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Properties

Label 15.13.8064956293...4032.1
Degree $15$
Signature $[13, 1]$
Discriminant $-\,2^{8}\cdot 3^{3}\cdot 11^{6}\cdot 19^{4}\cdot 131^{6}$
Root discriminant $72.52$
Ramified primes $2, 3, 11, 19, 131$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T90

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-55404, -118098, 47709, 215190, 36504, -115050, -28396, 27217, 2554, -5430, 183, 744, -14, -46, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 46*x^13 - 14*x^12 + 744*x^11 + 183*x^10 - 5430*x^9 + 2554*x^8 + 27217*x^7 - 28396*x^6 - 115050*x^5 + 36504*x^4 + 215190*x^3 + 47709*x^2 - 118098*x - 55404)
 
gp: K = bnfinit(x^15 - 46*x^13 - 14*x^12 + 744*x^11 + 183*x^10 - 5430*x^9 + 2554*x^8 + 27217*x^7 - 28396*x^6 - 115050*x^5 + 36504*x^4 + 215190*x^3 + 47709*x^2 - 118098*x - 55404, 1)
 

Normalized defining polynomial

\( x^{15} - 46 x^{13} - 14 x^{12} + 744 x^{11} + 183 x^{10} - 5430 x^{9} + 2554 x^{8} + 27217 x^{7} - 28396 x^{6} - 115050 x^{5} + 36504 x^{4} + 215190 x^{3} + 47709 x^{2} - 118098 x - 55404 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[13, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-8064956293708945042260364032=-\,2^{8}\cdot 3^{3}\cdot 11^{6}\cdot 19^{4}\cdot 131^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.52$
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:  $2, 3, 11, 19, 131$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{5} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{36} a^{11} + \frac{2}{9} a^{9} + \frac{1}{9} a^{8} + \frac{5}{12} a^{7} - \frac{5}{12} a^{6} - \frac{1}{3} a^{5} - \frac{1}{18} a^{4} - \frac{2}{9} a^{3} + \frac{17}{36} a^{2} - \frac{1}{3} a$, $\frac{1}{108} a^{12} + \frac{2}{27} a^{10} - \frac{7}{54} a^{9} + \frac{5}{36} a^{8} - \frac{11}{36} a^{7} + \frac{2}{9} a^{6} - \frac{19}{54} a^{5} - \frac{13}{54} a^{4} - \frac{19}{108} a^{3} - \frac{5}{18} a^{2}$, $\frac{1}{648} a^{13} - \frac{1}{648} a^{11} - \frac{7}{324} a^{10} + \frac{17}{216} a^{9} - \frac{41}{216} a^{8} + \frac{35}{216} a^{7} + \frac{259}{648} a^{6} + \frac{95}{324} a^{5} - \frac{217}{648} a^{4} + \frac{4}{27} a^{3} + \frac{19}{72} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{790058929613724612648} a^{14} + \frac{15650448355212899}{65838244134477051054} a^{13} + \frac{529490816376788015}{790058929613724612648} a^{12} + \frac{2672342163577170013}{197514732403431153162} a^{11} - \frac{9904459681222923095}{263352976537908204216} a^{10} + \frac{29376136404554485207}{263352976537908204216} a^{9} - \frac{41691664948318845709}{263352976537908204216} a^{8} - \frac{295088222165152039283}{790058929613724612648} a^{7} + \frac{95426416474144215529}{197514732403431153162} a^{6} - \frac{250504023852744036025}{790058929613724612648} a^{5} - \frac{860747040094089871}{2438453486462113002} a^{4} + \frac{2469309102778261327}{9753813945848452008} a^{3} + \frac{3035151995194802927}{7315360459386339006} a^{2} - \frac{79484879251465847}{406408914410352167} a - \frac{153052755215853532}{406408914410352167}$

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:  $13$
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:  \( 8604109121.98 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T90:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 466560
The 72 conjugacy class representatives for [S(3)^5]A(5)=S(3)wrA(5) are not computed
Character table for [S(3)^5]A(5)=S(3)wrA(5) is not computed

Intermediate fields

5.5.8305924.1

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

Sibling fields

Degree 30 siblings: 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 ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ $15$ R $15$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.6.6.7$x^{6} + 2 x^{2} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.9.6.1$x^{9} - 121 x^{3} + 3993$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.6.4.1$x^{6} + 57 x^{3} + 1444$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$131$131.2.0.1$x^{2} - x + 14$$1$$2$$0$$C_2$$[\ ]^{2}$
131.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
131.9.6.1$x^{9} - 17161 x^{3} + 20232819$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$