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Properties

Label 15.1.27100824264...3767.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,11^{4}\cdot 17^{2}\cdot 23^{5}\cdot 997556191^{2}$
Root discriminant $124.60$
Ramified primes $11, 17, 23, 997556191$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T60

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4096, 76800, -477440, 961600, 179744, -246544, -13720, 12820, -6982, 7789, 693, -425, -1, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 + 3*x^13 - x^12 - 425*x^11 + 693*x^10 + 7789*x^9 - 6982*x^8 + 12820*x^7 - 13720*x^6 - 246544*x^5 + 179744*x^4 + 961600*x^3 - 477440*x^2 + 76800*x - 4096)
 
gp: K = bnfinit(x^15 - 3*x^14 + 3*x^13 - x^12 - 425*x^11 + 693*x^10 + 7789*x^9 - 6982*x^8 + 12820*x^7 - 13720*x^6 - 246544*x^5 + 179744*x^4 + 961600*x^3 - 477440*x^2 + 76800*x - 4096, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} + 3 x^{13} - x^{12} - 425 x^{11} + 693 x^{10} + 7789 x^{9} - 6982 x^{8} + 12820 x^{7} - 13720 x^{6} - 246544 x^{5} + 179744 x^{4} + 961600 x^{3} - 477440 x^{2} + 76800 x - 4096 \)

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:  \(-27100824264108518067277197633767=-\,11^{4}\cdot 17^{2}\cdot 23^{5}\cdot 997556191^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $124.60$
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:  $11, 17, 23, 997556191$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{5}{12} a^{4} - \frac{5}{12} a^{3} - \frac{1}{12} a^{2} + \frac{1}{3} a$, $\frac{1}{24} a^{9} - \frac{1}{24} a^{8} + \frac{1}{24} a^{7} - \frac{7}{24} a^{6} - \frac{7}{24} a^{5} + \frac{7}{24} a^{4} - \frac{5}{24} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{48} a^{10} - \frac{1}{48} a^{9} + \frac{1}{48} a^{8} + \frac{1}{48} a^{7} + \frac{3}{16} a^{6} + \frac{7}{48} a^{5} - \frac{5}{48} a^{4} + \frac{1}{12} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{96} a^{11} - \frac{1}{96} a^{10} + \frac{1}{96} a^{9} + \frac{1}{96} a^{8} - \frac{7}{96} a^{7} - \frac{25}{96} a^{6} - \frac{5}{96} a^{5} - \frac{1}{2} a^{4} + \frac{5}{24} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{192} a^{12} - \frac{1}{192} a^{11} + \frac{1}{192} a^{10} + \frac{1}{192} a^{9} - \frac{7}{192} a^{8} + \frac{7}{192} a^{7} - \frac{37}{192} a^{6} - \frac{1}{4} a^{5} + \frac{5}{48} a^{4} - \frac{1}{12} a^{3} - \frac{1}{12} a^{2} - \frac{1}{3}$, $\frac{1}{16896} a^{13} - \frac{9}{5632} a^{12} - \frac{23}{5632} a^{11} - \frac{25}{16896} a^{10} - \frac{43}{5632} a^{9} - \frac{35}{16896} a^{8} - \frac{571}{16896} a^{7} + \frac{11}{768} a^{6} - \frac{113}{384} a^{5} - \frac{249}{704} a^{4} - \frac{109}{1056} a^{3} + \frac{101}{528} a^{2} + \frac{27}{88} a - \frac{1}{66}$, $\frac{1}{1050336477022818301341696} a^{14} + \frac{9352930922570068033}{1050336477022818301341696} a^{13} - \frac{429672868094882706233}{1050336477022818301341696} a^{12} - \frac{223560825875977605949}{116704053002535366815744} a^{11} - \frac{6695770879855386695549}{1050336477022818301341696} a^{10} - \frac{10931220690447228233087}{1050336477022818301341696} a^{9} + \frac{14088864372727706985905}{1050336477022818301341696} a^{8} + \frac{5678943641506817927903}{525168238511409150670848} a^{7} - \frac{1351712854265832809975}{23871283568700415939584} a^{6} - \frac{3388611260717279150767}{6910108401465909877248} a^{5} + \frac{18203582561313804605}{1151684733577651646208} a^{4} + \frac{1451305501623918186929}{2983910446087551992448} a^{3} + \frac{7441261907777163677}{165772802560419555136} a^{2} + \frac{156485008650989633651}{2051438431685191994808} a + \frac{374556712012554153985}{1025719215842595997404}$

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:  $7$
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:  \( 8571116531.23 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T60:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6000
The 40 conjugacy class representatives for [D(5)^3]S(3)=D(5)wrS(3)
Character table for [D(5)^3]S(3)=D(5)wrS(3) is not computed

Intermediate fields

3.1.23.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
997556191Data not computed