Properties

Label 15.1.723352547563839.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,3^{3}\cdot 6653\cdot 4026880369$
Root discriminant $9.79$
Ramified primes $3, 6653, 4026880369$
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, 1, 1, -2, 5, -4, -5, 4, -5, 5, 2, -2, 2, -2, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 2*x^13 + 2*x^12 - 2*x^11 + 2*x^10 + 5*x^9 - 5*x^8 + 4*x^7 - 5*x^6 - 4*x^5 + 5*x^4 - 2*x^3 + x^2 + x + 1)
gp: K = bnfinit(x^15 - x^14 - 2*x^13 + 2*x^12 - 2*x^11 + 2*x^10 + 5*x^9 - 5*x^8 + 4*x^7 - 5*x^6 - 4*x^5 + 5*x^4 - 2*x^3 + x^2 + x + 1, 1)

Normalized defining polynomial

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

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:  \(-723352547563839=-\,3^{3}\cdot 6653\cdot 4026880369\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.79$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 6653, 4026880369$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{139} a^{14} - \frac{19}{139} a^{13} + \frac{62}{139} a^{12} - \frac{2}{139} a^{11} + \frac{34}{139} a^{10} - \frac{54}{139} a^{9} + \frac{4}{139} a^{8} + \frac{62}{139} a^{7} - \frac{5}{139} a^{5} - \frac{53}{139} a^{4} - \frac{14}{139} a^{3} - \frac{28}{139} a^{2} - \frac{51}{139} a - \frac{54}{139}$

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:  \( \frac{93}{139} a^{14} + \frac{179}{139} a^{13} - \frac{72}{139} a^{12} - \frac{186}{139} a^{11} - \frac{174}{139} a^{10} - \frac{435}{139} a^{9} + \frac{94}{139} a^{8} + \frac{345}{139} a^{7} + a^{6} + \frac{647}{139} a^{5} - \frac{64}{139} a^{4} - \frac{190}{139} a^{3} + \frac{315}{139} a^{2} - \frac{156}{139} a - \frac{18}{139} \),  \( a \),  \( \frac{99}{139} a^{14} - \frac{74}{139} a^{13} - \frac{256}{139} a^{12} + \frac{80}{139} a^{11} - \frac{248}{139} a^{10} + \frac{75}{139} a^{9} + \frac{674}{139} a^{8} - \frac{117}{139} a^{7} + 4 a^{6} - \frac{217}{139} a^{5} - \frac{660}{139} a^{4} + \frac{143}{139} a^{3} - \frac{548}{139} a^{2} - \frac{184}{139} a - \frac{64}{139} \),  \( \frac{138}{139} a^{14} + \frac{19}{139} a^{13} - \frac{62}{139} a^{12} + \frac{141}{139} a^{11} - \frac{451}{139} a^{10} - \frac{85}{139} a^{9} + \frac{135}{139} a^{8} - \frac{479}{139} a^{7} + 6 a^{6} - \frac{134}{139} a^{5} + \frac{192}{139} a^{4} + \frac{570}{139} a^{3} - \frac{250}{139} a^{2} + \frac{190}{139} a - \frac{85}{139} \),  \( \frac{8}{139} a^{14} - \frac{13}{139} a^{13} - \frac{60}{139} a^{12} - \frac{16}{139} a^{11} - \frac{6}{139} a^{10} - \frac{15}{139} a^{9} + \frac{171}{139} a^{8} + \frac{79}{139} a^{7} + \frac{99}{139} a^{5} - \frac{146}{139} a^{4} - \frac{112}{139} a^{3} - \frac{85}{139} a^{2} - \frac{269}{139} a - \frac{15}{139} \),  \( \frac{99}{139} a^{14} - \frac{74}{139} a^{13} - \frac{256}{139} a^{12} + \frac{80}{139} a^{11} - \frac{248}{139} a^{10} + \frac{75}{139} a^{9} + \frac{674}{139} a^{8} - \frac{117}{139} a^{7} + 4 a^{6} - \frac{217}{139} a^{5} - \frac{660}{139} a^{4} + \frac{143}{139} a^{3} - \frac{548}{139} a^{2} - \frac{323}{139} a - \frac{64}{139} \),  \( \frac{208}{139} a^{14} + \frac{79}{139} a^{13} - \frac{309}{139} a^{12} + \frac{1}{139} a^{11} - \frac{434}{139} a^{10} - \frac{251}{139} a^{9} + \frac{693}{139} a^{8} - \frac{170}{139} a^{7} + 5 a^{6} + \frac{211}{139} a^{5} - \frac{599}{139} a^{4} + \frac{424}{139} a^{3} + \frac{14}{139} a^{2} - \frac{183}{139} a + \frac{27}{139} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 11.7410590074 \)
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 $15$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }$ $15$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.13.0.1}{13} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ $15$

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$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.9.0.1$x^{9} - x^{3} + x^{2} + 1$$1$$9$$0$$C_9$$[\ ]^{9}$
6653Data not computed
4026880369Data not computed