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Properties

Label 15.7.14471025819...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{18}\cdot 3^{21}\cdot 5^{6}\cdot 179^{4}\cdot 3671^{2}\cdot 49409^{2}$
Root discriminant $1024.94$
Ramified primes $2, 3, 5, 179, 3671, 49409$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T95

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-733184000, -3849216000, -8715724800, -11105446400, -8710225920, -4316787840, -1326742656, -228238176, -10539072, 4133160, 866592, 46872, -3888, -486, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 486*x^13 - 3888*x^12 + 46872*x^11 + 866592*x^10 + 4133160*x^9 - 10539072*x^8 - 228238176*x^7 - 1326742656*x^6 - 4316787840*x^5 - 8710225920*x^4 - 11105446400*x^3 - 8715724800*x^2 - 3849216000*x - 733184000)
 
gp: K = bnfinit(x^15 - 486*x^13 - 3888*x^12 + 46872*x^11 + 866592*x^10 + 4133160*x^9 - 10539072*x^8 - 228238176*x^7 - 1326742656*x^6 - 4316787840*x^5 - 8710225920*x^4 - 11105446400*x^3 - 8715724800*x^2 - 3849216000*x - 733184000, 1)
 

Normalized defining polynomial

\( x^{15} - 486 x^{13} - 3888 x^{12} + 46872 x^{11} + 866592 x^{10} + 4133160 x^{9} - 10539072 x^{8} - 228238176 x^{7} - 1326742656 x^{6} - 4316787840 x^{5} - 8710225920 x^{4} - 11105446400 x^{3} - 8715724800 x^{2} - 3849216000 x - 733184000 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1447102581933483211395771120066876223488000000=2^{18}\cdot 3^{21}\cdot 5^{6}\cdot 179^{4}\cdot 3671^{2}\cdot 49409^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1024.94$
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:  $2, 3, 5, 179, 3671, 49409$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{7} - \frac{1}{40} a^{5} + \frac{1}{20} a^{4} - \frac{1}{2} a$, $\frac{1}{800} a^{8} + \frac{3}{400} a^{6} + \frac{1}{100} a^{5} + \frac{2}{25} a^{4} - \frac{1}{50} a^{3} + \frac{7}{100} a^{2} - \frac{1}{5}$, $\frac{1}{3200} a^{9} - \frac{7}{1600} a^{7} - \frac{1}{100} a^{6} + \frac{3}{400} a^{5} + \frac{3}{25} a^{4} + \frac{57}{400} a^{3} - \frac{1}{10} a^{2} - \frac{1}{20} a$, $\frac{1}{6400} a^{10} + \frac{1}{3200} a^{8} - \frac{1}{200} a^{7} - \frac{1}{160} a^{6} - \frac{1}{50} a^{5} + \frac{1}{32} a^{4} + \frac{1}{100} a^{3} + \frac{43}{200} a^{2} - \frac{2}{5}$, $\frac{1}{128000} a^{11} + \frac{7}{64000} a^{9} - \frac{3}{8000} a^{8} + \frac{7}{8000} a^{7} - \frac{9}{4000} a^{6} - \frac{7}{3200} a^{5} - \frac{183}{2000} a^{4} - \frac{369}{2000} a^{3} + \frac{213}{1000} a^{2} - \frac{47}{200} a + \frac{1}{50}$, $\frac{1}{1280000} a^{12} - \frac{33}{640000} a^{10} + \frac{7}{80000} a^{9} - \frac{3}{80000} a^{8} + \frac{201}{40000} a^{7} - \frac{11}{6400} a^{6} + \frac{237}{20000} a^{5} - \frac{1259}{20000} a^{4} + \frac{1843}{10000} a^{3} - \frac{139}{2000} a^{2} + \frac{141}{500} a - \frac{6}{25}$, $\frac{1}{5120000000} a^{13} - \frac{31}{128000000} a^{12} + \frac{5637}{2560000000} a^{11} - \frac{10689}{160000000} a^{10} + \frac{33279}{640000000} a^{9} + \frac{14351}{160000000} a^{8} - \frac{147747}{25600000} a^{7} - \frac{31873}{5000000} a^{6} - \frac{746303}{160000000} a^{5} - \frac{3024687}{40000000} a^{4} + \frac{95871}{8000000} a^{3} - \frac{20919}{1000000} a^{2} + \frac{57879}{400000} a - \frac{18023}{100000}$, $\frac{1}{655360000000} a^{14} - \frac{11}{163840000000} a^{13} + \frac{60117}{327680000000} a^{12} - \frac{257293}{81920000000} a^{11} + \frac{3363103}{81920000000} a^{10} + \frac{521693}{5120000000} a^{9} - \frac{38190491}{81920000000} a^{8} - \frac{80236761}{20480000000} a^{7} + \frac{180360241}{20480000000} a^{6} - \frac{27466821}{1280000000} a^{5} + \frac{11789703}{5120000000} a^{4} + \frac{38106391}{256000000} a^{3} - \frac{17470853}{256000000} a^{2} - \frac{1589301}{6400000} a - \frac{326877}{3200000}$

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:  $10$
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:  \( 134777032173000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T95:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1296000
The 53 conjugacy class representatives for [A(5)^3:2]3 are not computed
Character table for [A(5)^3:2]3 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\)

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: 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 R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ $15$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.6$x^{6} + 4 x^{2} + 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.4$x^{6} + 4 x^{2} + 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$3$3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
179Data not computed
3671Data not computed
49409Data not computed