Properties

Label 1070.a.2140.1
Conductor $1070$
Discriminant $-2140$
Mordell-Weil group \(\Z \times \Z/{4}\Z\)
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

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

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

Invariants

Conductor: \( N \)  \(=\)  \(1070\) \(=\) \( 2 \cdot 5 \cdot 107 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-2140\) \(=\) \( - 2^{2} \cdot 5 \cdot 107 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(12\) \(=\)  \( 2^{2} \cdot 3 \)
\( I_4 \)  \(=\) \(3321\) \(=\)  \( 3^{4} \cdot 41 \)
\( I_6 \)  \(=\) \(141939\) \(=\)  \( 3^{3} \cdot 7 \cdot 751 \)
\( I_{10} \)  \(=\) \(273920\) \(=\)  \( 2^{9} \cdot 5 \cdot 107 \)
\( J_2 \)  \(=\) \(3\) \(=\)  \( 3 \)
\( J_4 \)  \(=\) \(-138\) \(=\)  \( - 2 \cdot 3 \cdot 23 \)
\( J_6 \)  \(=\) \(-1856\) \(=\)  \( - 2^{6} \cdot 29 \)
\( J_8 \)  \(=\) \(-6153\) \(=\)  \( - 3 \cdot 7 \cdot 293 \)
\( J_{10} \)  \(=\) \(2140\) \(=\)  \( 2^{2} \cdot 5 \cdot 107 \)
\( g_1 \)  \(=\) \(243/2140\)
\( g_2 \)  \(=\) \(-1863/1070\)
\( g_3 \)  \(=\) \(-4176/535\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((-2 : 1 : 1)\) \((1 : -2 : 1)\) \((-2 : 6 : 1)\)
All points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((-2 : 1 : 1)\) \((1 : -2 : 1)\) \((-2 : 6 : 1)\)
All points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : -2 : 1)\)
\((1 : 2 : 1)\) \((-2 : -5 : 1)\) \((-2 : 5 : 1)\)

magma: [C![-2,1,1],C![-2,6,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1]]; // minimal model
 
magma: [C![-2,-5,1],C![-2,5,1],C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

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: \(\Z \times \Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.054348\) \(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(4\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.054348\) \(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(4\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : 1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.054348\) \(\infty\)
\((0 : 1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + z^3\) \(0\) \(4\)

2-torsion field: 6.2.1431125.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.054348 \)
Real period: \( 27.74393 \)
Tamagawa product: \( 2 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.188481 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + T + 2 T^{2} )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 5 T^{2} )\)
\(107\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 16 T + 107 T^{2} )\)

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 \(\Q\).