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Properties

Label 15.15.8639058703...6889.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{16}\cdot 13^{6}\cdot 401^{6}$
Root discriminant $99.03$
Ramified primes $3, 13, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T34

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-112047, -195533, 210912, 433823, -151931, -359463, 57122, 137787, -12480, -25209, 1234, 2163, -31, -79, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 79*x^13 - 31*x^12 + 2163*x^11 + 1234*x^10 - 25209*x^9 - 12480*x^8 + 137787*x^7 + 57122*x^6 - 359463*x^5 - 151931*x^4 + 433823*x^3 + 210912*x^2 - 195533*x - 112047)
 
gp: K = bnfinit(x^15 - 79*x^13 - 31*x^12 + 2163*x^11 + 1234*x^10 - 25209*x^9 - 12480*x^8 + 137787*x^7 + 57122*x^6 - 359463*x^5 - 151931*x^4 + 433823*x^3 + 210912*x^2 - 195533*x - 112047, 1)
 

Normalized defining polynomial

\( x^{15} - 79 x^{13} - 31 x^{12} + 2163 x^{11} + 1234 x^{10} - 25209 x^{9} - 12480 x^{8} + 137787 x^{7} + 57122 x^{6} - 359463 x^{5} - 151931 x^{4} + 433823 x^{3} + 210912 x^{2} - 195533 x - 112047 \)

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:  \(863905870301926479068937836889=3^{16}\cdot 13^{6}\cdot 401^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.03$
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:  $3, 13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{13} a^{9} - \frac{1}{13} a^{7} - \frac{5}{13} a^{6} + \frac{5}{13} a^{5} - \frac{1}{13} a^{4} - \frac{2}{13} a^{3}$, $\frac{1}{13} a^{10} - \frac{1}{13} a^{8} - \frac{5}{13} a^{7} + \frac{5}{13} a^{6} - \frac{1}{13} a^{5} - \frac{2}{13} a^{4}$, $\frac{1}{13} a^{11} - \frac{5}{13} a^{8} + \frac{4}{13} a^{7} - \frac{6}{13} a^{6} + \frac{3}{13} a^{5} - \frac{1}{13} a^{4} - \frac{2}{13} a^{3}$, $\frac{1}{169} a^{12} - \frac{1}{169} a^{10} - \frac{5}{169} a^{9} + \frac{57}{169} a^{8} - \frac{27}{169} a^{7} + \frac{63}{169} a^{6} + \frac{6}{13} a^{5} + \frac{3}{13} a^{4} - \frac{4}{13} a^{3}$, $\frac{1}{169} a^{13} - \frac{1}{169} a^{11} - \frac{5}{169} a^{10} + \frac{5}{169} a^{9} - \frac{27}{169} a^{8} - \frac{54}{169} a^{7} - \frac{4}{13} a^{5} - \frac{5}{13} a^{3}$, $\frac{1}{702522421480626552751543} a^{14} - \frac{1304833648576370971663}{702522421480626552751543} a^{13} + \frac{305000167175310852769}{702522421480626552751543} a^{12} + \frac{12118887517640540701651}{702522421480626552751543} a^{11} + \frac{22997121332003229688117}{702522421480626552751543} a^{10} - \frac{2421918430277360658965}{702522421480626552751543} a^{9} - \frac{199438774420195097974897}{702522421480626552751543} a^{8} + \frac{237583833069004981277351}{702522421480626552751543} a^{7} - \frac{130525770068795930453726}{702522421480626552751543} a^{6} - \frac{8490351889601830806179}{54040186267740504057811} a^{5} - \frac{23454333476784884222067}{54040186267740504057811} a^{4} - \frac{15912030611746333825395}{54040186267740504057811} a^{3} - \frac{1114567090785320922466}{4156937405210808004447} a^{2} - \frac{1268712318994580986335}{4156937405210808004447} a - \frac{101863621311186544949}{244525729718282823791}$

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:  \( 55564346542.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T34:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 810
The 24 conjugacy class representatives for [3^4]D(5)
Character table for [3^4]D(5) is not computed

Intermediate fields

5.5.160801.1

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 30 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.6.8.9$x^{6} + 6 x^{5} + 9$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
3.6.8.9$x^{6} + 6 x^{5} + 9$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
401Data not computed