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Properties

Label 15.1.957206371560558183.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,3^{12}\cdot 23^{9}$
Root discriminant $15.80$
Ramified primes $3, 23$
Class number $1$
Class group Trivial
Galois group $A_5 \times S_3$ (as 15T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -5, 10, 4, -4, 0, 0, 23, 18, 5, -4, -6, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + x^13 - 6*x^12 - 4*x^11 + 5*x^10 + 18*x^9 + 23*x^8 - 4*x^5 + 4*x^4 + 10*x^3 - 5*x^2 + 1)
 
gp: K = bnfinit(x^15 - x^14 + x^13 - 6*x^12 - 4*x^11 + 5*x^10 + 18*x^9 + 23*x^8 - 4*x^5 + 4*x^4 + 10*x^3 - 5*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} + x^{13} - 6 x^{12} - 4 x^{11} + 5 x^{10} + 18 x^{9} + 23 x^{8} - 4 x^{5} + 4 x^{4} + 10 x^{3} - 5 x^{2} + 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-957206371560558183=-\,3^{12}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.80$
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:  $3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23} a^{12} + \frac{10}{23} a^{11} - \frac{6}{23} a^{10} + \frac{5}{23} a^{9} - \frac{2}{23} a^{8} - \frac{11}{23} a^{7} - \frac{5}{23} a^{6} + \frac{3}{23} a^{5} + \frac{11}{23} a^{4} - \frac{2}{23} a^{3} - \frac{10}{23} a^{2} - \frac{1}{23} a + \frac{9}{23}$, $\frac{1}{23} a^{13} + \frac{9}{23} a^{11} - \frac{4}{23} a^{10} - \frac{6}{23} a^{9} + \frac{9}{23} a^{8} - \frac{10}{23} a^{7} + \frac{7}{23} a^{6} + \frac{4}{23} a^{5} + \frac{3}{23} a^{4} + \frac{10}{23} a^{3} + \frac{7}{23} a^{2} - \frac{4}{23} a + \frac{2}{23}$, $\frac{1}{89263} a^{14} - \frac{61}{3881} a^{13} - \frac{660}{89263} a^{12} - \frac{25531}{89263} a^{11} + \frac{7757}{89263} a^{10} + \frac{22539}{89263} a^{9} - \frac{558}{89263} a^{8} - \frac{28790}{89263} a^{7} - \frac{18225}{89263} a^{6} + \frac{18351}{89263} a^{5} - \frac{32005}{89263} a^{4} + \frac{33821}{89263} a^{3} - \frac{30022}{89263} a^{2} - \frac{21892}{89263} a - \frac{29412}{89263}$

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:  $7$
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:  \( 627.503289611 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times A_5$ (as 15T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 15 conjugacy class representatives for $A_5 \times S_3$
Character table for $A_5 \times S_3$

Intermediate fields

3.1.23.1, 5.1.42849.1

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 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 $15$ R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.9.9.1$x^{9} + 54 x^{5} + 27 x^{3} + 189$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.6.5.2$x^{6} + 46$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$