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Properties

Label 15.15.1567340637...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{15}\cdot 5^{6}\cdot 7^{10}\cdot 29^{2}\cdot 965467^{4}\cdot 3851089843^{2}$
Root discriminant $16{,}330.93$
Ramified primes $2, 5, 7, 29, 965467, 3851089843$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T95

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-494319104000, -2595175296000, -5876218348800, -7487389678400, -5872510955520, -2913097505040, -903885633936, -166917328956, -16154451432, -440833995, 44214032, 2716937, -24768, -3096, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3096*x^13 - 24768*x^12 + 2716937*x^11 + 44214032*x^10 - 440833995*x^9 - 16154451432*x^8 - 166917328956*x^7 - 903885633936*x^6 - 2913097505040*x^5 - 5872510955520*x^4 - 7487389678400*x^3 - 5876218348800*x^2 - 2595175296000*x - 494319104000)
 
gp: K = bnfinit(x^15 - 3096*x^13 - 24768*x^12 + 2716937*x^11 + 44214032*x^10 - 440833995*x^9 - 16154451432*x^8 - 166917328956*x^7 - 903885633936*x^6 - 2913097505040*x^5 - 5872510955520*x^4 - 7487389678400*x^3 - 5876218348800*x^2 - 2595175296000*x - 494319104000, 1)
 

Normalized defining polynomial

\( x^{15} - 3096 x^{13} - 24768 x^{12} + 2716937 x^{11} + 44214032 x^{10} - 440833995 x^{9} - 16154451432 x^{8} - 166917328956 x^{7} - 903885633936 x^{6} - 2913097505040 x^{5} - 5872510955520 x^{4} - 7487389678400 x^{3} - 5876218348800 x^{2} - 2595175296000 x - 494319104000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1567340637715370810317104340867709615539728651698449490432000000=2^{15}\cdot 5^{6}\cdot 7^{10}\cdot 29^{2}\cdot 965467^{4}\cdot 3851089843^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16{,}330.93$
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:  $2, 5, 7, 29, 965467, 3851089843$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{10} a^{7} + \frac{2}{5} a^{4} + \frac{1}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{100} a^{8} - \frac{1}{25} a^{6} + \frac{2}{25} a^{5} + \frac{9}{100} a^{4} + \frac{11}{25} a^{3} - \frac{19}{100} a^{2} + \frac{2}{5}$, $\frac{1}{5600} a^{9} - \frac{1}{350} a^{8} + \frac{3}{175} a^{7} - \frac{4}{175} a^{6} - \frac{519}{5600} a^{5} + \frac{2}{7} a^{4} - \frac{2323}{5600} a^{3} - \frac{87}{700} a^{2} - \frac{33}{280} a + \frac{27}{70}$, $\frac{1}{1265600} a^{10} + \frac{43}{19775} a^{8} + \frac{1479}{39550} a^{7} - \frac{10011}{253120} a^{6} - \frac{2743}{79100} a^{5} - \frac{28631}{253120} a^{4} - \frac{71177}{158200} a^{3} + \frac{32779}{316400} a^{2} - \frac{197}{452} a - \frac{1453}{3955}$, $\frac{1}{50624000} a^{11} - \frac{1}{2531200} a^{10} - \frac{47}{1808000} a^{9} + \frac{211}{1582000} a^{8} - \frac{2146503}{50624000} a^{7} - \frac{212697}{12656000} a^{6} + \frac{994421}{10124800} a^{5} - \frac{2559843}{12656000} a^{4} + \frac{2564223}{6328000} a^{3} - \frac{181047}{395500} a^{2} + \frac{213109}{632800} a + \frac{34823}{158200}$, $\frac{1}{506240000} a^{12} + \frac{31}{126560000} a^{10} - \frac{1223}{31640000} a^{9} - \frac{287169}{72320000} a^{8} - \frac{155769}{4520000} a^{7} - \frac{8679211}{101248000} a^{6} + \frac{4852901}{63280000} a^{5} - \frac{906257}{63280000} a^{4} + \frac{7515107}{15820000} a^{3} - \frac{2628007}{6328000} a^{2} - \frac{122333}{395500} a + \frac{27707}{79100}$, $\frac{1}{1012480000000} a^{13} + \frac{19}{25312000000} a^{12} + \frac{1083}{126560000000} a^{11} - \frac{1227}{15820000000} a^{10} - \frac{33002103}{1012480000000} a^{9} - \frac{528287591}{126560000000} a^{8} - \frac{2953165151}{202496000000} a^{7} + \frac{1781631643}{63280000000} a^{6} + \frac{331680683}{36160000000} a^{5} + \frac{9672743649}{63280000000} a^{4} + \frac{218075969}{1808000000} a^{3} - \frac{245415237}{1582000000} a^{2} + \frac{105064767}{632800000} a + \frac{14459521}{158200000}$, $\frac{1}{3369533440000000} a^{14} + \frac{149}{842383360000000} a^{13} + \frac{51929}{60170240000000} a^{12} - \frac{368107}{105297920000000} a^{11} + \frac{1282052489}{3369533440000000} a^{10} - \frac{47997976959}{842383360000000} a^{9} - \frac{13333997258363}{3369533440000000} a^{8} - \frac{11225994675473}{842383360000000} a^{7} - \frac{396148613027}{842383360000000} a^{6} + \frac{1567652154641}{26324480000000} a^{5} + \frac{56524944926479}{210595840000000} a^{4} + \frac{432416692223}{10529792000000} a^{3} + \frac{3063299673571}{10529792000000} a^{2} + \frac{58306559587}{263244800000} a + \frac{46232636139}{131622400000}$

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:  $14$
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:  \( 1177083894170000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T95:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1296000
The 53 conjugacy class representatives for [A(5)^3:2]3 are not computed
Character table for [A(5)^3:2]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 sibling: 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $15$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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$$[\ ]^{3}$
2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.6.9.6$x^{6} + 4 x^{2} + 8$$2$$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$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
965467Data not computed
3851089843Data not computed