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Properties

Label 15.15.1419778908...6704.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 7^{14}\cdot 139^{2}\cdot 181^{3}\cdot 6679^{2}$
Root discriminant $189.15$
Ramified primes $2, 7, 139, 181, 6679$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T75

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2390177, 37429270, 7159929, -93826187, 40476996, 33584838, -21624778, -621438, 2212053, -202566, -81613, 11263, 1169, -196, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 196*x^13 + 1169*x^12 + 11263*x^11 - 81613*x^10 - 202566*x^9 + 2212053*x^8 - 621438*x^7 - 21624778*x^6 + 33584838*x^5 + 40476996*x^4 - 93826187*x^3 + 7159929*x^2 + 37429270*x - 2390177)
 
gp: K = bnfinit(x^15 - 5*x^14 - 196*x^13 + 1169*x^12 + 11263*x^11 - 81613*x^10 - 202566*x^9 + 2212053*x^8 - 621438*x^7 - 21624778*x^6 + 33584838*x^5 + 40476996*x^4 - 93826187*x^3 + 7159929*x^2 + 37429270*x - 2390177, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 196 x^{13} + 1169 x^{12} + 11263 x^{11} - 81613 x^{10} - 202566 x^{9} + 2212053 x^{8} - 621438 x^{7} - 21624778 x^{6} + 33584838 x^{5} + 40476996 x^{4} - 93826187 x^{3} + 7159929 x^{2} + 37429270 x - 2390177 \)

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:  \(14197789084358343049218550450376704=2^{12}\cdot 7^{14}\cdot 139^{2}\cdot 181^{3}\cdot 6679^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $189.15$
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:  $2, 7, 139, 181, 6679$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{52} a^{13} + \frac{3}{52} a^{12} + \frac{3}{26} a^{11} - \frac{1}{13} a^{10} + \frac{1}{52} a^{9} - \frac{1}{52} a^{8} - \frac{3}{13} a^{7} - \frac{1}{26} a^{6} + \frac{5}{13} a^{5} + \frac{3}{26} a^{4} + \frac{2}{13} a^{3} + \frac{1}{26} a^{2} + \frac{23}{52} a + \frac{17}{52}$, $\frac{1}{4707412140616804190562904008794215309877075023708} a^{14} - \frac{7852507642073322528794414469834772175720894149}{1176853035154201047640726002198553827469268755927} a^{13} + \frac{300393957518702018929418921337658515865281346869}{4707412140616804190562904008794215309877075023708} a^{12} - \frac{489864911316591949599691912127418997467876395791}{2353706070308402095281452004397107654938537511854} a^{11} - \frac{212012721297370488786665259555593990324836202687}{4707412140616804190562904008794215309877075023708} a^{10} - \frac{113468194979580781160817305512559772024254096299}{2353706070308402095281452004397107654938537511854} a^{9} + \frac{417254918776526564183258526162871965546067258167}{4707412140616804190562904008794215309877075023708} a^{8} - \frac{177115469014828289757077482715903354993871612003}{1176853035154201047640726002198553827469268755927} a^{7} - \frac{230812773023897148229892087696812115663365455274}{1176853035154201047640726002198553827469268755927} a^{6} + \frac{412820685129024692914080227135714468350393245657}{2353706070308402095281452004397107654938537511854} a^{5} - \frac{45183659149194696739117347448162041649978518859}{1176853035154201047640726002198553827469268755927} a^{4} - \frac{585482041968856271391087490989791800142812717846}{1176853035154201047640726002198553827469268755927} a^{3} + \frac{2233886229605926165776844080982673623009948823497}{4707412140616804190562904008794215309877075023708} a^{2} + \frac{219007798712660189328491302199446436305130645229}{1176853035154201047640726002198553827469268755927} a - \frac{211880459955440588088018614091833710176696350197}{4707412140616804190562904008794215309877075023708}$

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:  \( 2364264785700 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T75:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24000
The 55 conjugacy class representatives for [F(5)^3]3=F(5)wr3 are not computed
Character table for [F(5)^3]3=F(5)wr3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{5}$ R $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]^{3}$
2.12.12.12$x^{12} + 66 x^{10} - 93 x^{8} - 68 x^{6} - 41 x^{4} + 66 x^{2} - 123$$2$$6$$12$12T29$[2, 2, 2]^{6}$
7Data not computed
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.4.2.2$x^{4} - 139 x^{2} + 38642$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
139.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.4.3.2$x^{4} - 724$$4$$1$$3$$C_4$$[\ ]_{4}$
181.4.0.1$x^{4} - x + 54$$1$$4$$0$$C_4$$[\ ]^{4}$
181.4.0.1$x^{4} - x + 54$$1$$4$$0$$C_4$$[\ ]^{4}$
6679Data not computed