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Properties

Label 15.1.58245212806...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{15}\cdot 5^{17}\cdot 13^{12}$
Root discriminant $96.46$
Ramified primes $2, 5, 13$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2216, -15635, 33910, -24230, 15095, -6267, 6930, -2380, 40, -160, 18, 65, -40, 20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 20*x^13 - 40*x^12 + 65*x^11 + 18*x^10 - 160*x^9 + 40*x^8 - 2380*x^7 + 6930*x^6 - 6267*x^5 + 15095*x^4 - 24230*x^3 + 33910*x^2 - 15635*x + 2216)
 
gp: K = bnfinit(x^15 - 5*x^14 + 20*x^13 - 40*x^12 + 65*x^11 + 18*x^10 - 160*x^9 + 40*x^8 - 2380*x^7 + 6930*x^6 - 6267*x^5 + 15095*x^4 - 24230*x^3 + 33910*x^2 - 15635*x + 2216, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 20 x^{13} - 40 x^{12} + 65 x^{11} + 18 x^{10} - 160 x^{9} + 40 x^{8} - 2380 x^{7} + 6930 x^{6} - 6267 x^{5} + 15095 x^{4} - 24230 x^{3} + 33910 x^{2} - 15635 x + 2216 \)

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:  \(-582452128062025000000000000000=-\,2^{15}\cdot 5^{17}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.46$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{5} a^{7} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{3}$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{8} + \frac{1}{25} a^{7} - \frac{1}{25} a^{6} + \frac{1}{25} a^{5} - \frac{4}{25} a^{4} + \frac{4}{25} a^{3} - \frac{4}{25} a^{2} + \frac{4}{25} a - \frac{4}{25}$, $\frac{1}{25} a^{10} + \frac{2}{25} a^{5} + \frac{1}{25}$, $\frac{1}{25} a^{11} + \frac{2}{25} a^{6} + \frac{1}{25} a$, $\frac{1}{125} a^{12} + \frac{2}{125} a^{11} + \frac{1}{125} a^{10} - \frac{8}{125} a^{7} + \frac{9}{125} a^{6} - \frac{8}{125} a^{5} + \frac{16}{125} a^{2} + \frac{57}{125} a + \frac{16}{125}$, $\frac{1}{125} a^{13} + \frac{2}{125} a^{11} - \frac{2}{125} a^{10} - \frac{8}{125} a^{8} + \frac{9}{125} a^{6} - \frac{9}{125} a^{5} + \frac{16}{125} a^{3} + \frac{57}{125} a - \frac{57}{125}$, $\frac{1}{339510171444564875} a^{14} - \frac{216052613096132}{339510171444564875} a^{13} - \frac{821423886020013}{339510171444564875} a^{12} - \frac{5351869028215051}{339510171444564875} a^{11} + \frac{3734961413345209}{339510171444564875} a^{10} - \frac{3733896102722478}{339510171444564875} a^{9} + \frac{419265966308951}{339510171444564875} a^{8} + \frac{12837101448431384}{339510171444564875} a^{7} + \frac{11871888210053578}{339510171444564875} a^{6} + \frac{20903345486591233}{339510171444564875} a^{5} + \frac{101160959460177521}{339510171444564875} a^{4} - \frac{64975893343051692}{339510171444564875} a^{3} + \frac{63584589577071172}{339510171444564875} a^{2} + \frac{123124156900272254}{339510171444564875} a + \frac{160467870702079474}{339510171444564875}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 627205507.271632 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.200.1, 5.1.89253125.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.15.17.3$x^{15} - 5 x^{13} + 10 x^{12} + 10 x^{11} + 10 x^{9} - 5 x^{8} - 10 x^{6} + 10 x^{5} + 5 x^{3} + 10$$15$$1$$17$$F_5 \times S_3$$[5/4]_{12}^{2}$
$13$13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
13.10.8.1$x^{10} - 13 x^{5} + 338$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$