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Properties

Label 15.1.64007262884...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 5^{10}\cdot 41^{13}\cdot 61^{5}$
Root discriminant $832.56$
Ramified primes $2, 5, 41, 61$
Class number $75$ (GRH)
Class group $[5, 15]$ (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![-844596301, -456912753, -600364745, -760159369, -566354805, -12647679, -60011861, -1140151, -60303, -31523, 47979, 3215, 233, 115, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + 115*x^13 + 233*x^12 + 3215*x^11 + 47979*x^10 - 31523*x^9 - 60303*x^8 - 1140151*x^7 - 60011861*x^6 - 12647679*x^5 - 566354805*x^4 - 760159369*x^3 - 600364745*x^2 - 456912753*x - 844596301)
 
gp: K = bnfinit(x^15 - x^14 + 115*x^13 + 233*x^12 + 3215*x^11 + 47979*x^10 - 31523*x^9 - 60303*x^8 - 1140151*x^7 - 60011861*x^6 - 12647679*x^5 - 566354805*x^4 - 760159369*x^3 - 600364745*x^2 - 456912753*x - 844596301, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} + 115 x^{13} + 233 x^{12} + 3215 x^{11} + 47979 x^{10} - 31523 x^{9} - 60303 x^{8} - 1140151 x^{7} - 60011861 x^{6} - 12647679 x^{5} - 566354805 x^{4} - 760159369 x^{3} - 600364745 x^{2} - 456912753 x - 844596301 \)

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:  \(-64007262884567863139459214673592320000000000=-\,2^{23}\cdot 5^{10}\cdot 41^{13}\cdot 61^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $832.56$
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:  $2, 5, 41, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{61} a^{8} + \frac{16}{61} a^{7} + \frac{21}{61} a^{6} - \frac{20}{61} a^{5} + \frac{8}{61} a^{4} - \frac{14}{61} a^{3} + \frac{20}{61} a^{2}$, $\frac{1}{61} a^{9} + \frac{9}{61} a^{7} + \frac{10}{61} a^{6} + \frac{23}{61} a^{5} - \frac{20}{61} a^{4} - \frac{15}{61} a^{2}$, $\frac{1}{122} a^{10} - \frac{1}{122} a^{8} - \frac{14}{61} a^{7} - \frac{2}{61} a^{6} + \frac{29}{61} a^{5} + \frac{21}{61} a^{4} - \frac{29}{61} a^{3} - \frac{17}{122} a^{2} - \frac{1}{2}$, $\frac{1}{7442} a^{11} - \frac{1}{7442} a^{10} - \frac{7}{7442} a^{9} - \frac{11}{7442} a^{8} + \frac{967}{3721} a^{7} + \frac{352}{3721} a^{6} - \frac{1701}{3721} a^{5} + \frac{1721}{3721} a^{4} + \frac{31}{122} a^{3} - \frac{43}{122} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{453962} a^{12} - \frac{1}{453962} a^{11} - \frac{983}{453962} a^{10} + \frac{1331}{453962} a^{9} - \frac{131}{226981} a^{8} + \frac{50494}{226981} a^{7} + \frac{107611}{226981} a^{6} - \frac{42687}{226981} a^{5} - \frac{1131}{7442} a^{4} - \frac{2291}{7442} a^{3} + \frac{37}{122} a^{2} + \frac{53}{122} a$, $\frac{1}{2269810} a^{13} - \frac{1}{1134905} a^{12} - \frac{67}{2269810} a^{11} - \frac{1161}{1134905} a^{10} - \frac{1544}{226981} a^{9} + \frac{2801}{1134905} a^{8} + \frac{49219}{226981} a^{7} + \frac{339227}{1134905} a^{6} - \frac{67093}{453962} a^{5} + \frac{4434}{18605} a^{4} + \frac{4853}{37210} a^{3} + \frac{28}{61} a^{2} - \frac{118}{305} a - \frac{2}{5}$, $\frac{1}{586699757985538999652978838071124519721077370} a^{14} - \frac{7616488546430168470957391008663809627}{293349878992769499826489419035562259860538685} a^{13} + \frac{229031297748131269635756911939649077636}{293349878992769499826489419035562259860538685} a^{12} + \frac{728792759032087077941772808424625454667}{586699757985538999652978838071124519721077370} a^{11} - \frac{268405626793222298116033422043503846896493}{293349878992769499826489419035562259860538685} a^{10} - \frac{244273446637842530702607447631865944819743}{586699757985538999652978838071124519721077370} a^{9} - \frac{343219924406776009289445436239306992231319}{586699757985538999652978838071124519721077370} a^{8} + \frac{68853275245210981719238212873249838114945957}{293349878992769499826489419035562259860538685} a^{7} - \frac{3275049380587661239727324732305420706237643}{9618028819435065568081620296247942946247170} a^{6} - \frac{217172773170994530347567288042894098358331}{4809014409717532784040810148123971473123585} a^{5} - \frac{18650765898656521352068697924997599796964}{78836301798648078426898527018425761854485} a^{4} - \frac{62147741725234810004905229904590680716341}{157672603597296156853797054036851523708970} a^{3} - \frac{496678178046444715871387113680757276968}{1292398390141771777490139787187307571385} a^{2} - \frac{1127279277514347218467414053664400080027}{2584796780283543554980279574374615142770} a - \frac{7937581753638072441514446328448400597}{42373717709566287786561960235649428570}$

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

Class group and class number

$C_{5}\times C_{15}$, which has order $75$ (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:  \( 529297939111376.9 \) (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

3.1.100040.1, 5.1.5651522000.2

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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ R ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.19.33$x^{10} - 6 x^{4} + 4 x^{2} - 14$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.5.4.2$x^{5} + 246$$5$$1$$4$$C_5$$[\ ]_{5}$
41.10.9.4$x^{10} - 1912896$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$