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Properties

Label 15.1.18494950052...1504.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 13^{5}\cdot 53^{2}\cdot 1453943^{2}$
Root discriminant $76.64$
Ramified primes $2, 13, 53, 1453943$
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![-107657, 359739, -426463, 424387, -263536, 86954, -12671, 12239, -9201, 1407, 319, 77, -54, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 4*x^13 - 54*x^12 + 77*x^11 + 319*x^10 + 1407*x^9 - 9201*x^8 + 12239*x^7 - 12671*x^6 + 86954*x^5 - 263536*x^4 + 424387*x^3 - 426463*x^2 + 359739*x - 107657)
 
gp: K = bnfinit(x^15 - x^14 - 4*x^13 - 54*x^12 + 77*x^11 + 319*x^10 + 1407*x^9 - 9201*x^8 + 12239*x^7 - 12671*x^6 + 86954*x^5 - 263536*x^4 + 424387*x^3 - 426463*x^2 + 359739*x - 107657, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 4 x^{13} - 54 x^{12} + 77 x^{11} + 319 x^{10} + 1407 x^{9} - 9201 x^{8} + 12239 x^{7} - 12671 x^{6} + 86954 x^{5} - 263536 x^{4} + 424387 x^{3} - 426463 x^{2} + 359739 x - 107657 \)

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:  \(-18494950052095344498131861504=-\,2^{23}\cdot 13^{5}\cdot 53^{2}\cdot 1453943^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.64$
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:  $2, 13, 53, 1453943$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} + \frac{3}{8} a$, $\frac{1}{32} a^{12} - \frac{1}{16} a^{11} - \frac{1}{32} a^{10} + \frac{1}{16} a^{9} - \frac{15}{32} a^{6} + \frac{7}{16} a^{5} + \frac{1}{16} a^{4} + \frac{3}{16} a^{3} - \frac{1}{8} a^{2} - \frac{3}{8} a + \frac{9}{32}$, $\frac{1}{256} a^{13} + \frac{1}{256} a^{12} - \frac{3}{256} a^{11} + \frac{3}{256} a^{10} + \frac{11}{128} a^{9} - \frac{3}{32} a^{8} - \frac{71}{256} a^{7} + \frac{25}{256} a^{6} - \frac{7}{32} a^{5} - \frac{9}{32} a^{4} - \frac{55}{128} a^{3} - \frac{17}{64} a^{2} + \frac{105}{256} a - \frac{9}{256}$, $\frac{1}{26758120360236508774425731667968} a^{14} - \frac{6373117467405589219552331291}{3344765045029563596803216458496} a^{13} - \frac{65743104818130027249020698023}{6689530090059127193606432916992} a^{12} - \frac{694062759558702458720776444569}{13379060180118254387212865833984} a^{11} + \frac{627771274311360664905349321291}{26758120360236508774425731667968} a^{10} - \frac{1629123444592311208600672533087}{13379060180118254387212865833984} a^{9} + \frac{842702451590406845793109048593}{26758120360236508774425731667968} a^{8} - \frac{13592438301783612042033906919}{3344765045029563596803216458496} a^{7} - \frac{7888998022739739608623776605737}{26758120360236508774425731667968} a^{6} - \frac{8681342100321439055905313489}{1672382522514781798401608229248} a^{5} + \frac{1734410737006138192498162520973}{13379060180118254387212865833984} a^{4} - \frac{5778781506725809488964347042467}{13379060180118254387212865833984} a^{3} - \frac{6081943293869420188473471272819}{26758120360236508774425731667968} a^{2} - \frac{392333284080801752020661136293}{13379060180118254387212865833984} a - \frac{11827448699841473733756786564767}{26758120360236508774425731667968}$

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:  $7$
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:  \( 508319455.179 \) (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

3.1.104.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ $15$ R ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.10.5.2$x^{10} - 57122 x^{2} + 2227758$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
53Data not computed
1453943Data not computed