Properties

Label 15.3.12853809661090929.1
Degree $15$
Signature $[3, 6]$
Discriminant $3\cdot 4284603220363643$
Root discriminant $11.86$
Ramified primes $3, 4284603220363643$
Class number $1$
Class group Trivial
Galois Group 15T104

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 0, -1, 2, -1, -7, 2, 4, -4, -1, 0, 3, -1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - x^13 + 3*x^12 - x^10 - 4*x^9 + 4*x^8 + 2*x^7 - 7*x^6 - x^5 + 2*x^4 - x^3 + 2*x + 1)
gp: K = bnfinit(x^15 - x^14 - x^13 + 3*x^12 - x^10 - 4*x^9 + 4*x^8 + 2*x^7 - 7*x^6 - x^5 + 2*x^4 - x^3 + 2*x + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut x^{14} \) \(\mathstrut -\mathstrut x^{13} \) \(\mathstrut +\mathstrut 3 x^{12} \) \(\mathstrut -\mathstrut x^{10} \) \(\mathstrut -\mathstrut 4 x^{9} \) \(\mathstrut +\mathstrut 4 x^{8} \) \(\mathstrut +\mathstrut 2 x^{7} \) \(\mathstrut -\mathstrut 7 x^{6} \) \(\mathstrut -\mathstrut x^{5} \) \(\mathstrut +\mathstrut 2 x^{4} \) \(\mathstrut -\mathstrut x^{3} \) \(\mathstrut +\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(12853809661090929=3\cdot 4284603220363643\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.86$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 4284603220363643$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{709} a^{14} + \frac{128}{709} a^{13} + \frac{204}{709} a^{12} + \frac{86}{709} a^{11} - \frac{250}{709} a^{10} - \frac{346}{709} a^{9} + \frac{29}{709} a^{8} + \frac{200}{709} a^{7} + \frac{278}{709} a^{6} - \frac{304}{709} a^{5} - \frac{222}{709} a^{4} - \frac{276}{709} a^{3} - \frac{155}{709} a^{2} - \frac{143}{709} a - \frac{11}{709}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( a \),  \( \frac{1453}{709} a^{14} - \frac{2610}{709} a^{13} + \frac{759}{709} a^{12} + \frac{3719}{709} a^{11} - \frac{3078}{709} a^{10} + \frac{1361}{709} a^{9} - \frac{6784}{709} a^{8} + \frac{11254}{709} a^{7} - \frac{6577}{709} a^{6} - \frac{4968}{709} a^{5} + \frac{2865}{709} a^{4} - \frac{443}{709} a^{3} - \frac{1880}{709} a^{2} + \frac{667}{709} a + \frac{1742}{709} \),  \( \frac{1514}{709} a^{14} - \frac{2601}{709} a^{13} + \frac{441}{709} a^{12} + \frac{4002}{709} a^{11} - \frac{2730}{709} a^{10} + \frac{816}{709} a^{9} - \frac{7142}{709} a^{8} + \frac{11401}{709} a^{7} - \frac{5217}{709} a^{6} - \frac{5787}{709} a^{5} + \frac{2085}{709} a^{4} + \frac{446}{709} a^{3} - \frac{700}{709} a^{2} + \frac{452}{709} a + \frac{1780}{709} \),  \( \frac{2077}{709} a^{14} - \frac{3564}{709} a^{13} + \frac{435}{709} a^{12} + \frac{5626}{709} a^{11} - \frac{3807}{709} a^{10} + \frac{993}{709} a^{9} - \frac{9958}{709} a^{8} + \frac{14815}{709} a^{7} - \frac{6101}{709} a^{6} - \frac{8906}{709} a^{5} + \frac{3301}{709} a^{4} + \frac{329}{709} a^{3} - \frac{758}{709} a^{2} + \frac{2187}{709} a + \frac{2677}{709} \),  \( \frac{291}{709} a^{14} - \frac{329}{709} a^{13} - \frac{192}{709} a^{12} + \frac{920}{709} a^{11} - \frac{432}{709} a^{10} - \frac{8}{709} a^{9} - \frac{778}{709} a^{8} + \frac{771}{709} a^{7} + \frac{72}{709} a^{6} - \frac{1966}{709} a^{5} + \frac{1335}{709} a^{4} - \frac{199}{709} a^{3} - \frac{1147}{709} a^{2} + \frac{927}{709} a + \frac{1053}{709} \),  \( \frac{2833}{709} a^{14} - \frac{4638}{709} a^{13} + \frac{97}{709} a^{12} + \frac{8250}{709} a^{11} - \frac{4922}{709} a^{10} + \frac{329}{709} a^{9} - \frac{12140}{709} a^{8} + \frac{19252}{709} a^{7} - \frac{6506}{709} a^{6} - \frac{14686}{709} a^{5} + \frac{5629}{709} a^{4} + \frac{2246}{709} a^{3} - \frac{3789}{709} a^{2} + \frac{2556}{709} a + \frac{4287}{709} \),  \( \frac{3490}{709} a^{14} - \frac{5622}{709} a^{13} + \frac{124}{709} a^{12} + \frac{10159}{709} a^{11} - \frac{6102}{709} a^{10} + \frac{596}{709} a^{9} - \frac{14357}{709} a^{8} + \frac{23032}{709} a^{7} - \frac{8200}{709} a^{6} - \frac{18021}{709} a^{5} + \frac{7247}{709} a^{4} + \frac{2418}{709} a^{3} - \frac{5655}{709} a^{2} + \frac{2902}{709} a + \frac{5568}{709} \),  \( \frac{2786}{709} a^{14} - \frac{4982}{709} a^{13} + \frac{435}{709} a^{12} + \frac{8462}{709} a^{11} - \frac{5934}{709} a^{10} + \frac{284}{709} a^{9} - \frac{12085}{709} a^{8} + \frac{20487}{709} a^{7} - \frac{7519}{709} a^{6} - \frac{15287}{709} a^{5} + \frac{7555}{709} a^{4} + \frac{2456}{709} a^{3} - \frac{2885}{709} a^{2} + \frac{2896}{709} a + \frac{4095}{709} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 108.091851993 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T104:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 1307674368000
The 176 conjugacy class representatives for S15 are not computed
Character table for S15 is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 30 sibling: 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 $15$ R $15$ ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ $15$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
4284603220363643Data not computed