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Properties

Label 15.15.4989963380...0625.2
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 5^{24}\cdot 7^{4}$
Root discriminant $95.47$
Ramified primes $3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^4:C_5$ (as 15T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12005, 51450, 0, -256025, -320355, 42651, 248430, 68865, -61830, -26825, 5205, 2970, -130, -105, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 105*x^13 - 130*x^12 + 2970*x^11 + 5205*x^10 - 26825*x^9 - 61830*x^8 + 68865*x^7 + 248430*x^6 + 42651*x^5 - 320355*x^4 - 256025*x^3 + 51450*x + 12005)
 
gp: K = bnfinit(x^15 - 105*x^13 - 130*x^12 + 2970*x^11 + 5205*x^10 - 26825*x^9 - 61830*x^8 + 68865*x^7 + 248430*x^6 + 42651*x^5 - 320355*x^4 - 256025*x^3 + 51450*x + 12005, 1)
 

Normalized defining polynomial

\( x^{15} - 105 x^{13} - 130 x^{12} + 2970 x^{11} + 5205 x^{10} - 26825 x^{9} - 61830 x^{8} + 68865 x^{7} + 248430 x^{6} + 42651 x^{5} - 320355 x^{4} - 256025 x^{3} + 51450 x + 12005 \)

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:  \(498996338057577610015869140625=3^{20}\cdot 5^{24}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.47$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{14} a^{10} + \frac{3}{7} a^{9} + \frac{1}{14} a^{8} - \frac{5}{14} a^{7} - \frac{3}{14} a^{5} + \frac{1}{7} a^{4} - \frac{1}{14} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{14} a^{11} - \frac{1}{2} a^{9} + \frac{3}{14} a^{8} + \frac{1}{7} a^{7} - \frac{3}{14} a^{6} + \frac{3}{7} a^{5} + \frac{1}{14} a^{4} - \frac{1}{14} a^{3} - \frac{1}{2} a$, $\frac{1}{14} a^{12} + \frac{3}{14} a^{9} - \frac{5}{14} a^{8} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} - \frac{1}{14} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{490} a^{13} + \frac{1}{35} a^{12} + \frac{1}{70} a^{11} + \frac{17}{490} a^{10} + \frac{17}{49} a^{9} - \frac{213}{490} a^{8} - \frac{116}{245} a^{7} + \frac{99}{490} a^{6} + \frac{69}{490} a^{5} + \frac{3}{14} a^{3} - \frac{2}{7} a^{2}$, $\frac{1}{87199929311813228237502070} a^{14} + \frac{132726156336038562263}{1245713275883046117678601} a^{13} + \frac{61875385594783228750474}{6228566379415230588393005} a^{12} - \frac{1402964786114667350891761}{87199929311813228237502070} a^{11} + \frac{783097926330975734135407}{87199929311813228237502070} a^{10} + \frac{8456641350307397986478126}{43599964655906614118751035} a^{9} - \frac{2860261071807225505475740}{8719992931181322823750207} a^{8} + \frac{10764201475873875936046571}{43599964655906614118751035} a^{7} - \frac{8649509378198959404235977}{87199929311813228237502070} a^{6} - \frac{6088766225031545530125773}{12457132758830461176786010} a^{5} + \frac{270783839351660823595093}{1245713275883046117678601} a^{4} + \frac{611277269838327764093357}{1245713275883046117678601} a^{3} - \frac{3208774020070372776793}{50845439831961066027698} a^{2} - \frac{9521834307950969923095}{25422719915980533013849} a - \frac{546234192211025885876}{25422719915980533013849}$

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:  \( 45364940873.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^4:C_5$ (as 15T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 405
The 21 conjugacy class representatives for $C_3^4:C_5$
Character table for $C_3^4:C_5$ is not computed

Intermediate fields

5.5.390625.1

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 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 R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
3Data not computed
$5$5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$