↑

Properties

Label 15.15.2377542761...5216.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{6}\cdot 7^{10}\cdot 331^{6}$
Root discriminant $49.17$
Ramified primes $2, 7, 331$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\GL(2,4)$ (as 15T16)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, 267, 387, -2933, 349, 7386, -4815, -5065, 5132, 469, -1487, 176, 152, -28, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 28*x^13 + 152*x^12 + 176*x^11 - 1487*x^10 + 469*x^9 + 5132*x^8 - 5065*x^7 - 4815*x^6 + 7386*x^5 + 349*x^4 - 2933*x^3 + 387*x^2 + 267*x + 13)
 
gp: K = bnfinit(x^15 - 5*x^14 - 28*x^13 + 152*x^12 + 176*x^11 - 1487*x^10 + 469*x^9 + 5132*x^8 - 5065*x^7 - 4815*x^6 + 7386*x^5 + 349*x^4 - 2933*x^3 + 387*x^2 + 267*x + 13, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 28 x^{13} + 152 x^{12} + 176 x^{11} - 1487 x^{10} + 469 x^{9} + 5132 x^{8} - 5065 x^{7} - 4815 x^{6} + 7386 x^{5} + 349 x^{4} - 2933 x^{3} + 387 x^{2} + 267 x + 13 \)

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:  \(23775427618300208865265216=2^{6}\cdot 7^{10}\cdot 331^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.17$
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:  $2, 7, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{14} a^{12} - \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{14} a^{9} - \frac{5}{14} a^{8} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{3}{14} a^{5} + \frac{1}{7} a^{4} + \frac{5}{14} a^{3} + \frac{3}{7} a^{2} - \frac{1}{14} a - \frac{1}{14}$, $\frac{1}{182} a^{13} + \frac{2}{91} a^{12} - \frac{3}{91} a^{11} + \frac{55}{182} a^{10} - \frac{25}{182} a^{9} + \frac{20}{91} a^{8} - \frac{41}{91} a^{7} - \frac{53}{182} a^{6} - \frac{3}{7} a^{5} + \frac{5}{26} a^{4} + \frac{9}{91} a^{3} + \frac{47}{182} a^{2} + \frac{33}{182} a - \frac{3}{7}$, $\frac{1}{511282658031782} a^{14} - \frac{330318511178}{255641329015891} a^{13} + \frac{11701827441637}{511282658031782} a^{12} - \frac{12416363331637}{511282658031782} a^{11} - \frac{104451180125183}{511282658031782} a^{10} + \frac{84657499644313}{511282658031782} a^{9} + \frac{135781637501399}{511282658031782} a^{8} - \frac{119592942725895}{511282658031782} a^{7} + \frac{23127092883942}{255641329015891} a^{6} - \frac{116557021413666}{255641329015891} a^{5} - \frac{49168614775863}{255641329015891} a^{4} - \frac{117063964197814}{255641329015891} a^{3} + \frac{101465362946047}{511282658031782} a^{2} + \frac{185078603323645}{511282658031782} a + \frac{13078112742145}{39329435233214}$

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:  \( 105110387.126 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times A_5$ (as 15T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 180
The 15 conjugacy class representatives for $\GL(2,4)$
Character table for $\GL(2,4)$

Intermediate fields

\(\Q(\zeta_{7})^+\), 5.5.21473956.1

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 sibling: 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 $15$ $15$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ $15$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
331Data not computed