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Properties

Label 15.15.1973507101...3376.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 3^{16}\cdot 53^{10}$
Root discriminant $104.64$
Ramified primes $2, 3, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T42

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![488, 1712, -1224, -11836, -13992, 3732, 15360, 4716, -5244, -2806, 632, 510, -24, -38, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 38*x^13 - 24*x^12 + 510*x^11 + 632*x^10 - 2806*x^9 - 5244*x^8 + 4716*x^7 + 15360*x^6 + 3732*x^5 - 13992*x^4 - 11836*x^3 - 1224*x^2 + 1712*x + 488)
 
gp: K = bnfinit(x^15 - 38*x^13 - 24*x^12 + 510*x^11 + 632*x^10 - 2806*x^9 - 5244*x^8 + 4716*x^7 + 15360*x^6 + 3732*x^5 - 13992*x^4 - 11836*x^3 - 1224*x^2 + 1712*x + 488, 1)
 

Normalized defining polynomial

\( x^{15} - 38 x^{13} - 24 x^{12} + 510 x^{11} + 632 x^{10} - 2806 x^{9} - 5244 x^{8} + 4716 x^{7} + 15360 x^{6} + 3732 x^{5} - 13992 x^{4} - 11836 x^{3} - 1224 x^{2} + 1712 x + 488 \)

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:  \(1973507101352265840633401573376=2^{18}\cdot 3^{16}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.64$
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:  $2, 3, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{76} a^{13} - \frac{7}{76} a^{12} - \frac{1}{19} a^{11} + \frac{7}{38} a^{10} + \frac{4}{19} a^{9} + \frac{3}{38} a^{8} - \frac{5}{38} a^{7} + \frac{9}{38} a^{6} + \frac{7}{19} a^{5} + \frac{9}{19} a^{4} + \frac{5}{19} a^{3} - \frac{1}{19} a^{2} - \frac{6}{19} a - \frac{2}{19}$, $\frac{1}{4965867892} a^{14} - \frac{3971109}{1241466973} a^{13} + \frac{16828107}{4965867892} a^{12} + \frac{376952391}{2482933946} a^{11} + \frac{102947859}{2482933946} a^{10} - \frac{213240733}{2482933946} a^{9} + \frac{4031379}{65340367} a^{8} - \frac{95069660}{1241466973} a^{7} + \frac{306128903}{2482933946} a^{6} + \frac{15301126}{1241466973} a^{5} + \frac{400176082}{1241466973} a^{4} - \frac{602863691}{1241466973} a^{3} - \frac{327336232}{1241466973} a^{2} + \frac{508507514}{1241466973} a + \frac{1949783}{1241466973}$

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:  \( 80612912136.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T42:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1620
The 21 conjugacy class representatives for 1/2[3^4:2]F(5)
Character table for 1/2[3^4:2]F(5) is not computed

Intermediate fields

5.5.2382032.1

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

Sibling fields

Degree 15 sibling: 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 R R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.16.11$x^{12} + 20 x^{10} - 44 x^{8} - 4 x^{6} - 16 x^{4} - 48$$6$$2$$16$$C_3\times (C_3 : C_4)$$[2]_{3}^{6}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.12.16.16$x^{12} + 21 x^{11} - 72 x^{10} - 117 x^{9} - 27 x^{8} - 45 x^{6} - 81 x^{5} + 81 x^{4} - 108 x^{3} - 81 x^{2} + 81$$3$$4$$16$12T73$[2, 2, 2]^{4}$
53Data not computed