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Properties

Label 85280.d.682240.1
Conductor $85280$
Discriminant $682240$
Mordell-Weil group \(\Z \oplus \Z\)
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

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

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

Invariants

Conductor: \( N \)  \(=\)  \(85280\) \(=\) \( 2^{5} \cdot 5 \cdot 13 \cdot 41 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(682240\) \(=\) \( 2^{8} \cdot 5 \cdot 13 \cdot 41 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(74\) \(=\)  \( 2 \cdot 37 \)
\( I_4 \)  \(=\) \(-392\) \(=\)  \( - 2^{3} \cdot 7^{2} \)
\( I_6 \)  \(=\) \(-5370\) \(=\)  \( - 2 \cdot 3 \cdot 5 \cdot 179 \)
\( I_{10} \)  \(=\) \(-2665\) \(=\)  \( - 5 \cdot 13 \cdot 41 \)
\( J_2 \)  \(=\) \(148\) \(=\)  \( 2^{2} \cdot 37 \)
\( J_4 \)  \(=\) \(1958\) \(=\)  \( 2 \cdot 11 \cdot 89 \)
\( J_6 \)  \(=\) \(2716\) \(=\)  \( 2^{2} \cdot 7 \cdot 97 \)
\( J_8 \)  \(=\) \(-857949\) \(=\)  \( - 3 \cdot 285983 \)
\( J_{10} \)  \(=\) \(-682240\) \(=\)  \( - 2^{8} \cdot 5 \cdot 13 \cdot 41 \)
\( g_1 \)  \(=\) \(-277375828/2665\)
\( g_2 \)  \(=\) \(-49589287/5330\)
\( g_3 \)  \(=\) \(-929551/10660\)

  (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

Known points
\((1 : 0 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\) \((1 : -1 : 1)\)
\((1 : 1 : 1)\) \((1 : -46 : 4)\) \((1 : 46 : 4)\) \((4 : -339 : 9)\) \((4 : 339 : 9)\)
Known points
\((1 : 0 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\) \((1 : -1 : 1)\)
\((1 : 1 : 1)\) \((1 : -46 : 4)\) \((1 : 46 : 4)\) \((4 : -339 : 9)\) \((4 : 339 : 9)\)
Known points
\((1 : 0 : 0)\) \((0 : -1/2 : 1)\) \((0 : 1/2 : 1)\) \((-1 : -1/2 : 1)\) \((-1 : 1/2 : 1)\) \((1 : -1/2 : 1)\)
\((1 : 1/2 : 1)\) \((1 : -23 : 4)\) \((1 : 23 : 4)\) \((4 : -339/2 : 9)\) \((4 : 339/2 : 9)\)

Copy content magma:[C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-46,4],C![1,-1,1],C![1,0,0],C![1,1,1],C![1,46,4],C![4,-339,9],C![4,339,9]]; // minimal model
 
Copy content magma:[C![-1,-1/2,1],C![-1,1/2,1],C![0,-1/2,1],C![0,1/2,1],C![1,-23,4],C![1,-1/2,1],C![1,0,0],C![1,1/2,1],C![1,23,4],C![4,-339/2,9],C![4,339/2,9]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

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: \(\Z \oplus \Z\)

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

Generator $D_0$ Height Order
\((0 : 1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 + z^3\) \(0.155688\) \(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.178641\) \(\infty\)
Generator $D_0$ Height Order
\((0 : 1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 + z^3\) \(0.155688\) \(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.178641\) \(\infty\)
Generator $D_0$ Height Order
\((0 : 1/2 : 1) + (1 : -1/2 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + 1/2z^3\) \(0.155688\) \(\infty\)
\((1 : -1/2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-1/2z^3\) \(0.178641\) \(\infty\)

2-torsion field: 5.1.2665.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.023037 \)
Real period: \( 12.12601 \)
Tamagawa product: \( 6 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.676108 \)
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?
\(2\) \(5\) \(8\) \(6\) \(-1^*\) \(1\) no
\(5\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + 3 T + 5 T^{2} )\) yes
\(13\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 3 T + 13 T^{2} )\) yes
\(41\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 41 T^{2} )\) 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?
\(2\) 2.6.1 no

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);