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Properties

Label 15.5.94712131745...0000.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{15}\cdot 5^{6}\cdot 7^{8}\cdot 23^{5}\cdot 53^{4}\cdot 359^{2}\cdot 7001803^{2}$
Root discriminant $1579.16$
Ramified primes $2, 5, 7, 23, 53, 359, 7001803$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T97

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1329664000, 0, -1678700800, -1292267200, 553472640, 836860640, 297750656, -23050860, -11113704, -2071125, 179064, 45388, -288, -387, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 387*x^13 - 288*x^12 + 45388*x^11 + 179064*x^10 - 2071125*x^9 - 11113704*x^8 - 23050860*x^7 + 297750656*x^6 + 836860640*x^5 + 553472640*x^4 - 1292267200*x^3 - 1678700800*x^2 + 1329664000)
 
gp: K = bnfinit(x^15 - 387*x^13 - 288*x^12 + 45388*x^11 + 179064*x^10 - 2071125*x^9 - 11113704*x^8 - 23050860*x^7 + 297750656*x^6 + 836860640*x^5 + 553472640*x^4 - 1292267200*x^3 - 1678700800*x^2 + 1329664000, 1)
 

Normalized defining polynomial

\( x^{15} - 387 x^{13} - 288 x^{12} + 45388 x^{11} + 179064 x^{10} - 2071125 x^{9} - 11113704 x^{8} - 23050860 x^{7} + 297750656 x^{6} + 836860640 x^{5} + 553472640 x^{4} - 1292267200 x^{3} - 1678700800 x^{2} + 1329664000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-947121317450814515044219243193351148649984000000=-\,2^{15}\cdot 5^{6}\cdot 7^{8}\cdot 23^{5}\cdot 53^{4}\cdot 359^{2}\cdot 7001803^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1579.16$
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:  $2, 5, 7, 23, 53, 359, 7001803$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{7} + \frac{3}{8} a^{3}$, $\frac{1}{560} a^{10} - \frac{1}{40} a^{9} - \frac{51}{560} a^{8} - \frac{67}{280} a^{7} - \frac{7}{20} a^{6} - \frac{12}{35} a^{5} + \frac{29}{80} a^{4} + \frac{1}{40} a^{3} + \frac{3}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{1120} a^{11} + \frac{33}{1120} a^{9} - \frac{1}{140} a^{8} - \frac{1}{10} a^{7} + \frac{9}{70} a^{6} - \frac{7}{32} a^{5} + \frac{1}{20} a^{4} - \frac{9}{20} a^{2}$, $\frac{1}{2240} a^{12} + \frac{1}{2240} a^{10} - \frac{3}{56} a^{9} - \frac{1}{14} a^{8} + \frac{8}{35} a^{7} - \frac{19}{320} a^{6} + \frac{15}{56} a^{5} - \frac{2}{5} a^{4} + \frac{13}{40} a^{3} - \frac{9}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{44800} a^{13} + \frac{13}{44800} a^{11} + \frac{1}{1400} a^{10} - \frac{473}{11200} a^{9} + \frac{49}{800} a^{8} + \frac{559}{8960} a^{7} + \frac{1747}{5600} a^{6} + \frac{185}{448} a^{5} + \frac{17}{100} a^{4} + \frac{11}{40} a^{3} - \frac{1}{10} a^{2} + \frac{1}{4} a$, $\frac{1}{1896352534110788544684170747900881507459760671820800} a^{14} + \frac{575580546567372447003799712908829994459909453}{237044066763848568085521343487610188432470083977600} a^{13} - \frac{366157407724696208460763213869056448111110709907}{1896352534110788544684170747900881507459760671820800} a^{12} - \frac{89147704226170144279522608707195631316468170987}{237044066763848568085521343487610188432470083977600} a^{11} + \frac{412619823409134517463478548051932963503153953679}{474088133527697136171042686975220376864940167955200} a^{10} + \frac{11161774491856755374869407326832070894371418883947}{237044066763848568085521343487610188432470083977600} a^{9} + \frac{23755407383883647121366694338795084211448091279211}{1896352534110788544684170747900881507459760671820800} a^{8} + \frac{19021400470455704534890643250823629227458231769841}{118522033381924284042760671743805094216235041988800} a^{7} + \frac{29430112622766045336686400559837299040444778905641}{474088133527697136171042686975220376864940167955200} a^{6} - \frac{12151428038576740845191717391789091602558860244047}{59261016690962142021380335871902547108117520994400} a^{5} + \frac{2296697113527640805352490306236751061027112275561}{8465859527280306003054333695986078158302502999200} a^{4} + \frac{106434414189786101618817436805088847037778478281}{423292976364015300152716684799303907915125149960} a^{3} + \frac{85100126893491077374752545372252734235290563051}{846585952728030600305433369598607815830250299920} a^{2} - \frac{6850380652206595407339156855115767955331506603}{42329297636401530015271668479930390791512514996} a + \frac{2435664083432414238497353860683544386356112077}{10582324409100382503817917119982597697878128749}$

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:  $9$
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:  \( 4953255666940000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T97:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 70 conjugacy class representatives for [A(5)^3:2]S(3) are not computed
Character table for [A(5)^3:2]S(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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ $15$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.9.4$x^{6} + 4 x^{2} + 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$5$$\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}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.5.0.1$x^{5} - x + 4$$1$$5$$0$$C_5$$[\ ]^{5}$
7.10.8.1$x^{10} - 7 x^{5} + 147$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
53Data not computed
359Data not computed
7001803Data not computed