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Properties

Label 104996.a.419984.1
Conductor $104996$
Discriminant $-419984$
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

$y^2 + y = x^6 - 2x^5 - 5x^4 + 4x^2 + 2x$ (homogenize, simplify)
$y^2 + z^3y = x^6 - 2x^5z - 5x^4z^2 + 4x^2z^4 + 2xz^5$ (dehomogenize, simplify)
$y^2 = 4x^6 - 8x^5 - 20x^4 + 16x^2 + 8x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 4, 0, -5, -2, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 4, 0, -5, -2, 1], R![1]);
 
sage: X = HyperellipticCurve(R([1, 8, 16, 0, -20, -8, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(104996\) \(=\) \( 2^{2} \cdot 26249 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-419984\) \(=\) \( - 2^{4} \cdot 26249 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(200\) \(=\)  \( 2^{3} \cdot 5^{2} \)
\( I_4 \)  \(=\) \(2293\) \(=\)  \( 2293 \)
\( I_6 \)  \(=\) \(100967\) \(=\)  \( 31 \cdot 3257 \)
\( I_{10} \)  \(=\) \(-52498\) \(=\)  \( - 2 \cdot 26249 \)
\( J_2 \)  \(=\) \(200\) \(=\)  \( 2^{3} \cdot 5^{2} \)
\( J_4 \)  \(=\) \(138\) \(=\)  \( 2 \cdot 3 \cdot 23 \)
\( J_6 \)  \(=\) \(13696\) \(=\)  \( 2^{7} \cdot 107 \)
\( J_8 \)  \(=\) \(680039\) \(=\)  \( 680039 \)
\( J_{10} \)  \(=\) \(-419984\) \(=\)  \( - 2^{4} \cdot 26249 \)
\( g_1 \)  \(=\) \(-20000000000/26249\)
\( g_2 \)  \(=\) \(-69000000/26249\)
\( g_3 \)  \(=\) \(-34240000/26249\)

  (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
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((-1 : -1 : 1)\) \((1 : -1 : 1)\) \((-1 : -3 : 2)\) \((-1 : -5 : 2)\) \((1 : 7 : 2)\) \((-2 : -7 : 3)\)
\((1 : -15 : 2)\) \((-3 : -15 : 4)\) \((-2 : -20 : 3)\) \((-3 : -49 : 4)\) \((-1 : -56 : 5)\) \((-1 : -69 : 5)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((-1 : -1 : 1)\) \((1 : -1 : 1)\) \((-1 : -3 : 2)\) \((-1 : -5 : 2)\) \((1 : 7 : 2)\) \((-2 : -7 : 3)\)
\((1 : -15 : 2)\) \((-3 : -15 : 4)\) \((-2 : -20 : 3)\) \((-3 : -49 : 4)\) \((-1 : -56 : 5)\) \((-1 : -69 : 5)\)
Known points
\((1 : -2 : 0)\) \((1 : 2 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : -1 : 1)\) \((1 : 1 : 1)\) \((-1 : -2 : 2)\) \((-1 : 2 : 2)\) \((-2 : -13 : 3)\) \((-2 : 13 : 3)\)
\((-1 : -13 : 5)\) \((-1 : 13 : 5)\) \((1 : -22 : 2)\) \((1 : 22 : 2)\) \((-3 : -34 : 4)\) \((-3 : 34 : 4)\)

magma: [C![-3,-49,4],C![-3,-15,4],C![-2,-20,3],C![-2,-7,3],C![-1,-69,5],C![-1,-56,5],C![-1,-5,2],C![-1,-3,2],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-15,2],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![1,7,2]]; // minimal model
 
magma: [C![-3,-34,4],C![-3,34,4],C![-2,-13,3],C![-2,13,3],C![-1,-13,5],C![-1,13,5],C![-1,-2,2],C![-1,2,2],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-22,2],C![1,-2,0],C![1,-1,1],C![1,1,1],C![1,2,0],C![1,22,2]]; // 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: \(\Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -3 : 2) - (1 : -1 : 0)\) \(z (2x + z)\) \(=\) \(0,\) \(4y\) \(=\) \(4x^3 - z^3\) \(0.358920\) \(\infty\)
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3\) \(0.281052\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0.192560\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : -3 : 2) - (1 : -1 : 0)\) \(z (2x + z)\) \(=\) \(0,\) \(4y\) \(=\) \(4x^3 - z^3\) \(0.358920\) \(\infty\)
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3\) \(0.281052\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0.192560\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -2 : 0)\) \(z (2x + z)\) \(=\) \(0,\) \(4y\) \(=\) \(8x^3 - z^3\) \(0.358920\) \(\infty\)
\((-1 : -1 : 1) - (1 : -2 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(2x^3 + z^3\) \(0.281052\) \(\infty\)
\((0 : -1 : 1) - (1 : -2 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(2x^3 - z^3\) \(0.192560\) \(\infty\)

2-torsion field: 6.4.6719744.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.017277 \)
Real period: \( 17.03709 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.883100 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(4\) \(3\) \(1 + 2 T + 2 T^{2}\)
\(26249\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 160 T + 26249 T^{2} )\)

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);