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Properties

Label 15.11.4146832628...4704.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{12}\cdot 3^{24}\cdot 17^{3}\cdot 37^{3}\cdot 379531^{2}$
Root discriminant $203.16$
Ramified primes $2, 3, 17, 37, 379531$
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![-2191593, 3584043, 13866012, 11810481, 2883087, -1637631, -1606206, -732105, -277812, -63720, 7464, 6096, 333, -135, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 - 135*x^13 + 333*x^12 + 6096*x^11 + 7464*x^10 - 63720*x^9 - 277812*x^8 - 732105*x^7 - 1606206*x^6 - 1637631*x^5 + 2883087*x^4 + 11810481*x^3 + 13866012*x^2 + 3584043*x - 2191593)
 
gp: K = bnfinit(x^15 - 6*x^14 - 135*x^13 + 333*x^12 + 6096*x^11 + 7464*x^10 - 63720*x^9 - 277812*x^8 - 732105*x^7 - 1606206*x^6 - 1637631*x^5 + 2883087*x^4 + 11810481*x^3 + 13866012*x^2 + 3584043*x - 2191593, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} - 135 x^{13} + 333 x^{12} + 6096 x^{11} + 7464 x^{10} - 63720 x^{9} - 277812 x^{8} - 732105 x^{7} - 1606206 x^{6} - 1637631 x^{5} + 2883087 x^{4} + 11810481 x^{3} + 13866012 x^{2} + 3584043 x - 2191593 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(41468326288645730146555835992264704=2^{12}\cdot 3^{24}\cdot 17^{3}\cdot 37^{3}\cdot 379531^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $203.16$
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:  $2, 3, 17, 37, 379531$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{968624585006437619646043283695486734956193956} a^{14} + \frac{27110813390592880335814199653683737606560633}{484312292503218809823021641847743367478096978} a^{13} + \frac{48281763777462687236408653613268230398560573}{968624585006437619646043283695486734956193956} a^{12} - \frac{68624867093272698677122082087250494191175367}{484312292503218809823021641847743367478096978} a^{11} + \frac{31927768164420702310605291040545572568905881}{484312292503218809823021641847743367478096978} a^{10} + \frac{44236783269670725096402908195352899727191196}{242156146251609404911510820923871683739048489} a^{9} - \frac{23113464762318929397859365200038109234360253}{242156146251609404911510820923871683739048489} a^{8} + \frac{20598762247952849831634553917065347216879533}{484312292503218809823021641847743367478096978} a^{7} - \frac{161997056829010320627072722707243106844158219}{968624585006437619646043283695486734956193956} a^{6} - \frac{43351498850324542725750712753591799335040356}{242156146251609404911510820923871683739048489} a^{5} + \frac{169298016352545927441182209298766171989655679}{968624585006437619646043283695486734956193956} a^{4} + \frac{46127787542731035093529475240759176329988610}{242156146251609404911510820923871683739048489} a^{3} + \frac{350864346630761649400992945247148137933171377}{968624585006437619646043283695486734956193956} a^{2} - \frac{197101141286473227619010084967182905230039089}{484312292503218809823021641847743367478096978} a - \frac{389165661395370416958261878871761629218368829}{968624585006437619646043283695486734956193956}$

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:  $12$
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:  \( 1771496479170 \) (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

\(\Q(\zeta_{9})^+\)

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 $15$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ R ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $15$ $15$ R ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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}$
3Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.5.0.1$x^{5} - x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
379531Data not computed