Properties

Label 15.1.57477829056511319.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,1609^{3}\cdot 13798511$
Root discriminant $13.10$
Ramified primes $1609, 13798511$
Class number $1$
Class group Trivial
Galois Group 15T93

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, -4, 4, -15, 7, -7, -9, 4, -14, 6, -3, -3, 3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 3*x^13 - 3*x^12 - 3*x^11 + 6*x^10 - 14*x^9 + 4*x^8 - 9*x^7 - 7*x^6 + 7*x^5 - 15*x^4 + 4*x^3 - 4*x^2 + 2*x - 1)
gp: K = bnfinit(x^15 - 2*x^14 + 3*x^13 - 3*x^12 - 3*x^11 + 6*x^10 - 14*x^9 + 4*x^8 - 9*x^7 - 7*x^6 + 7*x^5 - 15*x^4 + 4*x^3 - 4*x^2 + 2*x - 1, 1)

Normalized defining polynomial

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

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

Invariants

Degree:  $15$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[1, 7]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-57477829056511319=-\,1609^{3}\cdot 13798511\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.10$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $1609, 13798511$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{113902} a^{14} - \frac{3439}{56951} a^{13} + \frac{23801}{113902} a^{12} + \frac{21495}{113902} a^{11} - \frac{5889}{56951} a^{10} + \frac{606}{56951} a^{9} - \frac{9440}{56951} a^{8} - \frac{14698}{56951} a^{7} + \frac{3894}{56951} a^{6} + \frac{20298}{56951} a^{5} + \frac{35713}{113902} a^{4} - \frac{23421}{56951} a^{3} + \frac{27691}{113902} a^{2} + \frac{20412}{56951} a - \frac{25647}{56951}$

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:  \( \frac{16594}{56951} a^{14} - \frac{64407}{113902} a^{13} + \frac{55560}{56951} a^{12} - \frac{53034}{56951} a^{11} - \frac{45251}{56951} a^{10} + \frac{187303}{113902} a^{9} - \frac{527097}{113902} a^{8} + \frac{45042}{56951} a^{7} - \frac{374151}{113902} a^{6} - \frac{445367}{113902} a^{5} + \frac{46367}{56951} a^{4} - \frac{313655}{56951} a^{3} - \frac{9379}{113902} a^{2} - \frac{55640}{56951} a - \frac{39941}{56951} \),  \( \frac{7095}{56951} a^{14} - \frac{41757}{113902} a^{13} + \frac{73711}{113902} a^{12} - \frac{72457}{113902} a^{11} - \frac{17793}{56951} a^{10} + \frac{169931}{113902} a^{9} - \frac{175701}{56951} a^{8} + \frac{103844}{56951} a^{7} - \frac{43561}{56951} a^{6} - \frac{86489}{56951} a^{5} + \frac{122638}{56951} a^{4} - \frac{354565}{113902} a^{3} + \frac{258145}{113902} a^{2} + \frac{43939}{113902} a + \frac{28871}{113902} \),  \( a \),  \( \frac{33011}{113902} a^{14} - \frac{21486}{56951} a^{13} + \frac{27883}{56951} a^{12} - \frac{38015}{113902} a^{11} - \frac{84967}{56951} a^{10} + \frac{71816}{56951} a^{9} - \frac{374593}{113902} a^{8} - \frac{117169}{113902} a^{7} - \frac{107475}{56951} a^{6} - \frac{455033}{113902} a^{5} + \frac{103498}{56951} a^{4} - \frac{154708}{56951} a^{3} - \frac{183753}{113902} a^{2} + \frac{9551}{113902} a + \frac{449}{56951} \),  \( \frac{12545}{113902} a^{14} - \frac{30348}{56951} a^{13} + \frac{51677}{56951} a^{12} - \frac{61056}{56951} a^{11} + \frac{32835}{113902} a^{10} + \frac{226427}{113902} a^{9} - \frac{194524}{56951} a^{8} + \frac{440513}{113902} a^{7} - \frac{84407}{113902} a^{6} + \frac{10489}{56951} a^{5} + \frac{220838}{56951} a^{4} - \frac{411129}{113902} a^{3} + \frac{190576}{56951} a^{2} + \frac{16844}{56951} a - \frac{25416}{56951} \),  \( \frac{2893}{113902} a^{14} - \frac{22155}{113902} a^{13} + \frac{59485}{113902} a^{12} - \frac{31204}{56951} a^{11} + \frac{39895}{113902} a^{10} + \frac{44628}{56951} a^{9} - \frac{144293}{56951} a^{8} + \frac{134985}{56951} a^{7} - \frac{124858}{56951} a^{6} - \frac{51318}{56951} a^{5} + \frac{8595}{113902} a^{4} - \frac{483085}{113902} a^{3} + \frac{264761}{113902} a^{2} - \frac{69493}{113902} a + \frac{10382}{56951} \),  \( \frac{6919}{113902} a^{14} + \frac{11077}{56951} a^{13} - \frac{23173}{113902} a^{12} + \frac{12422}{56951} a^{11} - \frac{26026}{56951} a^{10} - \frac{78411}{56951} a^{9} + \frac{7437}{56951} a^{8} - \frac{246707}{113902} a^{7} - \frac{275229}{113902} a^{6} - \frac{56255}{56951} a^{5} - \frac{119973}{56951} a^{4} + \frac{122245}{113902} a^{3} + \frac{10865}{113902} a^{2} - \frac{64803}{56951} a + \frac{72397}{113902} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 145.805581456 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T93:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 933120
The 108 conjugacy class representatives for [S(3)^5]S(5)=S(3)wrS(5) are not computed
Character table for [S(3)^5]S(5)=S(3)wrS(5) is not computed

Intermediate fields

5.1.1609.1

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
Degree 45 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
1609Data not computed
13798511Data not computed