Properties

Label 784.b.25088.1
Conductor $784$
Discriminant $-25088$
Mordell-Weil group \(\Z/{2}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + 1)y = -x^6 - 3x^5 + 7x^4 + 2x^3 - 49x^2 + 41x - 9$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = -x^6 - 3x^5z + 7x^4z^2 + 2x^3z^3 - 49x^2z^4 + 41xz^5 - 9z^6$ (dehomogenize, simplify)
$y^2 = -4x^6 - 12x^5 + 29x^4 + 8x^3 - 194x^2 + 164x - 35$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-9, 41, -49, 2, 7, -3, -1]), R([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-9, 41, -49, 2, 7, -3, -1], R![1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-35, 164, -194, 8, 29, -12, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(784\) \(=\) \( 2^{4} \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-25088\) \(=\) \( - 2^{9} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2740\) \(=\)  \( 2^{2} \cdot 5 \cdot 137 \)
\( I_4 \)  \(=\) \(15382525\) \(=\)  \( 5^{2} \cdot 521 \cdot 1181 \)
\( I_6 \)  \(=\) \(36170522453\) \(=\)  \( 13 \cdot 23 \cdot 6211 \cdot 19477 \)
\( I_{10} \)  \(=\) \(3136\) \(=\)  \( 2^{6} \cdot 7^{2} \)
\( J_2 \)  \(=\) \(2740\) \(=\)  \( 2^{2} \cdot 5 \cdot 137 \)
\( J_4 \)  \(=\) \(-9942200\) \(=\)  \( - 2^{3} \cdot 5^{2} \cdot 49711 \)
\( J_6 \)  \(=\) \(-24298750736\) \(=\)  \( - 2^{4} \cdot 13 \cdot 61 \cdot 149 \cdot 12853 \)
\( J_8 \)  \(=\) \(-41356479464160\) \(=\)  \( - 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 1531 \cdot 8039501 \)
\( J_{10} \)  \(=\) \(25088\) \(=\)  \( 2^{9} \cdot 7^{2} \)
\( g_1 \)  \(=\) \(301635777856250/49\)
\( g_2 \)  \(=\) \(-399451653071875/49\)
\( g_3 \)  \(=\) \(-712598832131225/98\)

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.
This curve has no rational points.
This curve has no rational points.

magma: []; // minimal model
 
magma: []; // simplified model
 

Number of rational Weierstrass points: \(0\)

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

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

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(4x^2 + 12xz - 5z^2\) \(=\) \(0,\) \(8y\) \(=\) \(12xz^2 - 9z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(4x^2 + 12xz - 5z^2\) \(=\) \(0,\) \(8y\) \(=\) \(12xz^2 - 9z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(4x^2 + 12xz - 5z^2\) \(=\) \(0,\) \(8y\) \(=\) \(x^2z + 24xz^2 - 17z^3\) \(0\) \(2\)

2-torsion field: 8.0.3211264.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 0.626116 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.313058 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(9\) \(1\) \(1 + T\)
\(7\) \(2\) \(2\) \(1\) \(( 1 - T )^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 14.a
  Elliptic curve isogeny class 56.b

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 \(\Q\).