Properties

Label 154006.a.308012.1
Conductor $154006$
Discriminant $-308012$
Mordell-Weil group \(\Z \oplus \Z \oplus \Z\)
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

$y^2 + xy = x^6 - 3x^5 + x^4 + 2x^3 + x^2 - 2x$ (homogenize, simplify)
$y^2 + xz^2y = x^6 - 3x^5z + x^4z^2 + 2x^3z^3 + x^2z^4 - 2xz^5$ (dehomogenize, simplify)
$y^2 = 4x^6 - 12x^5 + 4x^4 + 8x^3 + 5x^2 - 8x$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 1, 2, 1, -3, 1]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 1, 2, 1, -3, 1], R![0, 1]);
 
sage: X = HyperellipticCurve(R([0, -8, 5, 8, 4, -12, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(154006\) \(=\) \( 2 \cdot 77003 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-308012\) \(=\) \( - 2^{2} \cdot 77003 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(976\) \(=\)  \( 2^{4} \cdot 61 \)
\( I_4 \)  \(=\) \(5992\) \(=\)  \( 2^{3} \cdot 7 \cdot 107 \)
\( I_6 \)  \(=\) \(1214103\) \(=\)  \( 3 \cdot 11 \cdot 36791 \)
\( I_{10} \)  \(=\) \(-1232048\) \(=\)  \( - 2^{4} \cdot 77003 \)
\( J_2 \)  \(=\) \(488\) \(=\)  \( 2^{3} \cdot 61 \)
\( J_4 \)  \(=\) \(8924\) \(=\)  \( 2^{2} \cdot 23 \cdot 97 \)
\( J_6 \)  \(=\) \(269489\) \(=\)  \( 11 \cdot 24499 \)
\( J_8 \)  \(=\) \(12968214\) \(=\)  \( 2 \cdot 3 \cdot 7 \cdot 131 \cdot 2357 \)
\( J_{10} \)  \(=\) \(-308012\) \(=\)  \( - 2^{2} \cdot 77003 \)
\( g_1 \)  \(=\) \(-6918932897792/77003\)
\( g_2 \)  \(=\) \(-259274040832/77003\)
\( g_3 \)  \(=\) \(-16044297104/77003\)

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
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\) \((2 : 0 : 1)\)
\((-1 : -2 : 1)\) \((2 : -2 : 1)\) \((-1 : 3 : 1)\) \((-2 : -12 : 1)\) \((-2 : 14 : 1)\) \((-1 : -19 : 3)\)
\((-1 : 28 : 3)\) \((8 : -68 : 5)\) \((8 : -132 : 5)\) \((18 : -966 : 11)\) \((18 : -1212 : 11)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\) \((2 : 0 : 1)\)
\((-1 : -2 : 1)\) \((2 : -2 : 1)\) \((-1 : 3 : 1)\) \((-2 : -12 : 1)\) \((-2 : 14 : 1)\) \((-1 : -19 : 3)\)
\((-1 : 28 : 3)\) \((8 : -68 : 5)\) \((8 : -132 : 5)\) \((18 : -966 : 11)\) \((18 : -1212 : 11)\)
Known points
\((1 : -2 : 0)\) \((1 : 2 : 0)\) \((0 : 0 : 1)\) \((1 : -1 : 1)\) \((1 : 1 : 1)\) \((2 : -2 : 1)\)
\((2 : 2 : 1)\) \((-1 : -5 : 1)\) \((-1 : 5 : 1)\) \((-2 : -26 : 1)\) \((-2 : 26 : 1)\) \((-1 : -47 : 3)\)
\((-1 : 47 : 3)\) \((8 : -64 : 5)\) \((8 : 64 : 5)\) \((18 : -246 : 11)\) \((18 : 246 : 11)\)

magma: [C![-2,-12,1],C![-2,14,1],C![-1,-19,3],C![-1,-2,1],C![-1,3,1],C![-1,28,3],C![0,0,1],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![2,-2,1],C![2,0,1],C![8,-132,5],C![8,-68,5],C![18,-1212,11],C![18,-966,11]]; // minimal model
 
magma: [C![-2,-26,1],C![-2,26,1],C![-1,-47,3],C![-1,-5,1],C![-1,5,1],C![-1,47,3],C![0,0,1],C![1,-2,0],C![1,-1,1],C![1,1,1],C![1,2,0],C![2,-2,1],C![2,2,1],C![8,-64,5],C![8,64,5],C![18,-246,11],C![18,246,11]]; // 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: \(\Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) + (2 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.494631\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + z^3\) \(0.368691\) \(\infty\)
\((0 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.338870\) \(\infty\)
Generator $D_0$ Height Order
\((0 : 0 : 1) + (2 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.494631\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + z^3\) \(0.368691\) \(\infty\)
\((0 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.338870\) \(\infty\)
Generator $D_0$ Height Order
\((0 : 0 : 1) + (2 : 2 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0.494631\) \(\infty\)
\(D_0 - (1 : -2 : 0) - (1 : 2 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + 2z^3\) \(0.368691\) \(\infty\)
\((0 : 0 : 1) + (1 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0.338870\) \(\infty\)

2-torsion field: 5.3.1232048.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.044393 \)
Real period: \( 15.80830 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.403564 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 2 T^{2} )\)
\(77003\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 229 T + 77003 T^{2} )\)

Galois representations

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

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.6.1 no

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);