4} - 18 x^{3} - 18 x^{2} - 3 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(41809707512822766609=3^{24}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.33$
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:  $3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{98779231} a^{14} - \frac{19960887}{98779231} a^{13} + \frac{17093634}{98779231} a^{12} + \frac{12039915}{98779231} a^{11} + \frac{16737629}{98779231} a^{10} + \frac{572835}{5810543} a^{9} - \frac{3917943}{98779231} a^{8} + \frac{25948863}{98779231} a^{7} + \frac{9602357}{98779231} a^{6} - \frac{46840583}{98779231} a^{5} - \frac{14751877}{98779231} a^{4} - \frac{41438432}{98779231} a^{3} + \frac{14426004}{98779231} a^{2} + \frac{44427160}{98779231} a + \frac{40254692}{98779231}$

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:  $8$
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:  \( 14785.4641372 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times A_5$ (as 15T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 180
The 15 conjugacy class representatives for $\GL(2,4)$
Character table for $\GL(2,4)$

Intermediate fields

\(\Q(\zeta_{9})^+\), 5.1.42849.1

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

Sibling fields

Degree 15 siblings: data not computed
Degree 18 sibling: data not computed
Degree 30 sibling: 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 $15$ $15$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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$3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.9.15.13$x^{9} + 3 x^{7} + 6 x^{6} + 6 x^{3} + 3$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
$23$23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
23.9.6.1$x^{9} - 529 x^{3} + 48668$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$