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Properties

Label 440509.a.440509.1
Conductor $440509$
Discriminant $440509$
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

$y^2 + (x^3 + x + 1)y = x^5 - x^4 - 5x^3 + 9x + 6$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = x^5z - x^4z^2 - 5x^3z^3 + 9xz^5 + 6z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 2x^4 - 18x^3 + x^2 + 38x + 25$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([6, 9, 0, -5, -1, 1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![6, 9, 0, -5, -1, 1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([25, 38, 1, -18, -2, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(440509\) \(=\) \( 440509 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(440509\) \(=\) \( 440509 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1028\) \(=\)  \( 2^{2} \cdot 257 \)
\( I_4 \)  \(=\) \(71593\) \(=\)  \( 71593 \)
\( I_6 \)  \(=\) \(17110373\) \(=\)  \( 7 \cdot 79 \cdot 30941 \)
\( I_{10} \)  \(=\) \(56385152\) \(=\)  \( 2^{7} \cdot 440509 \)
\( J_2 \)  \(=\) \(257\) \(=\)  \( 257 \)
\( J_4 \)  \(=\) \(-231\) \(=\)  \( - 3 \cdot 7 \cdot 11 \)
\( J_6 \)  \(=\) \(14605\) \(=\)  \( 5 \cdot 23 \cdot 127 \)
\( J_8 \)  \(=\) \(925031\) \(=\)  \( 173 \cdot 5347 \)
\( J_{10} \)  \(=\) \(440509\) \(=\)  \( 440509 \)
\( g_1 \)  \(=\) \(1121154893057/440509\)
\( g_2 \)  \(=\) \(-3921130983/440509\)
\( g_3 \)  \(=\) \(964645645/440509\)

  (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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((-1 : 1 : 1)\) \((0 : 2 : 1)\) \((2 : 0 : 1)\)
\((1 : 2 : 1)\) \((0 : -3 : 1)\) \((-2 : 4 : 1)\) \((1 : -5 : 1)\) \((-2 : 5 : 1)\) \((7 : 5 : 3)\)
\((-1 : 10 : 2)\) \((-3 : 10 : 2)\) \((2 : -11 : 1)\) \((-1 : -13 : 2)\) \((-3 : 14 : 1)\) \((-3 : 15 : 1)\)
\((-3 : 21 : 2)\) \((-4 : 22 : 1)\) \((-4 : 45 : 1)\) \((-7 : 94 : 6)\) \((-7 : 285 : 6)\) \((7 : -438 : 3)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((-1 : 1 : 1)\) \((0 : 2 : 1)\) \((2 : 0 : 1)\)
\((1 : 2 : 1)\) \((0 : -3 : 1)\) \((-2 : 4 : 1)\) \((1 : -5 : 1)\) \((-2 : 5 : 1)\) \((7 : 5 : 3)\)
\((-1 : 10 : 2)\) \((-3 : 10 : 2)\) \((2 : -11 : 1)\) \((-1 : -13 : 2)\) \((-3 : 14 : 1)\) \((-3 : 15 : 1)\)
\((-3 : 21 : 2)\) \((-4 : 22 : 1)\) \((-4 : 45 : 1)\) \((-7 : 94 : 6)\) \((-7 : 285 : 6)\) \((7 : -438 : 3)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\) \((-2 : -1 : 1)\) \((-2 : 1 : 1)\)
\((-3 : -1 : 1)\) \((-3 : 1 : 1)\) \((0 : -5 : 1)\) \((0 : 5 : 1)\) \((1 : -7 : 1)\) \((1 : 7 : 1)\)
\((2 : -11 : 1)\) \((2 : 11 : 1)\) \((-3 : -11 : 2)\) \((-3 : 11 : 2)\) \((-1 : -23 : 2)\) \((-1 : 23 : 2)\)
\((-4 : -23 : 1)\) \((-4 : 23 : 1)\) \((-7 : -191 : 6)\) \((-7 : 191 : 6)\) \((7 : -443 : 3)\) \((7 : 443 : 3)\)

magma: [C![-7,94,6],C![-7,285,6],C![-4,22,1],C![-4,45,1],C![-3,10,2],C![-3,14,1],C![-3,15,1],C![-3,21,2],C![-2,4,1],C![-2,5,1],C![-1,-13,2],C![-1,0,1],C![-1,1,1],C![-1,10,2],C![0,-3,1],C![0,2,1],C![1,-5,1],C![1,-1,0],C![1,0,0],C![1,2,1],C![2,-11,1],C![2,0,1],C![7,-438,3],C![7,5,3]]; // minimal model
 
magma: [C![-7,-191,6],C![-7,191,6],C![-4,-23,1],C![-4,23,1],C![-3,-11,2],C![-3,-1,1],C![-3,1,1],C![-3,11,2],C![-2,-1,1],C![-2,1,1],C![-1,-23,2],C![-1,-1,1],C![-1,1,1],C![-1,23,2],C![0,-5,1],C![0,5,1],C![1,-7,1],C![1,-1,0],C![1,1,0],C![1,7,1],C![2,-11,1],C![2,11,1],C![7,-443,3],C![7,443,3]]; // 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
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(1.072693\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z\) \(0.404197\) \(\infty\)
\((-2 : 4 : 1) - (1 : 0 : 0)\) \(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 4z^3\) \(0.548892\) \(\infty\)
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.333349\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(1.072693\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z\) \(0.404197\) \(\infty\)
\((-2 : 4 : 1) - (1 : 0 : 0)\) \(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 4z^3\) \(0.548892\) \(\infty\)
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.333349\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 + 3z^3\) \(1.072693\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 2x^2z + xz^2 + z^3\) \(0.404197\) \(\infty\)
\((-2 : -1 : 1) - (1 : 1 : 0)\) \(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + xz^2 - 7z^3\) \(0.548892\) \(\infty\)
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 + z^3\) \(0.333349\) \(\infty\)

2-torsion field: 6.2.440509.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(4\)   (upper bound)
Mordell-Weil rank: \(4\)
2-Selmer rank:\(4\)
Regulator: \( 0.062575 \)
Real period: \( 17.94863 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.123153 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(440509\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 372 T + 440509 T^{2} )\)

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);
 

Additional information

The conductor $440509$ of the Jacobian of this curve is the smallest known for a genus $2$ curve with analytic rank $4$.