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Properties

Label 15.7.11967689218...9232.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{24}\cdot 3^{9}\cdot 107^{5}\cdot 137^{3}\cdot 317^{2}$
Root discriminant $160.40$
Ramified primes $2, 3, 107, 137, 317$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T82

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-120960561, -153874791, 39062877, 78014535, -11450085, -14942855, 2178233, 1255903, -170502, -62198, 4902, 2102, 117, -65, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 65*x^13 + 117*x^12 + 2102*x^11 + 4902*x^10 - 62198*x^9 - 170502*x^8 + 1255903*x^7 + 2178233*x^6 - 14942855*x^5 - 11450085*x^4 + 78014535*x^3 + 39062877*x^2 - 153874791*x - 120960561)
 
gp: K = bnfinit(x^15 - 5*x^14 - 65*x^13 + 117*x^12 + 2102*x^11 + 4902*x^10 - 62198*x^9 - 170502*x^8 + 1255903*x^7 + 2178233*x^6 - 14942855*x^5 - 11450085*x^4 + 78014535*x^3 + 39062877*x^2 - 153874791*x - 120960561, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 65 x^{13} + 117 x^{12} + 2102 x^{11} + 4902 x^{10} - 62198 x^{9} - 170502 x^{8} + 1255903 x^{7} + 2178233 x^{6} - 14942855 x^{5} - 11450085 x^{4} + 78014535 x^{3} + 39062877 x^{2} - 153874791 x - 120960561 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1196768921851523490591827382239232=2^{24}\cdot 3^{9}\cdot 107^{5}\cdot 137^{3}\cdot 317^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $160.40$
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:  $2, 3, 107, 137, 317$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{26} a^{13} + \frac{9}{26} a^{12} + \frac{6}{13} a^{11} + \frac{3}{13} a^{9} - \frac{3}{13} a^{8} + \frac{3}{13} a^{7} - \frac{3}{13} a^{6} + \frac{11}{26} a^{5} + \frac{11}{26} a^{4} + \frac{5}{13} a^{2} + \frac{11}{26} a + \frac{11}{26}$, $\frac{1}{70286361623649982551726775702736126550645958646109244} a^{14} - \frac{69191242938847816783945716977569729897472683127902}{17571590405912495637931693925684031637661489661527311} a^{13} - \frac{15736372520528303206006650730144350567642268682636521}{70286361623649982551726775702736126550645958646109244} a^{12} - \frac{317866366581541518796586549664802968777770982044029}{763982191561412853823117127203653549463543028762057} a^{11} - \frac{11211820883842411933312673576414040869477991605852573}{35143180811824991275863387851368063275322979323054622} a^{10} + \frac{3825565613576801532859155314916484639818917145275994}{17571590405912495637931693925684031637661489661527311} a^{9} + \frac{2508675337516219021535824985594448687667320177869565}{35143180811824991275863387851368063275322979323054622} a^{8} + \frac{690393830142098061317572979157984580936793402877303}{17571590405912495637931693925684031637661489661527311} a^{7} + \frac{8909345587546283637956068482362808054856157240598931}{70286361623649982551726775702736126550645958646109244} a^{6} - \frac{5269199747846864008951678131604677452494276051231672}{17571590405912495637931693925684031637661489661527311} a^{5} + \frac{992225432904576302871816566789069426307847104803159}{3055928766245651415292468508814614197854172115048228} a^{4} + \frac{4611357828362562420720766862999742824504452558287471}{17571590405912495637931693925684031637661489661527311} a^{3} + \frac{7640837495861749221872566500331604101998101238639823}{70286361623649982551726775702736126550645958646109244} a^{2} - \frac{8295699377601026460360678594123610335637342614652564}{17571590405912495637931693925684031637661489661527311} a - \frac{27647863132929478367819946541405804650894483894076095}{70286361623649982551726775702736126550645958646109244}$

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:  $10$
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:  \( 131964333996 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T82:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48000
The 65 conjugacy class representatives for [F(5)^3]S(3)=F(5)wrS(3) are not computed
Character table for [F(5)^3]S(3)=F(5)wrS(3) is not computed

Intermediate fields

3.3.321.1

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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.24.291$x^{12} + 12 x^{10} + 12 x^{9} - 14 x^{8} - 8 x^{6} - 8 x^{4} + 16 x^{3} - 8 x^{2} + 16 x - 8$$4$$3$$24$12T55$[2, 2, 3, 3, 3]^{3}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
107Data not computed
$137$$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
137.4.3.2$x^{4} - 1233$$4$$1$$3$$C_4$$[\ ]_{4}$
137.10.0.1$x^{10} - x + 53$$1$$10$$0$$C_{10}$$[\ ]^{10}$
317Data not computed