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Properties

Label 15.15.3003305448...2352.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{22}\cdot 3^{16}\cdot 53^{9}\cdot 71^{2}$
Root discriminant $170.55$
Ramified primes $2, 3, 53, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T85

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18176, -136320, -408960, -599088, -373968, 55728, 189464, 53361, -28944, -13868, 1812, 1359, -40, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 60*x^13 - 40*x^12 + 1359*x^11 + 1812*x^10 - 13868*x^9 - 28944*x^8 + 53361*x^7 + 189464*x^6 + 55728*x^5 - 373968*x^4 - 599088*x^3 - 408960*x^2 - 136320*x - 18176)
 
gp: K = bnfinit(x^15 - 60*x^13 - 40*x^12 + 1359*x^11 + 1812*x^10 - 13868*x^9 - 28944*x^8 + 53361*x^7 + 189464*x^6 + 55728*x^5 - 373968*x^4 - 599088*x^3 - 408960*x^2 - 136320*x - 18176, 1)
 

Normalized defining polynomial

\( x^{15} - 60 x^{13} - 40 x^{12} + 1359 x^{11} + 1812 x^{10} - 13868 x^{9} - 28944 x^{8} + 53361 x^{7} + 189464 x^{6} + 55728 x^{5} - 373968 x^{4} - 599088 x^{3} - 408960 x^{2} - 136320 x - 18176 \)

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:  \(3003305448427704785700521458532352=2^{22}\cdot 3^{16}\cdot 53^{9}\cdot 71^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $170.55$
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:  $2, 3, 53, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} - \frac{1}{16} a^{10} - \frac{17}{64} a^{8} + \frac{11}{32} a^{7} - \frac{3}{16} a^{6} - \frac{1}{8} a^{5} + \frac{1}{64} a^{4} + \frac{11}{32} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{512} a^{13} - \frac{1}{128} a^{12} - \frac{1}{16} a^{11} + \frac{5}{64} a^{10} + \frac{47}{512} a^{9} + \frac{7}{64} a^{8} - \frac{27}{64} a^{7} + \frac{15}{32} a^{6} - \frac{111}{512} a^{5} - \frac{59}{128} a^{4} + \frac{11}{128} a^{3} - \frac{9}{32} a^{2} - \frac{7}{16} a - \frac{3}{8}$, $\frac{1}{20480} a^{14} - \frac{3}{10240} a^{13} - \frac{3}{2560} a^{12} + \frac{13}{2560} a^{11} + \frac{147}{4096} a^{10} - \frac{1299}{10240} a^{9} + \frac{43}{512} a^{8} + \frac{53}{640} a^{7} + \frac{445}{4096} a^{6} - \frac{4103}{10240} a^{5} + \frac{641}{5120} a^{4} - \frac{29}{2560} a^{3} - \frac{59}{320} a^{2} + \frac{11}{80} a - \frac{77}{160}$

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:  \( 4578084145130 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T85:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 77760
The 45 conjugacy class representatives for [1/2.S(3)^5]F(5)
Character table for [1/2.S(3)^5]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 30 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 ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ R $15$

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$$[\ ]_{3}^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.16.36$x^{8} + 4 x^{6} + 20 x^{4} + 208$$4$$2$$16$$C_8:C_2$$[2, 3, 3]^{2}$
$3$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.12.12.1$x^{12} + 27 x^{6} + 81 x^{5} - 54 x^{3} - 81 x - 81$$3$$4$$12$12T173$[3/2, 3/2, 3/2, 3/2]_{2}^{4}$
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
71Data not computed