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Properties

Label 15.11.4611380930...1728.1
Degree $15$
Signature $[11, 2]$
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, 984, 234, -1053, -1304, -522, -54, 13, 120, 90, -12, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 18*x^13 - 12*x^12 + 90*x^11 + 120*x^10 + 13*x^9 - 54*x^8 - 522*x^7 - 1304*x^6 - 1053*x^5 + 234*x^4 + 984*x^3 + 720*x^2 + 240*x + 32)
 
gp: K = bnfinit(x^15 - 18*x^13 - 12*x^12 + 90*x^11 + 120*x^10 + 13*x^9 - 54*x^8 - 522*x^7 - 1304*x^6 - 1053*x^5 + 234*x^4 + 984*x^3 + 720*x^2 + 240*x + 32, 1)
 

Normalized defining polynomial

\( x^{15} - 18 x^{13} - 12 x^{12} + 90 x^{11} + 120 x^{10} + 13 x^{9} - 54 x^{8} - 522 x^{7} - 1304 x^{6} - 1053 x^{5} + 234 x^{4} + 984 x^{3} + 720 x^{2} + 240 x + 32 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} + \frac{3}{32} a^{10} - \frac{3}{8} a^{9} - \frac{7}{32} a^{8} - \frac{7}{16} a^{7} - \frac{19}{64} a^{6} + \frac{1}{4} a^{5} - \frac{3}{32} a^{4} - \frac{1}{16} a^{3} - \frac{13}{64} a^{2} + \frac{1}{16} a - \frac{5}{16}$, $\frac{1}{512} a^{13} - \frac{1}{128} a^{12} + \frac{5}{256} a^{11} - \frac{25}{128} a^{10} + \frac{49}{256} a^{9} + \frac{1}{8} a^{8} - \frac{91}{512} a^{7} + \frac{27}{256} a^{6} + \frac{109}{256} a^{5} + \frac{1}{64} a^{4} - \frac{5}{512} a^{3} - \frac{81}{256} a^{2} + \frac{9}{128} a + \frac{21}{64}$, $\frac{1}{4096} a^{14} + \frac{1}{2048} a^{13} - \frac{7}{2048} a^{12} - \frac{5}{512} a^{11} + \frac{5}{2048} a^{10} + \frac{35}{1024} a^{9} + \frac{293}{4096} a^{8} + \frac{133}{1024} a^{7} + \frac{271}{2048} a^{6} - \frac{55}{1024} a^{5} - \frac{1493}{4096} a^{4} + \frac{21}{64} a^{3} - \frac{53}{512} a^{2} - \frac{1}{32} a - \frac{1}{256}$

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:  $12$
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:  \( 1076539.51981 \) (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 ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R $15$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $15$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ $15$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.12$x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
3Data not computed
11Data not computed