Properties

Label 816.a.39168.1
Conductor $816$
Discriminant $39168$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z \times \Z/{6}\Z\)
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

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + 1)y = 3x^5 - 4x^3 - x^2 + x$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = 3x^5z - 4x^3z^3 - x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 12x^5 + x^4 - 16x^3 - 2x^2 + 4x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -1, -4, 0, 3]), R([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -1, -4, 0, 3], R![1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 4, -2, -16, 1, 12]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(816\) \(=\) \( 2^{4} \cdot 3 \cdot 17 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(39168\) \(=\) \( 2^{8} \cdot 3^{2} \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(436\) \(=\)  \( 2^{2} \cdot 109 \)
\( I_4 \)  \(=\) \(3373\) \(=\)  \( 3373 \)
\( I_6 \)  \(=\) \(434667\) \(=\)  \( 3 \cdot 144889 \)
\( I_{10} \)  \(=\) \(4896\) \(=\)  \( 2^{5} \cdot 3^{2} \cdot 17 \)
\( J_2 \)  \(=\) \(436\) \(=\)  \( 2^{2} \cdot 109 \)
\( J_4 \)  \(=\) \(5672\) \(=\)  \( 2^{3} \cdot 709 \)
\( J_6 \)  \(=\) \(77824\) \(=\)  \( 2^{12} \cdot 19 \)
\( J_8 \)  \(=\) \(439920\) \(=\)  \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \cdot 47 \)
\( J_{10} \)  \(=\) \(39168\) \(=\)  \( 2^{8} \cdot 3^{2} \cdot 17 \)
\( g_1 \)  \(=\) \(61544958196/153\)
\( g_2 \)  \(=\) \(1836351122/153\)
\( g_3 \)  \(=\) \(57789184/153\)

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

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

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -1 : 1),\, (1 : -1 : 1),\, (-1 : -15 : 3)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -1 : 1),\, (1 : -1 : 1),\, (-1 : -15 : 3)\)
All points: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 0 : 1),\, (-1 : 0 : 3)\)

magma: [C![-1,-15,3],C![-1,-1,1],C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0]]; // minimal model
 
magma: [C![-1,0,3],C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,0,1],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(4\)

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: \(\Z/{2}\Z \times \Z/{2}\Z \times \Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\((-1 : -15 : 3) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (3x + z)\) \(=\) \(0,\) \(3y\) \(=\) \(-xz^2 - 2z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(3x^2 - z^2\) \(=\) \(0,\) \(3y\) \(=\) \(-z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\((-1 : -15 : 3) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (3x + z)\) \(=\) \(0,\) \(3y\) \(=\) \(-xz^2 - 2z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(3x^2 - z^2\) \(=\) \(0,\) \(3y\) \(=\) \(-z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\((1 : 0 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \((x - z) (3x + z)\) \(=\) \(0,\) \(3y\) \(=\) \(x^2z - 2xz^2 - 3z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(3x^2 - z^2\) \(=\) \(0,\) \(3y\) \(=\) \(x^2z - z^3\) \(0\) \(6\)

2-torsion field: \(\Q(\sqrt{17}) \)

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(3\)
Regulator: \( 1 \)
Real period: \( 22.16669 \)
Tamagawa product: \( 8 \)
Torsion order:\( 24 \)
Leading coefficient: \( 0.307870 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(8\) \(4\) \(1 + T\)
\(3\) \(1\) \(2\) \(2\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(17\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 6 T + 17 T^{2} )\)

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 \(\Q\).