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Properties

Label 15.15.2250056984...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{34}\cdot 5^{5}\cdot 37^{5}\cdot 61^{2}\cdot 467^{2}\cdot 863^{2}$
Root discriminant $265.14$
Ramified primes $2, 5, 37, 61, 467, 863$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T82

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1153284124, -407455644, 2000962040, 1108467941, -460280152, -324217686, 18527692, 29116535, 471780, -1194256, -39488, 24835, 720, -254, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 - 254*x^13 + 720*x^12 + 24835*x^11 - 39488*x^10 - 1194256*x^9 + 471780*x^8 + 29116535*x^7 + 18527692*x^6 - 324217686*x^5 - 460280152*x^4 + 1108467941*x^3 + 2000962040*x^2 - 407455644*x - 1153284124)
 
gp: K = bnfinit(x^15 - 4*x^14 - 254*x^13 + 720*x^12 + 24835*x^11 - 39488*x^10 - 1194256*x^9 + 471780*x^8 + 29116535*x^7 + 18527692*x^6 - 324217686*x^5 - 460280152*x^4 + 1108467941*x^3 + 2000962040*x^2 - 407455644*x - 1153284124, 1)
 

Normalized defining polynomial

\( x^{15} - 4 x^{14} - 254 x^{13} + 720 x^{12} + 24835 x^{11} - 39488 x^{10} - 1194256 x^{9} + 471780 x^{8} + 29116535 x^{7} + 18527692 x^{6} - 324217686 x^{5} - 460280152 x^{4} + 1108467941 x^{3} + 2000962040 x^{2} - 407455644 x - 1153284124 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2250056984849498589655164413542400000=2^{34}\cdot 5^{5}\cdot 37^{5}\cdot 61^{2}\cdot 467^{2}\cdot 863^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $265.14$
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:  $2, 5, 37, 61, 467, 863$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{7} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{7} + \frac{3}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{208} a^{13} - \frac{3}{104} a^{12} - \frac{9}{208} a^{11} + \frac{1}{13} a^{10} - \frac{9}{104} a^{9} - \frac{3}{104} a^{7} - \frac{1}{52} a^{6} - \frac{19}{208} a^{5} - \frac{11}{104} a^{4} + \frac{63}{208} a^{3} + \frac{21}{52} a^{2} - \frac{23}{52} a + \frac{15}{52}$, $\frac{1}{20565218162186111812060872365001235481339936441817232} a^{14} + \frac{42716337109594201340947171820146428217512986331589}{20565218162186111812060872365001235481339936441817232} a^{13} - \frac{709636218915228130412019384095421912116664503608795}{20565218162186111812060872365001235481339936441817232} a^{12} - \frac{1117009262366834505803770957351570726566721983675923}{20565218162186111812060872365001235481339936441817232} a^{11} - \frac{1109098096097940393543315591875194266019661188864163}{10282609081093055906030436182500617740669968220908616} a^{10} - \frac{1027425154884059497740238129310758043367533286848683}{10282609081093055906030436182500617740669968220908616} a^{9} - \frac{632088346931936296785893273082641821812074212665841}{10282609081093055906030436182500617740669968220908616} a^{8} + \frac{210891459266895683343565697018016426441820981692769}{10282609081093055906030436182500617740669968220908616} a^{7} - \frac{1377119211456449215545472277081481078051883627006707}{20565218162186111812060872365001235481339936441817232} a^{6} - \frac{1681385387460770848222047987032446643536882689998815}{20565218162186111812060872365001235481339936441817232} a^{5} + \frac{2372681490710488478362519955214294276893252887246641}{20565218162186111812060872365001235481339936441817232} a^{4} + \frac{9295202030288574344528710172013633679587978974401017}{20565218162186111812060872365001235481339936441817232} a^{3} - \frac{210217775271792008957962243095501865618742892418196}{1285326135136631988253804522812577217583746027613577} a^{2} + \frac{1088190305657457286326475984124530578701756802710503}{2570652270273263976507609045625154435167492055227154} a - \frac{794210612996178219781329605573706586970061348706283}{5141304540546527953015218091250308870334984110454308}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 126266111719000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T82:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48000
The 65 conjugacy class representatives for [F(5)^3]S(3)=F(5)wrS(3) are not computed
Character table for [F(5)^3]S(3)=F(5)wrS(3) is not computed

Intermediate fields

3.3.148.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.32.291$x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} - 4 x^{8} - 4 x^{6} + 8 x^{5} + 4 x^{4} + 4 x^{2} + 8 x + 2$$12$$1$$32$12T98$[2, 8/3, 8/3, 11/3, 11/3]_{3}^{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.5.5.4$x^{5} + 10 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
37Data not computed
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.10.0.1$x^{10} + x^{2} - x + 10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
467Data not computed
863Data not computed