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Properties

Label 15.7.47176144634...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{10}\cdot 3^{4}\cdot 5^{15}\cdot 239^{6}$
Root discriminant $95.11$
Ramified primes $2, 3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T85

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-17056, -26160, 12240, 14120, -8970, -5643, 2440, 2205, -540, -470, 120, 90, -10, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 15*x^13 - 10*x^12 + 90*x^11 + 120*x^10 - 470*x^9 - 540*x^8 + 2205*x^7 + 2440*x^6 - 5643*x^5 - 8970*x^4 + 14120*x^3 + 12240*x^2 - 26160*x - 17056)
 
gp: K = bnfinit(x^15 - 15*x^13 - 10*x^12 + 90*x^11 + 120*x^10 - 470*x^9 - 540*x^8 + 2205*x^7 + 2440*x^6 - 5643*x^5 - 8970*x^4 + 14120*x^3 + 12240*x^2 - 26160*x - 17056, 1)
 

Normalized defining polynomial

\( x^{15} - 15 x^{13} - 10 x^{12} + 90 x^{11} + 120 x^{10} - 470 x^{9} - 540 x^{8} + 2205 x^{7} + 2440 x^{6} - 5643 x^{5} - 8970 x^{4} + 14120 x^{3} + 12240 x^{2} - 26160 x - 17056 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(471761446343357531250000000000=2^{10}\cdot 3^{4}\cdot 5^{15}\cdot 239^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.11$
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:  $2, 3, 5, 239$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{6} - \frac{1}{8} a^{4} - \frac{3}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{64} a^{8} - \frac{1}{64} a^{7} - \frac{1}{32} a^{5} - \frac{3}{64} a^{4} - \frac{1}{64} a^{3} + \frac{5}{32} a^{2} - \frac{3}{16} a - \frac{3}{8}$, $\frac{1}{384} a^{9} + \frac{1}{192} a^{8} + \frac{5}{384} a^{7} - \frac{5}{192} a^{6} + \frac{23}{384} a^{5} + \frac{11}{192} a^{4} - \frac{1}{384} a^{3} - \frac{19}{192} a^{2} - \frac{13}{32} a + \frac{19}{48}$, $\frac{1}{384} a^{10} + \frac{1}{384} a^{8} + \frac{1}{96} a^{7} - \frac{5}{384} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} + \frac{3}{32} a^{3} - \frac{5}{24} a^{2} - \frac{1}{24} a + \frac{5}{24}$, $\frac{1}{768} a^{11} - \frac{1}{768} a^{10} - \frac{1}{768} a^{9} - \frac{1}{768} a^{8} - \frac{19}{768} a^{7} + \frac{1}{768} a^{6} - \frac{19}{768} a^{5} - \frac{11}{768} a^{4} + \frac{13}{128} a^{3} + \frac{35}{192} a^{2} - \frac{7}{32} a$, $\frac{1}{1536} a^{12} + \frac{5}{768} a^{8} - \frac{1}{64} a^{7} + \frac{1}{96} a^{5} - \frac{49}{512} a^{4} + \frac{7}{64} a^{3} - \frac{3}{64} a^{2} - \frac{1}{48} a - \frac{43}{96}$, $\frac{1}{1536} a^{13} - \frac{1}{768} a^{9} - \frac{1}{128} a^{7} + \frac{5}{192} a^{6} + \frac{19}{512} a^{5} - \frac{1}{32} a^{4} - \frac{1}{128} a^{3} - \frac{43}{192} a^{2} - \frac{5}{48} a - \frac{3}{16}$, $\frac{1}{9216} a^{14} + \frac{1}{9216} a^{13} - \frac{1}{4608} a^{12} - \frac{1}{1536} a^{10} - \frac{1}{1536} a^{9} + \frac{1}{2304} a^{8} - \frac{1}{1152} a^{7} - \frac{131}{9216} a^{6} - \frac{307}{9216} a^{5} + \frac{187}{4608} a^{4} + \frac{29}{576} a^{3} - \frac{41}{576} a^{2} + \frac{7}{576} a - \frac{139}{288}$

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:  $10$
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:  \( 13673892773.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T85:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 77760
The 45 conjugacy class representatives for [1/2.S(3)^5]F(5)
Character table for [1/2.S(3)^5]F(5) is not computed

Intermediate fields

5.5.178503125.1

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

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $15$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.10$x^{8} + 2 x^{6} + 8 x^{3} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5Data not computed
239Data not computed