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Properties

Label 15.3.70978216464...2656.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{12}\cdot 3^{20}\cdot 89^{6}$
Root discriminant $45.37$
Ramified primes $2, 3, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\GL(2,4):C_2$ (as 15T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1062, -54, 333, 2802, 2808, 5889, 3272, 1017, -1677, -253, 30, 66, 27, -12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 12*x^13 + 27*x^12 + 66*x^11 + 30*x^10 - 253*x^9 - 1677*x^8 + 1017*x^7 + 3272*x^6 + 5889*x^5 + 2808*x^4 + 2802*x^3 + 333*x^2 - 54*x - 1062)
 
gp: K = bnfinit(x^15 - 3*x^14 - 12*x^13 + 27*x^12 + 66*x^11 + 30*x^10 - 253*x^9 - 1677*x^8 + 1017*x^7 + 3272*x^6 + 5889*x^5 + 2808*x^4 + 2802*x^3 + 333*x^2 - 54*x - 1062, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 12 x^{13} + 27 x^{12} + 66 x^{11} + 30 x^{10} - 253 x^{9} - 1677 x^{8} + 1017 x^{7} + 3272 x^{6} + 5889 x^{5} + 2808 x^{4} + 2802 x^{3} + 333 x^{2} - 54 x - 1062 \)

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:  \(7097821646486147478982656=2^{12}\cdot 3^{20}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.37$
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:  $2, 3, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{18} a^{12} - \frac{1}{3} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{4}{9} a^{6} - \frac{1}{2} a^{5} - \frac{1}{18} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{36} a^{13} - \frac{1}{36} a^{12} - \frac{5}{12} a^{11} - \frac{1}{2} a^{10} - \frac{1}{12} a^{9} - \frac{1}{2} a^{8} + \frac{7}{18} a^{7} + \frac{13}{36} a^{6} + \frac{1}{4} a^{5} + \frac{17}{36} a^{4} + \frac{5}{18} a^{3} + \frac{1}{12} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{15129745050146025490555248} a^{14} + \frac{3486062322497358987833}{840541391674779193919736} a^{13} - \frac{12968805487534026422441}{840541391674779193919736} a^{12} - \frac{1561551427004957184757669}{5043248350048675163518416} a^{11} + \frac{1835729781414446630701741}{5043248350048675163518416} a^{10} - \frac{988200578753783629561589}{5043248350048675163518416} a^{9} + \frac{733452611709188756798875}{1891218131268253186319406} a^{8} - \frac{1494505006588975900932023}{5043248350048675163518416} a^{7} + \frac{102012140208491974569089}{315203021878042197719901} a^{6} - \frac{339684193741374692043245}{1891218131268253186319406} a^{5} + \frac{336686806296252400039457}{1681082783349558387839472} a^{4} + \frac{1847833762460787738794087}{5043248350048675163518416} a^{3} + \frac{1267165135951447460815577}{5043248350048675163518416} a^{2} - \frac{30523571538798725549843}{420270695837389596959868} a + \frac{112825896172724790492199}{840541391674779193919736}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $8$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 23490657.801 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_5$ (as 15T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 12 conjugacy class representatives for $\GL(2,4):C_2$
Character table for $\GL(2,4):C_2$

Intermediate fields

3.1.324.1, 5.3.31684.1

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 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 R R $15$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.9.12.16$x^{9} + 9 x^{5} + 18 x^{3} + 27 x^{2} + 27$$3$$3$$12$$S_3\times C_3$$[2]^{6}$
$89$89.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
89.9.6.1$x^{9} - 7921 x^{3} + 4934783$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$