Properties

Label 197317.a.197317.1
Conductor $197317$
Discriminant $197317$
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

$y^2 + y = x^5 - 3x^4 - x^3 + 7x^2 - 2x$ (homogenize, simplify)
$y^2 + z^3y = x^5z - 3x^4z^2 - x^3z^3 + 7x^2z^4 - 2xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 - 12x^4 - 4x^3 + 28x^2 - 8x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 7, -1, -3, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 7, -1, -3, 1], R![1]);
 
sage: X = HyperellipticCurve(R([1, -8, 28, -4, -12, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(197317\) \(=\) \( 23^{2} \cdot 373 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(197317\) \(=\) \( 23^{2} \cdot 373 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1048\) \(=\)  \( 2^{3} \cdot 131 \)
\( I_4 \)  \(=\) \(31648\) \(=\)  \( 2^{5} \cdot 23 \cdot 43 \)
\( I_6 \)  \(=\) \(8995208\) \(=\)  \( 2^{3} \cdot 19 \cdot 23 \cdot 31 \cdot 83 \)
\( I_{10} \)  \(=\) \(789268\) \(=\)  \( 2^{2} \cdot 23^{2} \cdot 373 \)
\( J_2 \)  \(=\) \(524\) \(=\)  \( 2^{2} \cdot 131 \)
\( J_4 \)  \(=\) \(6166\) \(=\)  \( 2 \cdot 3083 \)
\( J_6 \)  \(=\) \(101340\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 563 \)
\( J_8 \)  \(=\) \(3770651\) \(=\)  \( 17 \cdot 137 \cdot 1619 \)
\( J_{10} \)  \(=\) \(197317\) \(=\)  \( 23^{2} \cdot 373 \)
\( g_1 \)  \(=\) \(39505397402624/197317\)
\( g_2 \)  \(=\) \(887150662784/197317\)
\( g_3 \)  \(=\) \(27825531840/197317\)

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
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 1 : 1)\) \((2 : 0 : 1)\) \((-1 : 2 : 1)\)
\((1 : -2 : 1)\) \((2 : -1 : 1)\) \((-1 : -3 : 1)\) \((3 : 5 : 1)\) \((3 : -6 : 1)\) \((-3 : 110 : 4)\)
\((-3 : -174 : 4)\) \((11 : 341 : 1)\) \((11 : -342 : 1)\)
Known points
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 1 : 1)\) \((2 : 0 : 1)\) \((-1 : 2 : 1)\)
\((1 : -2 : 1)\) \((2 : -1 : 1)\) \((-1 : -3 : 1)\) \((3 : 5 : 1)\) \((3 : -6 : 1)\) \((-3 : 110 : 4)\)
\((-3 : -174 : 4)\) \((11 : 341 : 1)\) \((11 : -342 : 1)\)
Known points
\((1 : 0 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((2 : -1 : 1)\) \((2 : 1 : 1)\) \((1 : -3 : 1)\)
\((1 : 3 : 1)\) \((-1 : -5 : 1)\) \((-1 : 5 : 1)\) \((3 : -11 : 1)\) \((3 : 11 : 1)\) \((-3 : -284 : 4)\)
\((-3 : 284 : 4)\) \((11 : -683 : 1)\) \((11 : 683 : 1)\)

magma: [C![-3,-174,4],C![-3,110,4],C![-1,-3,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,0,0],C![1,1,1],C![2,-1,1],C![2,0,1],C![3,-6,1],C![3,5,1],C![11,-342,1],C![11,341,1]]; // minimal model
 
magma: [C![-3,-284,4],C![-3,284,4],C![-1,-5,1],C![-1,5,1],C![0,-1,1],C![0,1,1],C![1,-3,1],C![1,0,0],C![1,3,1],C![2,-1,1],C![2,1,1],C![3,-11,1],C![3,11,1],C![11,-683,1],C![11,683,1]]; // 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: \(\Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.745872\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.384210\) \(\infty\)
\((0 : -1 : 1) + (2 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.276908\) \(\infty\)
Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.745872\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.384210\) \(\infty\)
\((0 : -1 : 1) + (2 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.276908\) \(\infty\)
Generator $D_0$ Height Order
\((1 : -3 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-3z^3\) \(0.745872\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.384210\) \(\infty\)
\((0 : -1 : 1) + (2 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.276908\) \(\infty\)

2-torsion field: 5.1.3157072.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.070224 \)
Real period: \( 19.57504 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.374638 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(23\) \(2\) \(2\) \(1\) \(1 + 8 T + 23 T^{2}\)
\(373\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 26 T + 373 T^{2} )\)

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);