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Properties

Label 15.7.43296861695...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{14}\cdot 5^{16}\cdot 7^{10}\cdot 19^{10}$
Root discriminant $276.97$
Ramified primes $2, 5, 7, 19$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T98

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 15360, 19200, 8000, 0, -704, -1760, -1100, 0, 0, -4, -5, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^11 - 4*x^10 - 1100*x^7 - 1760*x^6 - 704*x^5 + 8000*x^3 + 19200*x^2 + 15360*x + 4096)
 
gp: K = bnfinit(x^15 - 5*x^11 - 4*x^10 - 1100*x^7 - 1760*x^6 - 704*x^5 + 8000*x^3 + 19200*x^2 + 15360*x + 4096, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{11} - 4 x^{10} - 1100 x^{7} - 1760 x^{6} - 704 x^{5} + 8000 x^{3} + 19200 x^{2} + 15360 x + 4096 \)

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:  \(4329686169519589168622500000000000000=2^{14}\cdot 5^{16}\cdot 7^{10}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $276.97$
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:  $2, 5, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{10} + \frac{3}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{11} + \frac{3}{32} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{128} a^{12} - \frac{1}{64} a^{11} - \frac{1}{32} a^{10} - \frac{1}{16} a^{9} + \frac{11}{128} a^{8} + \frac{3}{64} a^{7} + \frac{7}{32} a^{6} - \frac{1}{16} a^{5} + \frac{1}{32} a^{4} - \frac{1}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{4096} a^{13} - \frac{1}{512} a^{12} - \frac{1}{256} a^{11} - \frac{1}{32} a^{10} - \frac{5}{4096} a^{9} - \frac{119}{1024} a^{8} + \frac{7}{256} a^{7} + \frac{11}{64} a^{6} - \frac{147}{1024} a^{5} - \frac{5}{32} a^{4} - \frac{7}{16} a^{3} + \frac{1}{8} a^{2} + \frac{13}{64} a - \frac{7}{16}$, $\frac{1}{1048576} a^{14} - \frac{13}{262144} a^{13} + \frac{169}{65536} a^{12} - \frac{149}{16384} a^{11} - \frac{28421}{1048576} a^{10} + \frac{141}{4096} a^{9} - \frac{41}{1024} a^{8} - \frac{11}{256} a^{7} + \frac{61165}{262144} a^{6} + \frac{15753}{65536} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{8067}{16384} a^{2} + \frac{249}{2048} a - \frac{315}{1024}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 190235235181000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T98:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 71 conjugacy class representatives for [1/2.S(5)^3]3 are not computed
Character table for [1/2.S(5)^3]3 is not computed

Intermediate fields

3.3.17689.2

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 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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R R $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ $15$ R ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $15$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
$5$5.5.6.4$x^{5} + 20 x^{2} + 5$$5$$1$$6$$F_5$$[3/2]_{2}^{2}$
5.5.5.3$x^{5} + 15 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
7Data not computed
$19$19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$