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Properties

Label 731.a.12427.1
Conductor $731$
Discriminant $-12427$
Mordell-Weil group \(\Z/{10}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-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)
$y^2 = x^6 + 6x^5 + 9x^4 - 4x - 12$ (homogenize, minimize)

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

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(480\) \(=\)  \( 2^{5} \cdot 3 \cdot 5 \)
\( I_4 \)  \(=\) \(-21564\) \(=\)  \( - 2^{2} \cdot 3^{2} \cdot 599 \)
\( I_6 \)  \(=\) \(-3373785\) \(=\)  \( - 3^{3} \cdot 5 \cdot 67 \cdot 373 \)
\( I_{10} \)  \(=\) \(-49708\) \(=\)  \( - 2^{2} \cdot 17^{2} \cdot 43 \)
\( J_2 \)  \(=\) \(240\) \(=\)  \( 2^{4} \cdot 3 \cdot 5 \)
\( J_4 \)  \(=\) \(5994\) \(=\)  \( 2 \cdot 3^{4} \cdot 37 \)
\( J_6 \)  \(=\) \(167265\) \(=\)  \( 3^{4} \cdot 5 \cdot 7 \cdot 59 \)
\( J_8 \)  \(=\) \(1053891\) \(=\)  \( 3^{5} \cdot 4337 \)
\( J_{10} \)  \(=\) \(-12427\) \(=\)  \( - 17^{2} \cdot 43 \)
\( g_1 \)  \(=\) \(-796262400000/12427\)
\( g_2 \)  \(=\) \(-82861056000/12427\)
\( g_3 \)  \(=\) \(-9634464000/12427\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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

Automorphism group

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

Rational points

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

Copy content magma:[C![-3,9,1],C![1,-1,0],C![1,-1,1],C![1,0,0]]; // minimal model
 
Copy content magma:[C![-3,0,1],C![1,-1,0],C![1,0,1],C![1,1,0]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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

This curve is locally solvable everywhere.

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

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 + 8xz + 7z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-8xz^2 - 11z^3\) \(0\) \(10\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 + 8xz + 7z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-8xz^2 - 11z^3\) \(0\) \(10\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(2x^2 + 8xz + 7z^2\) \(=\) \(0,\) \(2y\) \(=\) \(x^3 + x^2z - 16xz^2 - 22z^3\) \(0\) \(10\)

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 Root number L-factor Cluster picture Tame reduction?
\(17\) \(1\) \(2\) \(2\) \(1\) \(( 1 + T )( 1 - 3 T + 17 T^{2} )\) yes
\(43\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 43 T^{2} )\) yes

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.30.3 yes
\(5\) not computed yes

Sato-Tate group

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

Decomposition of the Jacobian

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

Copy content magma:HeuristicDecompositionFactors(C);
 

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 \(\Q\).

Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

Copy content magma:HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

Copy content magma:HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);