Properties

Label 68209.a.68209.1
Conductor 68209
Discriminant 68209
Mordell-Weil group \(\Z \times \Z \times \Z\)
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

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = x^5 - 3x^4 + 3x^3 - x$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^5z - 3x^4z^2 + 3x^3z^3 - xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 12x^4 + 14x^3 - 4x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 3, -3, 1], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 3, -3, 1]), R([1, 0, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, -4, 0, 14, -12, 4, 1]))
 

Invariants

Conductor: \( N \)  =  \(68209\) = \( 68209 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(68209\) = \( 68209 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(296\) =  \( 2^{3} \cdot 37 \)
\( I_4 \)  = \(17572\) =  \( 2^{2} \cdot 23 \cdot 191 \)
\( I_6 \)  = \(-5910616\) =  \( - 2^{3} \cdot 738827 \)
\( I_{10} \)  = \(279384064\) =  \( 2^{12} \cdot 68209 \)
\( J_2 \)  = \(37\) =  \( 37 \)
\( J_4 \)  = \(-126\) =  \( - 2 \cdot 3^{2} \cdot 7 \)
\( J_6 \)  = \(12260\) =  \( 2^{2} \cdot 5 \cdot 613 \)
\( J_8 \)  = \(109436\) =  \( 2^{2} \cdot 109 \cdot 251 \)
\( J_{10} \)  = \(68209\) =  \( 68209 \)
\( g_1 \)  = \(69343957/68209\)
\( g_2 \)  = \(-6382278/68209\)
\( g_3 \)  = \(16783940/68209\)

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

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

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((-1 : -1 : 2)\)
\((1 : -2 : 1)\) \((1 : -3 : 2)\) \((-1 : -6 : 2)\) \((1 : -6 : 2)\) \((2 : -6 : 3)\) \((2 : -29 : 3)\)
\((2 : -55 : 5)\) \((2 : -78 : 5)\)

magma: [C![-1,-6,2],C![-1,-1,2],C![0,-1,1],C![0,0,1],C![1,-6,2],C![1,-3,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-78,5],C![2,-55,5],C![2,-29,3],C![2,-6,3]];
 

Number of rational Weierstrass points: \(0\)

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: \(\Z \times \Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((-x + z) x\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2\) \(0.404526\) \(\infty\)
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0.326742\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.375390\) \(\infty\)

2-torsion field: 6.2.4365376.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.044349 \)
Real period: \( 18.00406 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.798465 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(68209\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 164 T + 68209 T^{2} )\)

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 \(\Q\).