Properties

Label 15.1.3703260525677583.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,3\cdot 13^{3}\cdot 561866260913$
Root discriminant $10.91$
Ramified primes $3, 13, 561866260913$
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, -3, -3, -5, -4, 3, -3, 6, 2, -3, 4, -5, -1, 1, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + x^13 - x^12 - 5*x^11 + 4*x^10 - 3*x^9 + 2*x^8 + 6*x^7 - 3*x^6 + 3*x^5 - 4*x^4 - 5*x^3 - 3*x^2 - 3*x - 1)
gp: K = bnfinit(x^15 - 2*x^14 + x^13 - x^12 - 5*x^11 + 4*x^10 - 3*x^9 + 2*x^8 + 6*x^7 - 3*x^6 + 3*x^5 - 4*x^4 - 5*x^3 - 3*x^2 - 3*x - 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut +\mathstrut x^{13} \) \(\mathstrut -\mathstrut x^{12} \) \(\mathstrut -\mathstrut 5 x^{11} \) \(\mathstrut +\mathstrut 4 x^{10} \) \(\mathstrut -\mathstrut 3 x^{9} \) \(\mathstrut +\mathstrut 2 x^{8} \) \(\mathstrut +\mathstrut 6 x^{7} \) \(\mathstrut -\mathstrut 3 x^{6} \) \(\mathstrut +\mathstrut 3 x^{5} \) \(\mathstrut -\mathstrut 4 x^{4} \) \(\mathstrut -\mathstrut 5 x^{3} \) \(\mathstrut -\mathstrut 3 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut -\mathstrut 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[1, 7]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-3703260525677583=-\,3\cdot 13^{3}\cdot 561866260913\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.91$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 13, 561866260913$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{659} a^{14} + \frac{147}{659} a^{13} + \frac{157}{659} a^{12} + \frac{327}{659} a^{11} - \frac{48}{659} a^{10} + \frac{101}{659} a^{9} - \frac{111}{659} a^{8} - \frac{62}{659} a^{7} - \frac{6}{659} a^{6} - \frac{238}{659} a^{5} + \frac{127}{659} a^{4} - \frac{192}{659} a^{3} - \frac{276}{659} a^{2} - \frac{269}{659} a + \frac{115}{659}$

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:  $7$
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:  \( a \),  \( \frac{334}{659} a^{14} - \frac{986}{659} a^{13} + \frac{1036}{659} a^{12} - \frac{835}{659} a^{11} - \frac{875}{659} a^{10} + \frac{2102}{659} a^{9} - \frac{2147}{659} a^{8} + \frac{1698}{659} a^{7} - \frac{27}{659} a^{6} - \frac{1071}{659} a^{5} + \frac{1560}{659} a^{4} - \frac{2182}{659} a^{3} + \frac{735}{659} a^{2} - \frac{881}{659} a - \frac{471}{659} \),  \( \frac{295}{659} a^{14} - \frac{129}{659} a^{13} - \frac{1133}{659} a^{12} + \frac{910}{659} a^{11} - \frac{1639}{659} a^{10} - \frac{1178}{659} a^{9} + \frac{3500}{659} a^{8} - \frac{1815}{659} a^{7} + \frac{2184}{659} a^{6} + \frac{1621}{659} a^{5} - \frac{3393}{659} a^{4} + \frac{693}{659} a^{3} - \frac{2340}{659} a^{2} - \frac{1593}{659} a + \frac{316}{659} \),  \( \frac{893}{659} a^{14} - \frac{2506}{659} a^{13} + \frac{2470}{659} a^{12} - \frac{1903}{659} a^{11} - \frac{3324}{659} a^{10} + \frac{6500}{659} a^{9} - \frac{6204}{659} a^{8} + \frac{4603}{659} a^{7} + \frac{2550}{659} a^{6} - \frac{5608}{659} a^{5} + \frac{5994}{659} a^{4} - \frac{6047}{659} a^{3} - \frac{661}{659} a^{2} - \frac{341}{659} a - \frac{1427}{659} \),  \( \frac{830}{659} a^{14} - \frac{2541}{659} a^{13} + \frac{3123}{659} a^{12} - \frac{2734}{659} a^{11} - \frac{2936}{659} a^{10} + \frac{7386}{659} a^{9} - \frac{8437}{659} a^{8} + \frac{5873}{659} a^{7} + \frac{2269}{659} a^{6} - \frac{6430}{659} a^{5} + \frac{7219}{659} a^{4} - \frac{6472}{659} a^{3} + \frac{252}{659} a^{2} - \frac{528}{659} a - \frac{1423}{659} \),  \( \frac{1058}{659} a^{14} - \frac{2634}{659} a^{13} + \frac{2015}{659} a^{12} - \frac{1327}{659} a^{11} - \frac{4654}{659} a^{10} + \frac{6031}{659} a^{9} - \frac{4749}{659} a^{8} + \frac{3599}{659} a^{7} + \frac{4196}{659} a^{6} - \frac{4679}{659} a^{5} + \frac{4543}{659} a^{4} - \frac{6095}{659} a^{3} - \frac{2048}{659} a^{2} - \frac{1891}{659} a - \frac{1563}{659} \),  \( \frac{759}{659} a^{14} - \frac{1775}{659} a^{13} + \frac{1202}{659} a^{12} - \frac{909}{659} a^{11} - \frac{3482}{659} a^{10} + \frac{4169}{659} a^{9} - \frac{2533}{659} a^{8} + \frac{1708}{659} a^{7} + \frac{4013}{659} a^{6} - \frac{3371}{659} a^{5} + \frac{1497}{659} a^{4} - \frac{3384}{659} a^{3} - \frac{2558}{659} a^{2} - \frac{1199}{659} a - \frac{1021}{659} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 39.9823344692 \)
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 ${\href{/LocalNumberField/5.11.0.1}{11} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ $15$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R $15$ $15$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$[\ ]_{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.9.0.1$x^{9} - x^{3} + x^{2} + 1$$1$$9$$0$$C_9$$[\ ]^{9}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
13.9.0.1$x^{9} - 2 x + 2$$1$$9$$0$$C_9$$[\ ]^{9}$
561866260913Data not computed