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Properties

Label 15.3.13351009166...5872.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{24}\cdot 3^{8}\cdot 47^{2}\cdot 257^{5}\cdot 2213^{2}$
Root discriminant $161.57$
Ramified primes $2, 3, 47, 257, 2213$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T74

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13203, -37098, 76131, -136251, 125694, -90216, 54480, -29354, 9749, -5052, 2757, -1197, 185, -22, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 22*x^13 + 185*x^12 - 1197*x^11 + 2757*x^10 - 5052*x^9 + 9749*x^8 - 29354*x^7 + 54480*x^6 - 90216*x^5 + 125694*x^4 - 136251*x^3 + 76131*x^2 - 37098*x - 13203)
 
gp: K = bnfinit(x^15 - x^14 - 22*x^13 + 185*x^12 - 1197*x^11 + 2757*x^10 - 5052*x^9 + 9749*x^8 - 29354*x^7 + 54480*x^6 - 90216*x^5 + 125694*x^4 - 136251*x^3 + 76131*x^2 - 37098*x - 13203, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 22 x^{13} + 185 x^{12} - 1197 x^{11} + 2757 x^{10} - 5052 x^{9} + 9749 x^{8} - 29354 x^{7} + 54480 x^{6} - 90216 x^{5} + 125694 x^{4} - 136251 x^{3} + 76131 x^{2} - 37098 x - 13203 \)

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:  \(1335100916632526059288147758415872=2^{24}\cdot 3^{8}\cdot 47^{2}\cdot 257^{5}\cdot 2213^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $161.57$
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:  $2, 3, 47, 257, 2213$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{12} + \frac{1}{9} a^{11} + \frac{4}{9} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{27575060489499831914878907565833334} a^{14} + \frac{56962134110082729221954884648877}{27575060489499831914878907565833334} a^{13} - \frac{427387043637952340713350221496335}{13787530244749915957439453782916667} a^{12} + \frac{373539228191400844854418878470497}{13787530244749915957439453782916667} a^{11} - \frac{694326768802692241358627813123291}{9191686829833277304959635855277778} a^{10} - \frac{4285064514512996430379516713107075}{9191686829833277304959635855277778} a^{9} + \frac{1613813763894934079602066959959572}{4595843414916638652479817927638889} a^{8} - \frac{4976162163949431592230881786913185}{13787530244749915957439453782916667} a^{7} + \frac{5592427325952580088423056093679261}{13787530244749915957439453782916667} a^{6} + \frac{181016283836989214827220366790352}{1531947804972212884159939309212963} a^{5} - \frac{141258744541226354656902925430614}{1531947804972212884159939309212963} a^{4} - \frac{165632761251827190357249807827645}{1531947804972212884159939309212963} a^{3} + \frac{1205580548925258674678729264666759}{3063895609944425768319878618425926} a^{2} + \frac{1060504639334276627527403792068721}{3063895609944425768319878618425926} a + \frac{13367003601338085139983431918824}{510649268324070961386646436404321}$

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:  $8$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 64106831798.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T74:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24000
The 40 conjugacy class representatives for [1/2.F(5)^3]S(3)
Character table for [1/2.F(5)^3]S(3) is not computed

Intermediate fields

3.3.257.1

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 R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ R ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.24.225$x^{12} - 12 x^{11} + 16 x^{10} - 4 x^{9} - 10 x^{8} + 16 x^{7} - 8 x^{4} - 8 x^{2} + 8$$4$$3$$24$12T55$[2, 2, 3, 3]^{6}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.8.0.1$x^{8} - x + 20$$1$$8$$0$$C_8$$[\ ]^{8}$
257Data not computed
2213Data not computed