Properties

Label 587.a.587.1
Conductor 587
Discriminant 587
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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This genus 2 curve conjecturally has the smallest conductor among curves whose Jacobian has positive analytic rank.

Minimal equation

Minimal equation

Simplified equation

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, -2, -3, 2, 2, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(587\) \(=\) \( 587 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(587\) \(=\) \( 587 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-120\) \(=\)  \( - 2^{3} \cdot 3 \cdot 5 \)
\( I_4 \)  \(=\) \(5604\) \(=\)  \( 2^{2} \cdot 3 \cdot 467 \)
\( I_6 \)  \(=\) \(-433176\) \(=\)  \( - 2^{3} \cdot 3 \cdot 18049 \)
\( I_{10} \)  \(=\) \(2404352\) \(=\)  \( 2^{12} \cdot 587 \)
\( J_2 \)  \(=\) \(-15\) \(=\)  \( - 3 \cdot 5 \)
\( J_4 \)  \(=\) \(-49\) \(=\)  \( - 7^{2} \)
\( J_6 \)  \(=\) \(501\) \(=\)  \( 3 \cdot 167 \)
\( J_8 \)  \(=\) \(-2479\) \(=\)  \( - 37 \cdot 67 \)
\( J_{10} \)  \(=\) \(587\) \(=\)  \( 587 \)
\( g_1 \)  \(=\) \(-759375/587\)
\( g_2 \)  \(=\) \(165375/587\)
\( g_3 \)  \(=\) \(112725/587\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : -1 : 1)\) \((1 : -2 : 1)\) \((-12 : 28 : 5)\) \((-12 : 1875 : 5)\)

magma: [C![-12,28,5],C![-12,1875,5],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0]];
 

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
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.003773\) \(\infty\)

2-torsion field: 6.2.37568.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.003773 \)
Real period: \( 29.51096 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.111351 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(587\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 36 T + 587 T^{2} )\)

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 \(\Q\).