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Properties

Label 15.15.8995776251...5561.1
Degree $15$
Signature $[15, 0]$
Discriminant $17^{6}\cdot 97^{10}\cdot 131^{6}$
Root discriminant $460.90$
Ramified primes $17, 97, 131$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $A_8$ (as 15T72)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-24112791818, -63519704967, -5570564384, 13548454466, 1588829161, -1201212236, -133259637, 57970302, 5430859, -1652403, -118255, 27855, 1324, -257, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 - 257*x^13 + 1324*x^12 + 27855*x^11 - 118255*x^10 - 1652403*x^9 + 5430859*x^8 + 57970302*x^7 - 133259637*x^6 - 1201212236*x^5 + 1588829161*x^4 + 13548454466*x^3 - 5570564384*x^2 - 63519704967*x - 24112791818)
 
gp: K = bnfinit(x^15 - 6*x^14 - 257*x^13 + 1324*x^12 + 27855*x^11 - 118255*x^10 - 1652403*x^9 + 5430859*x^8 + 57970302*x^7 - 133259637*x^6 - 1201212236*x^5 + 1588829161*x^4 + 13548454466*x^3 - 5570564384*x^2 - 63519704967*x - 24112791818, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} - 257 x^{13} + 1324 x^{12} + 27855 x^{11} - 118255 x^{10} - 1652403 x^{9} + 5430859 x^{8} + 57970302 x^{7} - 133259637 x^{6} - 1201212236 x^{5} + 1588829161 x^{4} + 13548454466 x^{3} - 5570564384 x^{2} - 63519704967 x - 24112791818 \)

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:  \(8995776251690372238394631846990504365561=17^{6}\cdot 97^{10}\cdot 131^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $460.90$
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:  $17, 97, 131$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{312360664998219959086828380611081839712326549} a^{14} - \frac{122247677044185801607530123561682947648456615}{312360664998219959086828380611081839712326549} a^{13} + \frac{17871418476837897466387073040547383789164619}{44622952142602851298118340087297405673189507} a^{12} + \frac{41301568675412850336495187916201999217848848}{312360664998219959086828380611081839712326549} a^{11} + \frac{72992471963102361576306321629610578740924501}{312360664998219959086828380611081839712326549} a^{10} - \frac{125599200018648571201176251820434393551503463}{312360664998219959086828380611081839712326549} a^{9} + \frac{46106912666074670902280162871770328509026700}{312360664998219959086828380611081839712326549} a^{8} - \frac{121917585186820541768929379712809974013796752}{312360664998219959086828380611081839712326549} a^{7} - \frac{6412814787865166983163082826604435913960905}{44622952142602851298118340087297405673189507} a^{6} - \frac{1739695366712408898788222234651043201509369}{44622952142602851298118340087297405673189507} a^{5} - \frac{9353630828530943034628094473894509121411634}{44622952142602851298118340087297405673189507} a^{4} - \frac{40112509090115232794892643587836271749627760}{312360664998219959086828380611081839712326549} a^{3} - \frac{99381125940700168123470979785815960650605115}{312360664998219959086828380611081839712326549} a^{2} + \frac{1345471886523433214371967385056210093088469}{6645971595706807640145284693852805100262267} a + \frac{43877071719914016364896985386955624748204448}{312360664998219959086828380611081839712326549}$

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:  $14$
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:  \( 2167432538320000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_8$ (as 15T72):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 20160
The 14 conjugacy class representatives for $A_8$
Character table for $A_8$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 8 sibling: data not computed
Degree 28 sibling: data not computed
Degree 35 sibling: data not computed
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ R ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$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}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed
$131$$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
131.2.0.1$x^{2} - x + 14$$1$$2$$0$$C_2$$[\ ]^{2}$
131.4.2.2$x^{4} - 131 x^{2} + 240254$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
131.8.4.1$x^{8} + 205932 x^{4} - 2248091 x^{2} + 10601997156$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$