Properties

Label 1145.a.1145.1
Conductor 1145
Discriminant 1145
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

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

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

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

Invariants

\( N \)  =  \(1145\) = \( 5 \cdot 229 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(1145\) = \( 5 \cdot 229 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(936\) =  \( 2^{3} \cdot 3^{2} \cdot 13 \)
\( I_4 \)  = \(21348\) =  \( 2^{2} \cdot 3^{2} \cdot 593 \)
\( I_6 \)  = \(6169320\) =  \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 17137 \)
\( I_{10} \)  = \(4689920\) =  \( 2^{12} \cdot 5 \cdot 229 \)
\( J_2 \)  = \(117\) =  \( 3^{2} \cdot 13 \)
\( J_4 \)  = \(348\) =  \( 2^{2} \cdot 3 \cdot 29 \)
\( J_6 \)  = \(224\) =  \( 2^{5} \cdot 7 \)
\( J_8 \)  = \(-23724\) =  \( - 2^{2} \cdot 3^{2} \cdot 659 \)
\( J_{10} \)  = \(1145\) =  \( 5 \cdot 229 \)
\( g_1 \)  = \(21924480357/1145\)
\( g_2 \)  = \(557361324/1145\)
\( g_3 \)  = \(3066336/1145\)

Automorphism group

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

Rational points

magma: [C![0,-1,1],C![0,0,1],C![1,-5,2],C![1,-4,2],C![1,-1,0],C![1,-1,1],C![1,0,0]];
 

Points: \((0 : 0 : 1),\, (1 : 0 : 0),\, (0 : -1 : 1),\, (1 : -1 : 0),\, (1 : -1 : 1),\, (1 : -4 : 2),\, (1 : -5 : 2)\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(1\)

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);
 

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z \times \Z/{2}\Z\)

Generator Height Order
\(z (2x - z)\) \(=\) \(0,\) \(2y\) \(=\) \(-z^3\) \(0.026503\) \(\infty\)
\(x^2 - 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-4xz^2 + z^3\) \(0\) \(2\)

2-torsion field: 6.6.6555125.1

BSD invariants

Analytic rank: \(1\)   (upper bound)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.026503 \)
Real period: \( 29.58514 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.196027 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 3 T + 5 T^{2} )\)
\(229\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 229 T^{2} )\)

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 \(\Q\).