Properties

Label 731.a.12427.1
Conductor 731
Discriminant -12427
Mordell-Weil group \(\Z/{10}\Z\)
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

$y^2 + (x^3 + x^2)y = x^5 + 2x^4 - x - 3$ (homogenize, simplify)
$y^2 + (x^3 + x^2z)y = x^5z + 2x^4z^2 - xz^5 - 3z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 6x^5 + 9x^4 - 4x - 12$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, -1, 0, 0, 2, 1]), R([0, 0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, -1, 0, 0, 2, 1], R![0, 0, 1, 1]);
 
sage: X = HyperellipticCurve(R([-12, -4, 0, 0, 9, 6, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(731\) = \( 17 \cdot 43 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-12427\) = \( - 17^{2} \cdot 43 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(1920\) =  \( 2^{7} \cdot 3 \cdot 5 \)
\( I_4 \)  = \(-345024\) =  \( - 2^{6} \cdot 3^{2} \cdot 599 \)
\( I_6 \)  = \(-215922240\) =  \( - 2^{6} \cdot 3^{3} \cdot 5 \cdot 67 \cdot 373 \)
\( I_{10} \)  = \(-50900992\) =  \( - 2^{12} \cdot 17^{2} \cdot 43 \)
\( J_2 \)  = \(240\) =  \( 2^{4} \cdot 3 \cdot 5 \)
\( J_4 \)  = \(5994\) =  \( 2 \cdot 3^{4} \cdot 37 \)
\( J_6 \)  = \(167265\) =  \( 3^{4} \cdot 5 \cdot 7 \cdot 59 \)
\( J_8 \)  = \(1053891\) =  \( 3^{5} \cdot 4337 \)
\( J_{10} \)  = \(-12427\) =  \( - 17^{2} \cdot 43 \)
\( g_1 \)  = \(-796262400000/12427\)
\( g_2 \)  = \(-82861056000/12427\)
\( g_3 \)  = \(-9634464000/12427\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -1 : 1),\, (-3 : 9 : 1)\)

magma: [C![-3,9,1],C![1,-1,0],C![1,-1,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(2\)

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: \(\Z/{10}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 + 8xz + 7z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-8xz^2 - 11z^3\) \(0\) \(10\)

2-torsion field: 4.2.688.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 14.92677 \)
Tamagawa product: \( 2 \)
Torsion order:\( 10 \)
Leading coefficient: \( 0.298535 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(17\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 - 3 T + 17 T^{2} )\)
\(43\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 43 T^{2} )\)

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 \(\Q\).