# 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

# Related objects

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)

## Normalizeddefining 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

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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $15$ 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$$[\ ]_{2} 3.3.0.1x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4} 3.6.0.1x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
4284603220363643Data not computed