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Properties

Label 15.15.8995776251...5561.2
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![42544769744, 121109770197, 99761473375, 18214687705, -10179835125, -3397215797, 356354865, 186322787, -4560961, -4867029, -5743, 65358, 549, -424, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 424*x^13 + 549*x^12 + 65358*x^11 - 5743*x^10 - 4867029*x^9 - 4560961*x^8 + 186322787*x^7 + 356354865*x^6 - 3397215797*x^5 - 10179835125*x^4 + 18214687705*x^3 + 99761473375*x^2 + 121109770197*x + 42544769744)
 
gp: K = bnfinit(x^15 - 3*x^14 - 424*x^13 + 549*x^12 + 65358*x^11 - 5743*x^10 - 4867029*x^9 - 4560961*x^8 + 186322787*x^7 + 356354865*x^6 - 3397215797*x^5 - 10179835125*x^4 + 18214687705*x^3 + 99761473375*x^2 + 121109770197*x + 42544769744, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 424 x^{13} + 549 x^{12} + 65358 x^{11} - 5743 x^{10} - 4867029 x^{9} - 4560961 x^{8} + 186322787 x^{7} + 356354865 x^{6} - 3397215797 x^{5} - 10179835125 x^{4} + 18214687705 x^{3} + 99761473375 x^{2} + 121109770197 x + 42544769744 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{121} a^{13} - \frac{4}{121} a^{12} - \frac{27}{121} a^{11} - \frac{28}{121} a^{10} - \frac{38}{121} a^{9} - \frac{1}{11} a^{8} + \frac{35}{121} a^{7} + \frac{1}{11} a^{6} + \frac{40}{121} a^{5} - \frac{9}{121} a^{4} + \frac{46}{121} a^{3} - \frac{50}{121} a^{2} - \frac{6}{121} a + \frac{27}{121}$, $\frac{1}{60319904845095362539509965474232407638316130008505091898691101} a^{14} - \frac{222374795899712904586754346471618689419402787219804015548206}{60319904845095362539509965474232407638316130008505091898691101} a^{13} - \frac{13556694730889900582000023794888925321448811203861769467081557}{60319904845095362539509965474232407638316130008505091898691101} a^{12} + \frac{26662170998431734847188234824969920499596611426313417756810319}{60319904845095362539509965474232407638316130008505091898691101} a^{11} + \frac{6326118837452886524738302390677615569084517501085052019809597}{60319904845095362539509965474232407638316130008505091898691101} a^{10} + \frac{29943139242028941727635609101355523906650381773251227102441995}{60319904845095362539509965474232407638316130008505091898691101} a^{9} + \frac{3749860690469181660994858530920922333140833058418138927132862}{60319904845095362539509965474232407638316130008505091898691101} a^{8} - \frac{19619694537851784890496682182297316320962651829129244248642763}{60319904845095362539509965474232407638316130008505091898691101} a^{7} + \frac{1170516928626789710699089472442091071068010632266797291873832}{60319904845095362539509965474232407638316130008505091898691101} a^{6} - \frac{2055654858729691827798000490874730960506033579133184517481081}{5483627713190487503591815043112037058028739091682281081699191} a^{5} - \frac{25842945328969397369479571639080854552290580682108341876391319}{60319904845095362539509965474232407638316130008505091898691101} a^{4} + \frac{24691395369640370967218702504725709851426789128273027823718144}{60319904845095362539509965474232407638316130008505091898691101} a^{3} - \frac{30147643212757234204219353257672969387421708484788876270064367}{60319904845095362539509965474232407638316130008505091898691101} a^{2} + \frac{28870920134249811720904161668530011393373561185250356850301811}{60319904845095362539509965474232407638316130008505091898691101} a + \frac{22741047145038491465204967712617277858731377484884594122229219}{60319904845095362539509965474232407638316130008505091898691101}$

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$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
17.8.4.1$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}$