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Properties

Label 15.15.1276678782...1696.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 887^{6}$
Root discriminant $34.71$
Ramified primes $2, 887$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_5$ (as 15T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -15, -80, -157, 78, 710, 700, -456, -1041, -255, 358, 183, -27, -25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 25*x^13 - 27*x^12 + 183*x^11 + 358*x^10 - 255*x^9 - 1041*x^8 - 456*x^7 + 700*x^6 + 710*x^5 + 78*x^4 - 157*x^3 - 80*x^2 - 15*x - 1)
 
gp: K = bnfinit(x^15 - 25*x^13 - 27*x^12 + 183*x^11 + 358*x^10 - 255*x^9 - 1041*x^8 - 456*x^7 + 700*x^6 + 710*x^5 + 78*x^4 - 157*x^3 - 80*x^2 - 15*x - 1, 1)
 

Normalized defining polynomial

\( x^{15} - 25 x^{13} - 27 x^{12} + 183 x^{11} + 358 x^{10} - 255 x^{9} - 1041 x^{8} - 456 x^{7} + 700 x^{6} + 710 x^{5} + 78 x^{4} - 157 x^{3} - 80 x^{2} - 15 x - 1 \)

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:  \(127667878299171450781696=2^{18}\cdot 887^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.71$
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:  $2, 887$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{57} a^{13} - \frac{16}{57} a^{12} - \frac{4}{57} a^{11} - \frac{22}{57} a^{10} - \frac{7}{57} a^{9} - \frac{1}{19} a^{8} + \frac{13}{57} a^{7} + \frac{26}{57} a^{6} + \frac{2}{19} a^{5} + \frac{23}{57} a^{4} + \frac{5}{19} a^{3} + \frac{1}{3} a^{2} + \frac{4}{57} a + \frac{8}{57}$, $\frac{1}{3249} a^{14} - \frac{8}{3249} a^{13} + \frac{13}{1083} a^{12} - \frac{113}{1083} a^{11} - \frac{118}{1083} a^{10} - \frac{59}{3249} a^{9} + \frac{217}{3249} a^{8} + \frac{472}{3249} a^{7} - \frac{983}{3249} a^{6} - \frac{1183}{3249} a^{5} + \frac{427}{3249} a^{4} - \frac{89}{3249} a^{3} + \frac{185}{1083} a^{2} - \frac{1271}{3249} a + \frac{406}{3249}$

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:  \( 5340385.5843 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_5$ (as 15T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 60
The 5 conjugacy class representatives for $A_5$
Character table for $A_5$

Intermediate fields

5.5.50353216.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 5 sibling: 5.5.50353216.1
Degree 6 sibling: 6.6.50353216.1
Degree 10 sibling: 10.10.2535446361542656.1
Degree 12 sibling: Deg 12
Degree 20 sibling: Deg 20
Degree 30 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.18.59$x^{12} - 2 x^{11} + 6 x^{10} + 4 x^{9} + 6 x^{8} + 12 x^{7} - 4 x^{6} - 8 x^{3} + 16 x^{2} - 8$$4$$3$$18$$A_4$$[2, 2]^{3}$
887Data not computed