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Properties

Label 15.15.7929522270...7125.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{3}\cdot 229^{6}\cdot 353^{3}$
Root discriminant $39.20$
Ramified primes $5, 229, 353$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_5 \times S_3$ (as 15T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -13, 123, 761, -1077, -2850, 2048, 3449, -1406, -1759, 337, 403, -18, -35, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 35*x^13 - 18*x^12 + 403*x^11 + 337*x^10 - 1759*x^9 - 1406*x^8 + 3449*x^7 + 2048*x^6 - 2850*x^5 - 1077*x^4 + 761*x^3 + 123*x^2 - 13*x - 1)
 
gp: K = bnfinit(x^15 - 35*x^13 - 18*x^12 + 403*x^11 + 337*x^10 - 1759*x^9 - 1406*x^8 + 3449*x^7 + 2048*x^6 - 2850*x^5 - 1077*x^4 + 761*x^3 + 123*x^2 - 13*x - 1, 1)
 

Normalized defining polynomial

\( x^{15} - 35 x^{13} - 18 x^{12} + 403 x^{11} + 337 x^{10} - 1759 x^{9} - 1406 x^{8} + 3449 x^{7} + 2048 x^{6} - 2850 x^{5} - 1077 x^{4} + 761 x^{3} + 123 x^{2} - 13 x - 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(792952227088888747277125=5^{3}\cdot 229^{6}\cdot 353^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.20$
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:  $5, 229, 353$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{179112219350045} a^{14} - \frac{4521400453559}{179112219350045} a^{13} - \frac{6173941070204}{179112219350045} a^{12} - \frac{27960820279646}{179112219350045} a^{11} + \frac{71757030825554}{179112219350045} a^{10} + \frac{47095511265561}{179112219350045} a^{9} - \frac{8134662121774}{35822443870009} a^{8} - \frac{67418547678789}{179112219350045} a^{7} - \frac{26746763861761}{179112219350045} a^{6} + \frac{5778104499297}{35822443870009} a^{5} + \frac{29889698213762}{179112219350045} a^{4} - \frac{21180549256823}{179112219350045} a^{3} - \frac{83805919830721}{179112219350045} a^{2} - \frac{45713983619046}{179112219350045} a - \frac{72865190622963}{179112219350045}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 11905413.804 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5 \times S_3$ (as 15T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 720
The 21 conjugacy class representatives for $S_5 \times S_3$
Character table for $S_5 \times S_3$ is not computed

Intermediate fields

3.3.229.1, 5.5.404185.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ $15$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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}$
229Data not computed
353Data not computed