Properties

Label 563011.a.563011.1
Conductor 563011
Discriminant -563011
Mordell-Weil group \(\Z \times \Z \times \Z \times \Z\)
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

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Minimal equation

Minimal equation

Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 4, 7, 1, -2], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 4, 7, 1, -2]), R([1, 1, 0, 1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 4, 7, 1, -2], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 18, 29, 6, -6, 0, 1]))
 

$y^2 + (x^3 + x + 1)y = -2x^4 + x^3 + 7x^2 + 4x$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -2x^4z^2 + x^3z^3 + 7x^2z^4 + 4xz^5$ (dehomogenize, simplify)
$y^2 = x^6 - 6x^4 + 6x^3 + 29x^2 + 18x + 1$ (minimize, homogenize)

Invariants

\( N \)  =  \(563011\) = \( 563011 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(-563011\) = \( - 563011 \)
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);
 

\( I_2 \)  = \(2760\) =  \( 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
\( I_4 \)  = \(188772\) =  \( 2^{2} \cdot 3 \cdot 15731 \)
\( I_6 \)  = \(145837416\) =  \( 2^{3} \cdot 3 \cdot 2287 \cdot 2657 \)
\( I_{10} \)  = \(-2306093056\) =  \( - 2^{12} \cdot 563011 \)
\( J_2 \)  = \(345\) =  \( 3 \cdot 5 \cdot 23 \)
\( J_4 \)  = \(2993\) =  \( 41 \cdot 73 \)
\( J_6 \)  = \(30309\) =  \( 3 \cdot 10103 \)
\( J_8 \)  = \(374639\) =  \( 374639 \)
\( J_{10} \)  = \(-563011\) =  \( - 563011 \)
\( g_1 \)  = \(-4887597965625/563011\)
\( g_2 \)  = \(-122903429625/563011\)
\( g_3 \)  = \(-3607528725/563011\)

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![-7,64,6],C![-7,315,6],C![-5,9,6],C![-5,80,6],C![-3,6,1],C![-3,15,2],C![-3,16,2],C![-3,23,1],C![-2,4,1],C![-2,5,1],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-226,5],C![1,-5,1],C![1,-1,0],C![1,0,0],C![1,2,1],C![1,75,5],C![2,-12,1],C![2,1,1],C![4,-64,1],C![4,-5,1],C![57,-203225,20],C![57,-12768,20]];
 

Known points
\((0 : 0 : 1)\) \((1 : 0 : 0)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : -1 : 0)\) \((-1 : 1 : 1)\)
\((1 : 2 : 1)\) \((2 : 1 : 1)\) \((-2 : 4 : 1)\) \((1 : -5 : 1)\) \((-2 : 5 : 1)\) \((4 : -5 : 1)\)
\((-3 : 6 : 1)\) \((-5 : 9 : 6)\) \((2 : -12 : 1)\) \((-3 : 15 : 2)\) \((-3 : 16 : 2)\) \((-3 : 23 : 1)\)
\((4 : -64 : 1)\) \((-7 : 64 : 6)\) \((1 : 75 : 5)\) \((-5 : 80 : 6)\) \((1 : -226 : 5)\) \((-7 : 315 : 6)\)
\((57 : -12768 : 20)\) \((57 : -203225 : 20)\)

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

Number of rational Weierstrass points: \(0\)

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

Generator Height Order
\(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 3z^3\) \(0.821823\) \(\infty\)
\(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.304182\) \(\infty\)
\(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.786746\) \(\infty\)
\(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.606468\) \(\infty\)

2-torsion field: 6.4.563011.1

BSD invariants

Analytic rank: \(4\)   (upper bound)
Mordell-Weil rank: \(4\)
2-Selmer rank:\(4\)
Regulator: \( 0.073355 \)
Real period: \( 16.33052 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.197932 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(563011\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 120 T + 563011 T^{2} )\)

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 \(\Q\).