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Properties

Label 15.9.46113809309...1728.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{10}\cdot 3^{15}\cdot 11^{12}$
Root discriminant $32.43$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T81

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, -240, -720, -1032, -522, 405, 672, 351, 162, 65, -48, -36, -2, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^13 - 2*x^12 - 36*x^11 - 48*x^10 + 65*x^9 + 162*x^8 + 351*x^7 + 672*x^6 + 405*x^5 - 522*x^4 - 1032*x^3 - 720*x^2 - 240*x - 32)
 
gp: K = bnfinit(x^15 - 3*x^13 - 2*x^12 - 36*x^11 - 48*x^10 + 65*x^9 + 162*x^8 + 351*x^7 + 672*x^6 + 405*x^5 - 522*x^4 - 1032*x^3 - 720*x^2 - 240*x - 32, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{13} - 2 x^{12} - 36 x^{11} - 48 x^{10} + 65 x^{9} + 162 x^{8} + 351 x^{7} + 672 x^{6} + 405 x^{5} - 522 x^{4} - 1032 x^{3} - 720 x^{2} - 240 x - 32 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-46113809309420128201728=-\,2^{10}\cdot 3^{15}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.43$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{8} a^{11} - \frac{3}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} + \frac{21}{64} a^{10} + \frac{5}{16} a^{9} + \frac{1}{8} a^{8} + \frac{3}{8} a^{7} + \frac{17}{64} a^{6} - \frac{1}{2} a^{5} - \frac{13}{64} a^{4} - \frac{15}{32} a^{3} + \frac{13}{64} a^{2} - \frac{1}{16} a + \frac{5}{16}$, $\frac{1}{512} a^{13} - \frac{1}{128} a^{12} + \frac{25}{512} a^{11} - \frac{75}{256} a^{10} - \frac{5}{16} a^{9} - \frac{23}{64} a^{8} - \frac{95}{512} a^{7} - \frac{65}{256} a^{6} + \frac{243}{512} a^{5} + \frac{47}{128} a^{4} + \frac{9}{512} a^{3} + \frac{81}{256} a^{2} - \frac{9}{128} a - \frac{21}{64}$, $\frac{1}{4096} a^{14} + \frac{1}{2048} a^{13} + \frac{1}{4096} a^{12} - \frac{9}{1024} a^{10} - \frac{15}{512} a^{9} - \frac{175}{4096} a^{8} - \frac{47}{1024} a^{7} - \frac{25}{4096} a^{6} + \frac{311}{2048} a^{5} + \frac{1649}{4096} a^{4} - \frac{165}{512} a^{3} + \frac{53}{512} a^{2} + \frac{1}{32} a + \frac{1}{256}$

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:  $11$
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:  \( 691102.83729 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T81:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 38880
The 63 conjugacy class representatives for [S(3)^5]5=S(3)wr5 are not computed
Character table for [S(3)^5]5=S(3)wr5 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\)

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 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 ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ $15$ R $15$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ $15$ $15$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.4$x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
3Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$