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Properties

Label 726.a.1452.1
Conductor $726$
Discriminant $-1452$
Mordell-Weil group \(\Z/{10}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + 1)y = 2x^5 + 2x^4 + 6x^3 - 2x^2 - x$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = 2x^5z + 2x^4z^2 + 6x^3z^3 - 2x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = 8x^5 + 9x^4 + 24x^3 - 6x^2 - 4x + 1$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -2, 6, 2, 2]), R([1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -2, 6, 2, 2], R![1, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([1, -4, -6, 24, 9, 8]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(726\) \(=\) \( 2 \cdot 3 \cdot 11^{2} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-1452\) \(=\) \( - 2^{2} \cdot 3 \cdot 11^{2} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(760\) \(=\)  \( 2^{3} \cdot 5 \cdot 19 \)
\( I_4 \)  \(=\) \(-69236\) \(=\)  \( - 2^{2} \cdot 19 \cdot 911 \)
\( I_6 \)  \(=\) \(-16142609\) \(=\)  \( - 7^{3} \cdot 19 \cdot 2477 \)
\( I_{10} \)  \(=\) \(-5808\) \(=\)  \( - 2^{4} \cdot 3 \cdot 11^{2} \)
\( J_2 \)  \(=\) \(380\) \(=\)  \( 2^{2} \cdot 5 \cdot 19 \)
\( J_4 \)  \(=\) \(17556\) \(=\)  \( 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 19 \)
\( J_6 \)  \(=\) \(702601\) \(=\)  \( 19 \cdot 36979 \)
\( J_8 \)  \(=\) \(-10306189\) \(=\)  \( - 19^{2} \cdot 28549 \)
\( J_{10} \)  \(=\) \(-1452\) \(=\)  \( - 2^{2} \cdot 3 \cdot 11^{2} \)
\( g_1 \)  \(=\) \(-1980879200000/363\)
\( g_2 \)  \(=\) \(-7297976000/11\)
\( g_3 \)  \(=\) \(-25363896100/363\)

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

Copy content magma:[C![0,-1,1],C![0,0,1],C![1,0,0]]; // minimal model
 
Copy content magma:[C![0,-1,1],C![0,1,1],C![1,0,0]]; // 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: \(\Z/{10}\Z\)

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

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

2-torsion field: 6.0.52272.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 15.12408 \)
Tamagawa product: \( 2 \)
Torsion order:\( 10 \)
Leading coefficient: \( 0.302481 \)
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?
\(2\) \(1\) \(2\) \(2\) \(-1^*\) \(( 1 - T )( 1 + 2 T + 2 T^{2} )\) yes
\(3\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + T + 3 T^{2} )\) yes
\(11\) \(2\) \(2\) \(1\) \(1\) \(( 1 - T )^{2}\) yes

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.60.1 yes
\(3\) 3.90.1 no

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 66.c
  Elliptic curve isogeny class 11.a

Copy content magma:HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(5\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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);