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Properties

Label 15.15.1087405383...5856.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{14}\cdot 3^{12}\cdot 7^{6}\cdot 101^{6}$
Root discriminant $63.45$
Ramified primes $2, 3, 7, 101$
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![-9, -81, 81, 1371, 1163, -4501, -5407, 3991, 4957, -1787, -1521, 381, 165, -35, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 35*x^13 + 165*x^12 + 381*x^11 - 1521*x^10 - 1787*x^9 + 4957*x^8 + 3991*x^7 - 5407*x^6 - 4501*x^5 + 1163*x^4 + 1371*x^3 + 81*x^2 - 81*x - 9)
 
gp: K = bnfinit(x^15 - 5*x^14 - 35*x^13 + 165*x^12 + 381*x^11 - 1521*x^10 - 1787*x^9 + 4957*x^8 + 3991*x^7 - 5407*x^6 - 4501*x^5 + 1163*x^4 + 1371*x^3 + 81*x^2 - 81*x - 9, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 35 x^{13} + 165 x^{12} + 381 x^{11} - 1521 x^{10} - 1787 x^{9} + 4957 x^{8} + 3991 x^{7} - 5407 x^{6} - 4501 x^{5} + 1163 x^{4} + 1371 x^{3} + 81 x^{2} - 81 x - 9 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1087405383368698039133945856=2^{14}\cdot 3^{12}\cdot 7^{6}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.45$
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:  $2, 3, 7, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{14} a^{12} + \frac{3}{14} a^{11} - \frac{1}{7} a^{10} + \frac{3}{14} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{5} - \frac{5}{14} a^{4} - \frac{1}{2} a^{3} + \frac{3}{7} a + \frac{5}{14}$, $\frac{1}{42} a^{13} + \frac{1}{42} a^{12} - \frac{4}{21} a^{11} - \frac{1}{14} a^{10} + \frac{1}{14} a^{9} - \frac{1}{14} a^{8} - \frac{10}{21} a^{7} + \frac{2}{21} a^{6} + \frac{1}{42} a^{5} + \frac{17}{42} a^{4} + \frac{1}{3} a^{3} - \frac{1}{42} a^{2} - \frac{1}{2} a - \frac{1}{14}$, $\frac{1}{3889128400965402} a^{14} + \frac{1498523198363}{1944564200482701} a^{13} + \frac{14849249279975}{1944564200482701} a^{12} - \frac{153080889520171}{648188066827567} a^{11} + \frac{154083877912077}{648188066827567} a^{10} - \frac{7141257034848}{648188066827567} a^{9} + \frac{210462435679973}{1944564200482701} a^{8} + \frac{216353994850376}{1944564200482701} a^{7} - \frac{262103780309309}{3889128400965402} a^{6} - \frac{30022130733653}{1944564200482701} a^{5} + \frac{648134270825092}{1944564200482701} a^{4} - \frac{660225296012741}{1944564200482701} a^{3} - \frac{219362271028849}{648188066827567} a^{2} - \frac{260443041650837}{648188066827567} a + \frac{285592397866871}{648188066827567}$

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:  $14$
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:  \( 2319150768.71 \) (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

3.3.404.1, 5.5.6413904.1

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 $15$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ $15$ $15$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2Data not computed
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
101Data not computed