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Properties

Label 15.3.16411954406...3536.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{18}\cdot 11^{10}\cdot 17^{6}$
Root discriminant $35.29$
Ramified primes $2, 11, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\GL(2,4):C_2$ (as 15T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-512, -704, 584, -1523, 1902, -1098, -74, 1255, 228, -204, -124, 75, 14, 6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 6*x^13 + 14*x^12 + 75*x^11 - 124*x^10 - 204*x^9 + 228*x^8 + 1255*x^7 - 74*x^6 - 1098*x^5 + 1902*x^4 - 1523*x^3 + 584*x^2 - 704*x - 512)
 
gp: K = bnfinit(x^15 - 2*x^14 + 6*x^13 + 14*x^12 + 75*x^11 - 124*x^10 - 204*x^9 + 228*x^8 + 1255*x^7 - 74*x^6 - 1098*x^5 + 1902*x^4 - 1523*x^3 + 584*x^2 - 704*x - 512, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} + 6 x^{13} + 14 x^{12} + 75 x^{11} - 124 x^{10} - 204 x^{9} + 228 x^{8} + 1255 x^{7} - 74 x^{6} - 1098 x^{5} + 1902 x^{4} - 1523 x^{3} + 584 x^{2} - 704 x - 512 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(164119544067243368513536=2^{18}\cdot 11^{10}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.29$
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:  $2, 11, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{3}{8} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{88} a^{11} - \frac{1}{88} a^{9} - \frac{1}{22} a^{8} + \frac{5}{44} a^{7} - \frac{2}{11} a^{6} - \frac{1}{44} a^{5} - \frac{1}{11} a^{4} + \frac{3}{8} a^{3} + \frac{1}{22} a^{2} - \frac{9}{88} a - \frac{1}{11}$, $\frac{1}{88} a^{12} - \frac{1}{88} a^{10} - \frac{1}{22} a^{9} + \frac{5}{44} a^{8} + \frac{3}{44} a^{7} - \frac{1}{44} a^{6} + \frac{7}{44} a^{5} - \frac{1}{8} a^{4} + \frac{13}{44} a^{3} + \frac{35}{88} a^{2} + \frac{7}{44} a$, $\frac{1}{176} a^{13} - \frac{1}{176} a^{12} - \frac{1}{176} a^{11} - \frac{3}{176} a^{10} - \frac{1}{22} a^{9} + \frac{9}{88} a^{8} + \frac{7}{88} a^{7} - \frac{3}{88} a^{6} - \frac{3}{176} a^{5} + \frac{15}{176} a^{4} + \frac{75}{176} a^{3} - \frac{87}{176} a^{2} - \frac{7}{88} a$, $\frac{1}{1139164868451798848} a^{14} + \frac{53917773541535}{569582434225899424} a^{13} - \frac{113585249140249}{569582434225899424} a^{12} - \frac{1899074012954681}{569582434225899424} a^{11} + \frac{70781396130672707}{1139164868451798848} a^{10} + \frac{14122373897127657}{284791217112949712} a^{9} + \frac{25661884806280241}{284791217112949712} a^{8} - \frac{9419804581745775}{284791217112949712} a^{7} - \frac{72384140614994729}{1139164868451798848} a^{6} + \frac{7366527227365675}{569582434225899424} a^{5} - \frac{101084492762588001}{569582434225899424} a^{4} + \frac{241285398437749431}{569582434225899424} a^{3} + \frac{473811753370201413}{1139164868451798848} a^{2} + \frac{34524012159084605}{142395608556474856} a + \frac{93804299558689}{17799451069559357}$

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:  $8$
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:  \( 1212431.30034 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_5$ (as 15T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 12 conjugacy class representatives for $\GL(2,4):C_2$
Character table for $\GL(2,4):C_2$

Intermediate fields

3.1.44.1, 5.3.6154544.1

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 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 $15$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.16.18$x^{12} + x^{10} + 6 x^{8} - 3 x^{6} + 6 x^{4} + x^{2} - 3$$6$$2$$16$$C_3 : C_4$$[2]_{3}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$