Properties

Label 270720.b.270720.1
Conductor 270720
Discriminant -270720
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1410, 0, 544, 0, 70, 0, 3], R![0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1410, 0, 544, 0, 70, 0, 3]), R([0, 1]))
 

$y^2 + xy = 3x^6 + 70x^4 + 544x^2 + 1410$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 270720 \)  =  \( 2^{7} \cdot 3^{2} \cdot 5 \cdot 47 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-270720\)  =  \( -1 \cdot 2^{7} \cdot 3^{2} \cdot 5 \cdot 47 \)

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  =  \(-25996160\)  =  \( -1 \cdot 2^{7} \cdot 5 \cdot 151 \cdot 269 \)
\( I_4 \)  =  \(3097408\)  =  \( 2^{6} \cdot 48397 \)
\( I_6 \)  =  \(-26839668372480\)  =  \( -1 \cdot 2^{10} \cdot 3^{2} \cdot 5 \cdot 17971 \cdot 32411 \)
\( I_{10} \)  =  \(-1108869120\)  =  \( -1 \cdot 2^{19} \cdot 3^{2} \cdot 5 \cdot 47 \)
\( J_2 \)  =  \(-3249520\)  =  \( -1 \cdot 2^{4} \cdot 5 \cdot 151 \cdot 269 \)
\( J_4 \)  =  \(439974144002\)  =  \( 2 \cdot 219987072001 \)
\( J_6 \)  =  \(-79428031708101120\)  =  \( -1 \cdot 2^{9} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 47 \cdot 6668079709 \)
\( J_8 \)  =  \(16131432551454029721599\)  =  \( 16131432551454029721599 \)
\( J_{10} \)  =  \(-270720\)  =  \( -1 \cdot 2^{7} \cdot 3^{2} \cdot 5 \cdot 47 \)
\( g_1 \)  =  \(566129906277843097800186880000/423\)
\( g_2 \)  =  \(23588744365074445774311454400/423\)
\( g_3 \)  =  \(3098074718373623337958400\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) \(V_4 \) (GAP id : [4,2])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(V_4 \) (GAP id : [4,2])

Rational points

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
 

This curve is locally solvable except over $\Q_{2}$ and $\Q_{7}$.

magma: [];
 

There are no rational points.

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(0\)

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);
 

2-Selmer rank: \(6\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 1.0

Real period: 3.3835115798391790478970284305

Tamagawa numbers: 1 (p = 2), 1 (p = 3), 1 (p = 5), 1 (p = 47)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
 

Torsion: \(\Z/{2}\Z \times \Z/{2}\Z\)

2-torsion field: splitting field of \(x^{8} - 4 x^{7} - 102 x^{6} + 320 x^{5} + 6277 x^{4} - 13092 x^{3} - 200796 x^{2} + 207396 x + 3390849\) with Galois group $C_2^3$

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 192.c2
  Elliptic curve 1410.j2

Endomorphisms

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).