↑

Properties

Label 15.5.982108001708984375.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,5^{16}\cdot 23^{5}$
Root discriminant $15.83$
Ramified primes $5, 23$
Class number $1$
Class group Trivial
Galois group 15T32

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43, 134, 154, 36, -120, -78, -7, -32, 7, 5, 30, 9, -11, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^13 - 11*x^12 + 9*x^11 + 30*x^10 + 5*x^9 + 7*x^8 - 32*x^7 - 7*x^6 - 78*x^5 - 120*x^4 + 36*x^3 + 154*x^2 + 134*x + 43)
 
gp: K = bnfinit(x^15 - 4*x^13 - 11*x^12 + 9*x^11 + 30*x^10 + 5*x^9 + 7*x^8 - 32*x^7 - 7*x^6 - 78*x^5 - 120*x^4 + 36*x^3 + 154*x^2 + 134*x + 43, 1)
 

Normalized defining polynomial

\( x^{15} - 4 x^{13} - 11 x^{12} + 9 x^{11} + 30 x^{10} + 5 x^{9} + 7 x^{8} - 32 x^{7} - 7 x^{6} - 78 x^{5} - 120 x^{4} + 36 x^{3} + 154 x^{2} + 134 x + 43 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-982108001708984375=-\,5^{16}\cdot 23^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.83$
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:  $5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{24412247885} a^{14} + \frac{1646829922}{24412247885} a^{13} - \frac{1315688103}{24412247885} a^{12} - \frac{356133432}{4882449577} a^{11} - \frac{435577617}{24412247885} a^{10} - \frac{369787707}{4882449577} a^{9} + \frac{6165042833}{24412247885} a^{8} - \frac{8742388874}{24412247885} a^{7} - \frac{5629932023}{24412247885} a^{6} - \frac{10156258087}{24412247885} a^{5} - \frac{1463424901}{24412247885} a^{4} + \frac{2623546591}{24412247885} a^{3} - \frac{4022754281}{24412247885} a^{2} - \frac{7981696253}{24412247885} a - \frac{23277162}{113545339}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1326.38565727 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T32:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 750
The 65 conjugacy class representatives for [5^3]S(3) are not computed
Character table for [5^3]S(3) is not computed

Intermediate fields

3.1.23.1

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 30 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 $15$ $15$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{5}$ R $15$ $15$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{5}$ $15$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ $15$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$5$5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
5.10.16.26$x^{10} + 30 x^{9} + 25 x^{8} + 10 x^{5} + 275 x^{4} + 25$$5$$2$$16$$D_5\times C_5$$[2, 2]^{2}$
$23$23.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
23.10.5.2$x^{10} - 279841 x^{2} + 12872686$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$