Properties

Label 784.b.76832.1
Conductor 784
Discriminant -76832
Mordell-Weil group \(\Z/{6}\Z\)
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

Related objects

Learn more about

Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x + 1)y = -x^6 + 4x^5 - 4x^4 - 2x^3 + 10x - 9$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = -x^6 + 4x^5z - 4x^4z^2 - 2x^3z^3 + 10xz^5 - 9z^6$ (dehomogenize, simplify)
$y^2 = -4x^6 + 16x^5 - 16x^4 - 8x^3 + x^2 + 42x - 35$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-9, 10, 0, -2, -4, 4, -1]), R([1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-9, 10, 0, -2, -4, 4, -1], R![1, 1]);
 
sage: X = HyperellipticCurve(R([-35, 42, 1, -8, -16, 16, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(784\) \(=\) \( 2^{4} \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-76832\) \(=\) \( - 2^{5} \cdot 7^{4} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-6080\) \(=\)  \( - 2^{6} \cdot 5 \cdot 19 \)
\( I_4 \)  \(=\) \(2116480\) \(=\)  \( 2^{7} \cdot 5 \cdot 3307 \)
\( I_6 \)  \(=\) \(-3262676224\) \(=\)  \( - 2^{8} \cdot 23 \cdot 554123 \)
\( I_{10} \)  \(=\) \(-314703872\) \(=\)  \( - 2^{17} \cdot 7^{4} \)
\( J_2 \)  \(=\) \(-760\) \(=\)  \( - 2^{3} \cdot 5 \cdot 19 \)
\( J_4 \)  \(=\) \(2020\) \(=\)  \( 2^{2} \cdot 5 \cdot 101 \)
\( J_6 \)  \(=\) \(-6076\) \(=\)  \( - 2^{2} \cdot 7^{2} \cdot 31 \)
\( J_8 \)  \(=\) \(134340\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 2239 \)
\( J_{10} \)  \(=\) \(-76832\) \(=\)  \( - 2^{5} \cdot 7^{4} \)
\( g_1 \)  \(=\) \(7923516800000/2401\)
\( g_2 \)  \(=\) \(27710360000/2401\)
\( g_3 \)  \(=\) \(2238200/49\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

This curve has no rational points.

magma: [];
 

Number of rational Weierstrass points: \(0\)

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

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

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);
 

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 - 3xz + 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - 3z^3\) \(0\) \(6\)

2-torsion field: 8.0.3211264.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 3.756700 \)
Tamagawa product: \( 3 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.313058 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(5\) \(1\) \(1 + T\)
\(7\) \(2\) \(4\) \(3\) \(( 1 - T )^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 56.b2
  Elliptic curve 14.a6

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(4\) 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 \(\Q\).