Properties

Label 743.a.743.1
Conductor 743
Discriminant -743
Mordell-Weil group \(\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = -x^4 + x^2$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 + x^2z^4$ (dehomogenize, simplify)
$y^2 = x^6 - 2x^4 + 2x^3 + 5x^2 + 2x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 1, 0, -1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 1, 0, -1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 2, 5, 2, -2, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(743\) \(=\) \( 743 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-743\) \(=\) \( -743 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-56\) \(=\)  \( - 2^{3} \cdot 7 \)
\( I_4 \)  \(=\) \(7780\) \(=\)  \( 2^{2} \cdot 5 \cdot 389 \)
\( I_6 \)  \(=\) \(-121752\) \(=\)  \( - 2^{3} \cdot 3^{2} \cdot 19 \cdot 89 \)
\( I_{10} \)  \(=\) \(-3043328\) \(=\)  \( - 2^{12} \cdot 743 \)
\( J_2 \)  \(=\) \(-7\) \(=\)  \( -7 \)
\( J_4 \)  \(=\) \(-79\) \(=\)  \( -79 \)
\( J_6 \)  \(=\) \(53\) \(=\)  \( 53 \)
\( J_8 \)  \(=\) \(-1653\) \(=\)  \( - 3 \cdot 19 \cdot 29 \)
\( J_{10} \)  \(=\) \(-743\) \(=\)  \( -743 \)
\( g_1 \)  \(=\) \(16807/743\)
\( g_2 \)  \(=\) \(-27097/743\)
\( g_3 \)  \(=\) \(-2597/743\)

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

All points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((-1 : 1 : 1)\) \((1 : -3 : 1)\)

magma: [C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1]];
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable everywhere.

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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.004576\) \(\infty\)

2-torsion field: 6.0.47552.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.004576 \)
Real period: \( 28.76539 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.131655 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(743\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 24 T + 743 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

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