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Properties

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

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Minimal equation

Minimal equation

Simplified equation

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

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: X = HyperellipticCurve(R([1, 18, 29, 6, -6, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1380\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
\( I_4 \)  \(=\) \(47193\) \(=\)  \( 3 \cdot 15731 \)
\( I_6 \)  \(=\) \(18229677\) \(=\)  \( 3 \cdot 2287 \cdot 2657 \)
\( I_{10} \)  \(=\) \(-72065408\) \(=\)  \( - 2^{7} \cdot 563011 \)
\( J_2 \)  \(=\) \(345\) \(=\)  \( 3 \cdot 5 \cdot 23 \)
\( J_4 \)  \(=\) \(2993\) \(=\)  \( 41 \cdot 73 \)
\( J_6 \)  \(=\) \(30309\) \(=\)  \( 3 \cdot 10103 \)
\( J_8 \)  \(=\) \(374639\) \(=\)  \( 374639 \)
\( J_{10} \)  \(=\) \(-563011\) \(=\)  \( -563011 \)
\( g_1 \)  \(=\) \(-4887597965625/563011\)
\( g_2 \)  \(=\) \(-122903429625/563011\)
\( g_3 \)  \(=\) \(-3607528725/563011\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 2 : 1)\) \((2 : 1 : 1)\) \((-2 : 4 : 1)\) \((1 : -5 : 1)\) \((-2 : 5 : 1)\) \((4 : -5 : 1)\)
\((-3 : 6 : 1)\) \((-5 : 9 : 6)\) \((2 : -12 : 1)\) \((-3 : 15 : 2)\) \((-3 : 16 : 2)\) \((-3 : 23 : 1)\)
\((4 : -64 : 1)\) \((-7 : 64 : 6)\) \((1 : 75 : 5)\) \((-5 : 80 : 6)\) \((1 : -226 : 5)\) \((-7 : 315 : 6)\)
\((57 : -12768 : 20)\) \((57 : -203225 : 20)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 2 : 1)\) \((2 : 1 : 1)\) \((-2 : 4 : 1)\) \((1 : -5 : 1)\) \((-2 : 5 : 1)\) \((4 : -5 : 1)\)
\((-3 : 6 : 1)\) \((-5 : 9 : 6)\) \((2 : -12 : 1)\) \((-3 : 15 : 2)\) \((-3 : 16 : 2)\) \((-3 : 23 : 1)\)
\((4 : -64 : 1)\) \((-7 : 64 : 6)\) \((1 : 75 : 5)\) \((-5 : 80 : 6)\) \((1 : -226 : 5)\) \((-7 : 315 : 6)\)
\((57 : -12768 : 20)\) \((57 : -203225 : 20)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((-2 : -1 : 1)\) \((-2 : 1 : 1)\) \((-3 : -1 : 2)\) \((-3 : 1 : 2)\) \((1 : -7 : 1)\) \((1 : 7 : 1)\)
\((2 : -13 : 1)\) \((2 : 13 : 1)\) \((-3 : -17 : 1)\) \((-3 : 17 : 1)\) \((4 : -59 : 1)\) \((4 : 59 : 1)\)
\((-5 : -71 : 6)\) \((-5 : 71 : 6)\) \((-7 : -251 : 6)\) \((-7 : 251 : 6)\) \((1 : -301 : 5)\) \((1 : 301 : 5)\)
\((57 : -190457 : 20)\) \((57 : 190457 : 20)\)

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]]; // minimal model
 
magma: [C![-7,-251,6],C![-7,251,6],C![-5,-71,6],C![-5,71,6],C![-3,-17,1],C![-3,-1,2],C![-3,1,2],C![-3,17,1],C![-2,-1,1],C![-2,1,1],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-301,5],C![1,-7,1],C![1,-1,0],C![1,1,0],C![1,7,1],C![1,301,5],C![2,-13,1],C![2,13,1],C![4,-59,1],C![4,59,1],C![57,-190457,20],C![57,190457,20]]; // simplified model
 

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: \(\Z \oplus \Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-2 : 5 : 1) - (1 : 0 : 0)\) \(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 3z^3\) \(0.821823\) \(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.304182\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.786746\) \(\infty\)
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.606468\) \(\infty\)
Generator $D_0$ Height Order
\((-2 : 5 : 1) - (1 : 0 : 0)\) \(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 3z^3\) \(0.821823\) \(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.304182\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.786746\) \(\infty\)
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.606468\) \(\infty\)
Generator $D_0$ Height Order
\((-2 : 1 : 1) - (1 : 1 : 0)\) \(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + xz^2 - 5z^3\) \(0.821823\) \(\infty\)
\((-1 : -1 : 1) - (1 : 1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + xz^2 - z^3\) \(0.304182\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 - z^3\) \(0.786746\) \(\infty\)
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 + z^3\) \(0.606468\) \(\infty\)

2-torsion field: 6.4.563011.1

BSD invariants

Hasse-Weil conjecture: unverified
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 Cluster picture
\(563011\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 120 T + 563011 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

magma: HeuristicDecompositionFactors(C);
 

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 \(\Q\).

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);