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Properties

Label 15.11.9788165887...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 13^{2}\cdot 43^{4}\cdot 433^{4}\cdot 631^{2}\cdot 323441^{2}$
Root discriminant $3975.39$
Ramified primes $2, 5, 7, 13, 43, 433, 631, 323441$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T98

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-76263424000, -400382976000, -906581452800, -1155152550400, -906009477120, -449403866240, -139353010816, -25615490336, -2397580992, -34227000, 12494272, 756052, -13248, -1656, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 1656*x^13 - 13248*x^12 + 756052*x^11 + 12494272*x^10 - 34227000*x^9 - 2397580992*x^8 - 25615490336*x^7 - 139353010816*x^6 - 449403866240*x^5 - 906009477120*x^4 - 1155152550400*x^3 - 906581452800*x^2 - 400382976000*x - 76263424000)
 
gp: K = bnfinit(x^15 - 1656*x^13 - 13248*x^12 + 756052*x^11 + 12494272*x^10 - 34227000*x^9 - 2397580992*x^8 - 25615490336*x^7 - 139353010816*x^6 - 449403866240*x^5 - 906009477120*x^4 - 1155152550400*x^3 - 906581452800*x^2 - 400382976000*x - 76263424000, 1)
 

Normalized defining polynomial

\( x^{15} - 1656 x^{13} - 13248 x^{12} + 756052 x^{11} + 12494272 x^{10} - 34227000 x^{9} - 2397580992 x^{8} - 25615490336 x^{7} - 139353010816 x^{6} - 449403866240 x^{5} - 906009477120 x^{4} - 1155152550400 x^{3} - 906581452800 x^{2} - 400382976000 x - 76263424000 \)

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:  \(978816588753356030957478208479888197420687527936000000=2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 13^{2}\cdot 43^{4}\cdot 433^{4}\cdot 631^{2}\cdot 323441^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3975.39$
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:  $2, 5, 7, 13, 43, 433, 631, 323441$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{7} + \frac{1}{20} a^{4} - \frac{1}{20} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{800} a^{8} - \frac{1}{200} a^{6} + \frac{1}{100} a^{5} + \frac{21}{200} a^{4} + \frac{9}{50} a^{3} + \frac{7}{100} a^{2} - \frac{1}{5}$, $\frac{1}{3200} a^{9} + \frac{1}{200} a^{7} - \frac{1}{100} a^{6} + \frac{1}{800} a^{5} - \frac{3}{100} a^{4} - \frac{3}{400} a^{3} + \frac{1}{5} a^{2} - \frac{1}{20} a$, $\frac{1}{83200} a^{10} + \frac{1}{5200} a^{8} + \frac{1}{650} a^{7} + \frac{241}{20800} a^{6} - \frac{7}{650} a^{5} - \frac{363}{10400} a^{4} - \frac{43}{260} a^{3} - \frac{33}{520} a^{2} + \frac{7}{26} a + \frac{1}{13}$, $\frac{1}{1664000} a^{11} - \frac{9}{416000} a^{9} - \frac{9}{52000} a^{8} - \frac{2047}{416000} a^{7} - \frac{27}{26000} a^{6} + \frac{31}{1664} a^{5} + \frac{2827}{26000} a^{4} + \frac{691}{26000} a^{3} + \frac{1683}{13000} a^{2} + \frac{153}{2600} a + \frac{7}{50}$, $\frac{1}{16640000} a^{12} - \frac{9}{4160000} a^{10} + \frac{47}{1040000} a^{9} + \frac{33}{4160000} a^{8} + \frac{883}{260000} a^{7} - \frac{2969}{416000} a^{6} + \frac{333}{65000} a^{5} + \frac{2251}{260000} a^{4} - \frac{863}{32500} a^{3} - \frac{6399}{26000} a^{2} + \frac{1}{250} a - \frac{2}{25}$, $\frac{1}{66560000000} a^{13} - \frac{31}{1664000000} a^{12} + \frac{1263}{8320000000} a^{11} + \frac{2339}{520000000} a^{10} - \frac{1494347}{16640000000} a^{9} - \frac{544809}{2080000000} a^{8} + \frac{5437501}{1664000000} a^{7} + \frac{3153401}{520000000} a^{6} - \frac{6310333}{2080000000} a^{5} - \frac{17924007}{520000000} a^{4} + \frac{15935431}{104000000} a^{3} - \frac{2920759}{13000000} a^{2} + \frac{1108319}{5200000} a + \frac{95697}{1300000}$, $\frac{1}{8519680000000} a^{14} - \frac{11}{2129920000000} a^{13} + \frac{14883}{1064960000000} a^{12} - \frac{63007}{266240000000} a^{11} - \frac{2704139}{2129920000000} a^{10} - \frac{18275371}{532480000000} a^{9} - \frac{586510751}{1064960000000} a^{8} - \frac{905450401}{266240000000} a^{7} - \frac{250073073}{20480000000} a^{6} + \frac{162029213}{33280000000} a^{5} + \frac{943260783}{66560000000} a^{4} - \frac{794206849}{3328000000} a^{3} + \frac{21984067}{3328000000} a^{2} - \frac{2503661}{83200000} a - \frac{384597}{41600000}$

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:  $12$
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:  \( 12988737439300000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T98:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 71 conjugacy class representatives for [1/2.S(5)^3]3 are not computed
Character table for [1/2.S(5)^3]3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.4$x^{6} + 4 x^{2} + 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
43Data not computed
433Data not computed
631Data not computed
323441Data not computed