Properties

Label 15.3.1788527348449625.1
Degree $15$
Signature $[3, 6]$
Discriminant $5^{3}\cdot 7^{10}\cdot 37^{3}$
Root discriminant $10.40$
Ramified primes $5, 7, 37$
Class number $1$
Class group Trivial
Galois Group $S_5 \times C_3$ (as 15T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 6, -16, 30, -44, 48, -44, 33, -23, 18, -15, 13, -9, 4, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 4*x^13 - 9*x^12 + 13*x^11 - 15*x^10 + 18*x^9 - 23*x^8 + 33*x^7 - 44*x^6 + 48*x^5 - 44*x^4 + 30*x^3 - 16*x^2 + 6*x - 1)
gp: K = bnfinit(x^15 - 2*x^14 + 4*x^13 - 9*x^12 + 13*x^11 - 15*x^10 + 18*x^9 - 23*x^8 + 33*x^7 - 44*x^6 + 48*x^5 - 44*x^4 + 30*x^3 - 16*x^2 + 6*x - 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut +\mathstrut 4 x^{13} \) \(\mathstrut -\mathstrut 9 x^{12} \) \(\mathstrut +\mathstrut 13 x^{11} \) \(\mathstrut -\mathstrut 15 x^{10} \) \(\mathstrut +\mathstrut 18 x^{9} \) \(\mathstrut -\mathstrut 23 x^{8} \) \(\mathstrut +\mathstrut 33 x^{7} \) \(\mathstrut -\mathstrut 44 x^{6} \) \(\mathstrut +\mathstrut 48 x^{5} \) \(\mathstrut -\mathstrut 44 x^{4} \) \(\mathstrut +\mathstrut 30 x^{3} \) \(\mathstrut -\mathstrut 16 x^{2} \) \(\mathstrut +\mathstrut 6 x \) \(\mathstrut -\mathstrut 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[3, 6]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(1788527348449625=5^{3}\cdot 7^{10}\cdot 37^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.40$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 7, 37$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7561} a^{14} - \frac{3367}{7561} a^{13} + \frac{3581}{7561} a^{12} + \frac{2160}{7561} a^{11} - \frac{2266}{7561} a^{10} + \frac{3587}{7561} a^{9} - \frac{2881}{7561} a^{8} + \frac{1340}{7561} a^{7} - \frac{2711}{7561} a^{6} - \frac{3656}{7561} a^{5} + \frac{741}{7561} a^{4} + \frac{1621}{7561} a^{3} - \frac{3154}{7561} a^{2} - \frac{2450}{7561} a + \frac{2766}{7561}$

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:  $8$
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:  \( 26.5068203244 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_3\times S_5$ (as 15T24):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 360
The 21 conjugacy class representatives for $S_5 \times C_3$
Character table for $S_5 \times C_3$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 5.1.9065.1

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 sibling: 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 $15$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ R ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$37$37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$