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Properties

Label 15.11.1584541601...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{22}\cdot 5^{6}\cdot 19^{2}\cdot 37^{5}\cdot 491^{4}\cdot 1307^{2}\cdot 3119^{2}$
Root discriminant $1031.16$
Ramified primes $2, 5, 19, 37, 491, 1307, 3119$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T96

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-125696000, 879872000, -1810022400, 416368000, 1889210880, -1146538240, -471272384, 382478112, -4014720, -12084336, 86128, 146384, -432, -684, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 684*x^13 - 432*x^12 + 146384*x^11 + 86128*x^10 - 12084336*x^9 - 4014720*x^8 + 382478112*x^7 - 471272384*x^6 - 1146538240*x^5 + 1889210880*x^4 + 416368000*x^3 - 1810022400*x^2 + 879872000*x - 125696000)
 
gp: K = bnfinit(x^15 - 684*x^13 - 432*x^12 + 146384*x^11 + 86128*x^10 - 12084336*x^9 - 4014720*x^8 + 382478112*x^7 - 471272384*x^6 - 1146538240*x^5 + 1889210880*x^4 + 416368000*x^3 - 1810022400*x^2 + 879872000*x - 125696000, 1)
 

Normalized defining polynomial

\( x^{15} - 684 x^{13} - 432 x^{12} + 146384 x^{11} + 86128 x^{10} - 12084336 x^{9} - 4014720 x^{8} + 382478112 x^{7} - 471272384 x^{6} - 1146538240 x^{5} + 1889210880 x^{4} + 416368000 x^{3} - 1810022400 x^{2} + 879872000 x - 125696000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1584541601971758745306546692340589068288000000=2^{22}\cdot 5^{6}\cdot 19^{2}\cdot 37^{5}\cdot 491^{4}\cdot 1307^{2}\cdot 3119^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1031.16$
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:  $2, 5, 19, 37, 491, 1307, 3119$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{32} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{320} a^{11} - \frac{1}{80} a^{9} + \frac{1}{40} a^{8} + \frac{1}{80} a^{7} + \frac{1}{40} a^{6} - \frac{1}{20} a^{5} + \frac{1}{10} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{640} a^{12} - \frac{1}{160} a^{10} + \frac{1}{80} a^{9} - \frac{1}{40} a^{8} + \frac{1}{80} a^{7} - \frac{1}{40} a^{6} + \frac{1}{20} a^{4} - \frac{1}{10} a^{3}$, $\frac{1}{6400} a^{13} - \frac{1}{1600} a^{11} - \frac{1}{200} a^{10} + \frac{1}{100} a^{9} - \frac{1}{200} a^{8} - \frac{1}{400} a^{7} + \frac{1}{20} a^{6} + \frac{1}{200} a^{5} - \frac{3}{50} a^{4} - \frac{1}{5} a^{3} - \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{18800109198937964347554447509148346456881789452800} a^{14} - \frac{406509581611104782425539432685563766150253939}{9400054599468982173777223754574173228440894726400} a^{13} + \frac{958946506869964345650360564948195444008161787}{2350013649867245543444305938643543307110223681600} a^{12} + \frac{188223766108736735205931127653121021261903419}{235001364986724554344430593864354330711022368160} a^{11} - \frac{763180166926716332658383591512992223825473397}{58750341246681138586107648466088582677755592040} a^{10} + \frac{590894635589158723582643280237445487866075351}{1175006824933622771722152969321771653555111840800} a^{9} + \frac{937910405877307525138906181085567983143560085}{47000272997344910868886118772870866142204473632} a^{8} - \frac{2524859549842980079693100482961756852944417771}{587503412466811385861076484660885826777555920400} a^{7} - \frac{32304550476394837483335866096227060732288791519}{587503412466811385861076484660885826777555920400} a^{6} - \frac{905172579491162899579011541682332907903871249}{14687585311670284646526912116522145669438898010} a^{5} + \frac{134795825152576471109891544840775366953798097}{8639756065688402733251124774424791570258175300} a^{4} - \frac{472491047793058113054209384869254556737055059}{14687585311670284646526912116522145669438898010} a^{3} + \frac{6660816741035138407718856275654567905137289069}{29375170623340569293053824233044291338877796020} a^{2} - \frac{607895693919236794787025405229265686207964751}{2937517062334056929305382423304429133887779602} a - \frac{556958113422324179277332244040687884225812790}{1468758531167028464652691211652214566943889801}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $12$
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:  \( 631360478866000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T96:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1296000
The 65 conjugacy class representatives for [A(5)^3]S(3)=A(5)wrS(3) are not computed
Character table for [A(5)^3]S(3)=A(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 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 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 R ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.20.69$x^{12} + 2 x^{11} + 2 x^{10} + 2 x^{9} + 2 x^{6} + 2 x^{4} + 2$$12$$1$$20$12T206$[4/3, 4/3, 4/3, 4/3, 2, 2]_{3}^{6}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$37$37.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
37.10.5.1$x^{10} - 2738 x^{6} + 1874161 x^{2} - 11719128733$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
491Data not computed
1307Data not computed
3119Data not computed