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Properties

Label 15.15.2605357388...8848.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 3^{15}\cdot 17\cdot 61^{3}\cdot 271^{2}\cdot 397^{3}$
Root discriminant $91.42$
Ramified primes $2, 3, 17, 61, 271, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T93

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8672, -65040, -195120, -284616, -171126, 43011, 106104, 28404, -18630, -8871, 1332, 999, -34, -51, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 51*x^13 - 34*x^12 + 999*x^11 + 1332*x^10 - 8871*x^9 - 18630*x^8 + 28404*x^7 + 106104*x^6 + 43011*x^5 - 171126*x^4 - 284616*x^3 - 195120*x^2 - 65040*x - 8672)
 
gp: K = bnfinit(x^15 - 51*x^13 - 34*x^12 + 999*x^11 + 1332*x^10 - 8871*x^9 - 18630*x^8 + 28404*x^7 + 106104*x^6 + 43011*x^5 - 171126*x^4 - 284616*x^3 - 195120*x^2 - 65040*x - 8672, 1)
 

Normalized defining polynomial

\( x^{15} - 51 x^{13} - 34 x^{12} + 999 x^{11} + 1332 x^{10} - 8871 x^{9} - 18630 x^{8} + 28404 x^{7} + 106104 x^{6} + 43011 x^{5} - 171126 x^{4} - 284616 x^{3} - 195120 x^{2} - 65040 x - 8672 \)

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:  \(260535738879584930879430478848=2^{10}\cdot 3^{15}\cdot 17\cdot 61^{3}\cdot 271^{2}\cdot 397^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.42$
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:  $2, 3, 17, 61, 271, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{8} a^{11} - \frac{3}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} - \frac{27}{64} a^{10} + \frac{5}{16} a^{9} + \frac{19}{64} a^{8} - \frac{13}{32} a^{7} - \frac{23}{64} a^{6} + \frac{3}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{3} + \frac{3}{64} a^{2} + \frac{1}{16} a - \frac{5}{16}$, $\frac{1}{512} a^{13} - \frac{1}{128} a^{12} - \frac{23}{512} a^{11} + \frac{5}{256} a^{10} + \frac{171}{512} a^{9} - \frac{1}{2} a^{8} - \frac{163}{512} a^{7} - \frac{29}{256} a^{6} + \frac{7}{64} a^{5} + \frac{7}{16} a^{4} - \frac{221}{512} a^{3} - \frac{65}{256} a^{2} - \frac{7}{128} a + \frac{5}{64}$, $\frac{1}{4096} a^{14} + \frac{1}{2048} a^{13} + \frac{17}{4096} a^{12} - \frac{1}{16} a^{11} - \frac{985}{4096} a^{10} + \frac{1}{2048} a^{9} - \frac{2019}{4096} a^{8} + \frac{477}{1024} a^{7} + \frac{327}{1024} a^{6} - \frac{93}{256} a^{5} - \frac{1437}{4096} a^{4} + \frac{5}{256} a^{3} + \frac{51}{512} a^{2} - \frac{15}{32} a - \frac{65}{256}$

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:  \( 9336261754.24 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T93:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 933120
The 108 conjugacy class representatives for [S(3)^5]S(5)=S(3)wrS(5) are not computed
Character table for [S(3)^5]S(5)=S(3)wrS(5) is not computed

Intermediate fields

5.5.24217.1

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
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 ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ R ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.14$x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
3Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
61Data not computed
271Data not computed
397Data not computed