Properties

Label 2650.a.5300.1
Conductor $2650$
Discriminant $5300$
Mordell-Weil group \(\Z \oplus \Z/{2}\Z\)
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

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

Copy content sage:R.<x> = PolynomialRing(QQ); Cmin = HyperellipticCurve(R([0, 0, 0, -1, 0, 1]), R([1, 1])) # minimal equation
 
Copy content magma:R<x> := PolynomialRing(Rationals()); Cmin := HyperellipticCurve(R![0, 0, 0, -1, 0, 1], R![1, 1]); // minimal equation
 
Copy content sage:Csim = HyperellipticCurve(R([1, 2, 1, -4, 0, 4])); Csim # simplified equation
 
Copy content magma:Csim, pi := SimplifiedModel(Cmin); Csim; // simplified equation
 

Invariants

Conductor: \( N \)  \(=\)  \(2650\) \(=\) \( 2 \cdot 5^{2} \cdot 53 \)
Copy content magma:Conductor(LSeries(Cmin)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(5300\) \(=\) \( 2^{2} \cdot 5^{2} \cdot 53 \)
Copy content magma:Discriminant(Cmin); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(104\) \(=\)  \( 2^{3} \cdot 13 \)
\( I_4 \)  \(=\) \(280\) \(=\)  \( 2^{3} \cdot 5 \cdot 7 \)
\( I_6 \)  \(=\) \(-1885\) \(=\)  \( - 5 \cdot 13 \cdot 29 \)
\( I_{10} \)  \(=\) \(21200\) \(=\)  \( 2^{4} \cdot 5^{2} \cdot 53 \)
\( J_2 \)  \(=\) \(52\) \(=\)  \( 2^{2} \cdot 13 \)
\( J_4 \)  \(=\) \(66\) \(=\)  \( 2 \cdot 3 \cdot 11 \)
\( J_6 \)  \(=\) \(1209\) \(=\)  \( 3 \cdot 13 \cdot 31 \)
\( J_8 \)  \(=\) \(14628\) \(=\)  \( 2^{2} \cdot 3 \cdot 23 \cdot 53 \)
\( J_{10} \)  \(=\) \(5300\) \(=\)  \( 2^{2} \cdot 5^{2} \cdot 53 \)
\( g_1 \)  \(=\) \(95051008/1325\)
\( g_2 \)  \(=\) \(2320032/1325\)
\( g_3 \)  \(=\) \(817284/1325\)

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

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

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (1 : 0 : 1),\, (1 : -2 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (1 : 0 : 1),\, (1 : -2 : 1)\)
All points: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : -2 : 1),\, (1 : 2 : 1)\)

Copy content magma:[Cmin![-1,0,1],Cmin![0,-1,1],Cmin![0,0,1],Cmin![1,-2,1],Cmin![1,0,0],Cmin![1,0,1]]; // minimal model
 
Copy content magma:[Csim![-1,0,1],Csim![0,-1,1],Csim![0,1,1],Csim![1,-2,1],Csim![1,0,0],Csim![1,2,1]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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

This curve is locally solvable everywhere.

Copy content magma:f,h:=HyperellipticPolynomials(Cmin); 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: \(\Z \oplus \Z/{2}\Z\)

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

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.037039\) \(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.037039\) \(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0.037039\) \(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + z^3\) \(0\) \(2\)

2-torsion field: 4.0.5300.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.037039 \)
Real period: \( 20.25811 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.375177 \)
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?
\(2\) \(1\) \(2\) \(2\) \(1^*\) \(( 1 + T )( 1 + 2 T^{2} )\) yes
\(5\) \(2\) \(2\) \(1\) \(1\) \(1 + 2 T + 5 T^{2}\) yes
\(53\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + 53 T^{2} )\) 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
\(3\) 3.40.1 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(Cmin);
 

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(Cmin); HeuristicEndomorphismDescription(Cmin); HeuristicEndomorphismFieldOfDefinition(Cmin);
 

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