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Properties

Label 15.1.75003483001...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{18}\cdot 3^{13}\cdot 5^{10}\cdot 179^{5}$
Root discriminant $98.10$
Ramified primes $2, 3, 5, 179$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4559605, 1440165, 1597287, 747683, 486393, 72609, 81103, 15195, -2409, 2935, -915, 33, -37, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 + 3*x^13 - 37*x^12 + 33*x^11 - 915*x^10 + 2935*x^9 - 2409*x^8 + 15195*x^7 + 81103*x^6 + 72609*x^5 + 486393*x^4 + 747683*x^3 + 1597287*x^2 + 1440165*x + 4559605)
 
gp: K = bnfinit(x^15 - 3*x^14 + 3*x^13 - 37*x^12 + 33*x^11 - 915*x^10 + 2935*x^9 - 2409*x^8 + 15195*x^7 + 81103*x^6 + 72609*x^5 + 486393*x^4 + 747683*x^3 + 1597287*x^2 + 1440165*x + 4559605, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} + 3 x^{13} - 37 x^{12} + 33 x^{11} - 915 x^{10} + 2935 x^{9} - 2409 x^{8} + 15195 x^{7} + 81103 x^{6} + 72609 x^{5} + 486393 x^{4} + 747683 x^{3} + 1597287 x^{2} + 1440165 x + 4559605 \)

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:  \(-750034830013451205120000000000=-\,2^{18}\cdot 3^{13}\cdot 5^{10}\cdot 179^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.10$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 179$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{24} a^{10} - \frac{1}{12} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{8} a^{2} + \frac{5}{12} a + \frac{1}{24}$, $\frac{1}{120} a^{11} - \frac{1}{60} a^{10} - \frac{1}{8} a^{9} + \frac{1}{20} a^{8} + \frac{1}{20} a^{7} + \frac{1}{10} a^{6} - \frac{3}{20} a^{5} - \frac{1}{5} a^{4} - \frac{1}{8} a^{3} + \frac{11}{60} a^{2} + \frac{5}{24} a - \frac{1}{4}$, $\frac{1}{120} a^{12} + \frac{1}{120} a^{10} - \frac{1}{30} a^{9} - \frac{1}{10} a^{8} - \frac{1}{20} a^{7} - \frac{1}{5} a^{6} - \frac{1}{4} a^{5} + \frac{9}{40} a^{4} + \frac{11}{60} a^{3} + \frac{13}{40} a^{2} + \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{240} a^{13} - \frac{1}{240} a^{12} + \frac{1}{120} a^{10} - \frac{1}{80} a^{9} - \frac{1}{16} a^{8} - \frac{1}{10} a^{7} + \frac{1}{20} a^{6} - \frac{3}{16} a^{5} + \frac{49}{240} a^{4} + \frac{2}{15} a^{3} - \frac{1}{40} a^{2} - \frac{5}{48} a - \frac{7}{16}$, $\frac{1}{3135897888470814093288931654800} a^{14} + \frac{2714633029606382682016373863}{1567948944235407046644465827400} a^{13} - \frac{1318987773682884089740028131}{1045299296156938031096310551600} a^{12} + \frac{2554289393473805579479020283}{1567948944235407046644465827400} a^{11} + \frac{33341139285187477381551027467}{3135897888470814093288931654800} a^{10} - \frac{15845655846066822214927342282}{195993618029425880830558228425} a^{9} - \frac{125757774013993280980965617521}{1045299296156938031096310551600} a^{8} - \frac{1830928455866245231256099309}{130662412019617253887038818950} a^{7} + \frac{3893739790713425557263747217}{1045299296156938031096310551600} a^{6} - \frac{92488463070026165150894650039}{1567948944235407046644465827400} a^{5} - \frac{350600820027335854920656145163}{3135897888470814093288931654800} a^{4} - \frac{185802499324458756151820828469}{522649648078469015548155275800} a^{3} + \frac{353165393510460070648826518577}{3135897888470814093288931654800} a^{2} + \frac{6599893229063937064510257937}{39198723605885176166111645685} a + \frac{56226022264797557077045095211}{627179577694162818657786330960}$

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

Class group and class number

$C_{5}$, which has order $5$ (assuming GRH)

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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1622981675.850239 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.10740.1, 5.1.162000.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.14.5$x^{10} - 2 x^{5} - 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$179$$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.4.2.1$x^{4} + 2327 x^{2} + 1570009$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
179.4.2.1$x^{4} + 2327 x^{2} + 1570009$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$