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Properties

Label 15.3.70877518646...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{6}\cdot 5^{5}\cdot 17^{2}\cdot 23^{7}\cdot 97^{4}\cdot 101^{2}\cdot 1997^{2}$
Root discriminant $245.49$
Ramified primes $2, 5, 17, 23, 97, 101, 1997$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T102

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-97000, -339500, -349200, 666875, 1457910, 436965, -1453352, -531618, 16110, 96111, 23004, 1114, -144, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 36*x^13 - 144*x^12 + 1114*x^11 + 23004*x^10 + 96111*x^9 + 16110*x^8 - 531618*x^7 - 1453352*x^6 + 436965*x^5 + 1457910*x^4 + 666875*x^3 - 349200*x^2 - 339500*x - 97000)
 
gp: K = bnfinit(x^15 - 36*x^13 - 144*x^12 + 1114*x^11 + 23004*x^10 + 96111*x^9 + 16110*x^8 - 531618*x^7 - 1453352*x^6 + 436965*x^5 + 1457910*x^4 + 666875*x^3 - 349200*x^2 - 339500*x - 97000, 1)
 

Normalized defining polynomial

\( x^{15} - 36 x^{13} - 144 x^{12} + 1114 x^{11} + 23004 x^{10} + 96111 x^{9} + 16110 x^{8} - 531618 x^{7} - 1453352 x^{6} + 436965 x^{5} + 1457910 x^{4} + 666875 x^{3} - 349200 x^{2} - 339500 x - 97000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(708775186466216358380486589441400000=2^{6}\cdot 5^{5}\cdot 17^{2}\cdot 23^{7}\cdot 97^{4}\cdot 101^{2}\cdot 1997^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $245.49$
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:  $2, 5, 17, 23, 97, 101, 1997$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{10} a^{6} - \frac{2}{5} a^{5} - \frac{1}{2} a^{4} - \frac{3}{10} a^{3} + \frac{3}{10} a^{2}$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{10} - \frac{1}{5} a^{9} - \frac{1}{20} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} + \frac{1}{10} a^{4} - \frac{1}{10} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{100} a^{13} - \frac{1}{100} a^{11} - \frac{1}{25} a^{10} + \frac{9}{100} a^{9} - \frac{4}{25} a^{8} + \frac{3}{50} a^{7} + \frac{1}{5} a^{6} + \frac{21}{50} a^{5} + \frac{19}{50} a^{4} - \frac{7}{20} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{8536329429107638953825180382155653087300} a^{14} - \frac{17069365901169875037423052406390793023}{4268164714553819476912590191077826543650} a^{13} + \frac{95514112402658315953205589134096571}{2134082357276909738456295095538913271825} a^{12} - \frac{45521215445964752717743957484162624569}{4268164714553819476912590191077826543650} a^{11} - \frac{63567259938531859593365904252204490128}{2134082357276909738456295095538913271825} a^{10} - \frac{34727079562364148002980819475749980393}{853632942910763895382518038215565308730} a^{9} + \frac{591660212870466263056346674818287374067}{8536329429107638953825180382155653087300} a^{8} + \frac{24723167200625006980187984450733401631}{2134082357276909738456295095538913271825} a^{7} + \frac{157488225482758115306951677704953877151}{4268164714553819476912590191077826543650} a^{6} - \frac{1887181969066323670684055957986634282237}{4268164714553819476912590191077826543650} a^{5} - \frac{3102799232213529767462691100773719489833}{8536329429107638953825180382155653087300} a^{4} + \frac{158013338865584891252316210032322644916}{426816471455381947691259019107782654365} a^{3} - \frac{559868282754897863497334159789772923593}{1707265885821527790765036076431130617460} a^{2} + \frac{34927885655671169550000814097160942594}{85363294291076389538251803821556530873} a + \frac{35357864560294242268277899264252106238}{85363294291076389538251803821556530873}$

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:  $8$
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:  \( 3756298719080 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T102:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10368000
The 140 conjugacy class representatives for [S(5)^3]S(3)=S(5)wrS(3) are not computed
Character table for [S(5)^3]S(3)=S(5)wrS(3) is not computed

Intermediate fields

3.1.23.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ $15$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{5}$

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$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.5.4.1$x^{5} - 97$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
101Data not computed
1997Data not computed