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Properties

Label 15.9.47985969425...4704.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{10}\cdot 3^{20}\cdot 11^{4}\cdot 13^{3}\cdot 347^{3}$
Root discriminant $70.05$
Ramified primes $2, 3, 11, 13, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T91

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3872, 34848, 52272, -69036, -81840, 9009, 49848, 10698, -9972, -3406, 840, 495, -26, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 36*x^13 - 26*x^12 + 495*x^11 + 840*x^10 - 3406*x^9 - 9972*x^8 + 10698*x^7 + 49848*x^6 + 9009*x^5 - 81840*x^4 - 69036*x^3 + 52272*x^2 + 34848*x - 3872)
 
gp: K = bnfinit(x^15 - 36*x^13 - 26*x^12 + 495*x^11 + 840*x^10 - 3406*x^9 - 9972*x^8 + 10698*x^7 + 49848*x^6 + 9009*x^5 - 81840*x^4 - 69036*x^3 + 52272*x^2 + 34848*x - 3872, 1)
 

Normalized defining polynomial

\( x^{15} - 36 x^{13} - 26 x^{12} + 495 x^{11} + 840 x^{10} - 3406 x^{9} - 9972 x^{8} + 10698 x^{7} + 49848 x^{6} + 9009 x^{5} - 81840 x^{4} - 69036 x^{3} + 52272 x^{2} + 34848 x - 3872 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-4798596942510120043666504704=-\,2^{10}\cdot 3^{20}\cdot 11^{4}\cdot 13^{3}\cdot 347^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.05$
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:  $2, 3, 11, 13, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{44} a^{12} + \frac{2}{11} a^{10} + \frac{9}{22} a^{9} + \frac{1}{4} a^{8} + \frac{1}{11} a^{7} - \frac{9}{22} a^{6} + \frac{4}{11} a^{5} + \frac{3}{22} a^{4} - \frac{1}{11} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{88} a^{13} + \frac{1}{11} a^{11} - \frac{13}{44} a^{10} - \frac{3}{8} a^{9} - \frac{5}{11} a^{8} - \frac{9}{44} a^{7} - \frac{7}{22} a^{6} - \frac{19}{44} a^{5} + \frac{5}{11} a^{4} + \frac{3}{8} a^{3}$, $\frac{1}{69074242899886954082731519856} a^{14} + \frac{45258553422020915651584675}{17268560724971738520682879964} a^{13} + \frac{83565127312978532350401159}{8634280362485869260341439982} a^{12} + \frac{369619067883810799959275013}{3139738313631225185578705448} a^{11} - \frac{27260560210919039488604067513}{69074242899886954082731519856} a^{10} - \frac{506660584615789663140988957}{1569869156815612592789352724} a^{9} - \frac{4415641974916351934637840757}{34537121449943477041365759928} a^{8} - \frac{604522685997054171496053231}{1569869156815612592789352724} a^{7} + \frac{661617267875944813799530845}{2656701649995652080105058456} a^{6} + \frac{1140049052828100252403508451}{4317140181242934630170719991} a^{5} + \frac{32546975762815292233578127305}{69074242899886954082731519856} a^{4} - \frac{93433664339609412123730365}{1328350824997826040052529228} a^{3} - \frac{46948713709147586375875822}{392467289203903148197338181} a^{2} - \frac{88449020671435096069420451}{784934578407806296394676362} a - \frac{4939208537242704726862418}{392467289203903148197338181}$

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:  $11$
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:  \( 233504331.052 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T91:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 466560
The 72 conjugacy class representatives for [1/2.S(3)^5]S(5) are not computed
Character table for [1/2.S(3)^5]S(5) is not computed

Intermediate fields

5.3.4511.1

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
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 R $15$ $15$ R R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
3Data not computed
$11$11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$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$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
347Data not computed