Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = 2x^4 + 2x^3 - x$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = 2x^4z^2 + 2x^3z^3 - xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 8x^4 + 10x^3 - 4x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 2, 2]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 2, 2], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([1, -4, 0, 10, 8, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(79154\) | \(=\) | \( 2 \cdot 19 \cdot 2083 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-158308\) | \(=\) | \( - 2^{2} \cdot 19 \cdot 2083 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(180\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 5 \) |
\( I_4 \) | \(=\) | \(31929\) | \(=\) | \( 3 \cdot 29 \cdot 367 \) |
\( I_6 \) | \(=\) | \(1243629\) | \(=\) | \( 3^{2} \cdot 138181 \) |
\( I_{10} \) | \(=\) | \(-20263424\) | \(=\) | \( - 2^{9} \cdot 19 \cdot 2083 \) |
\( J_2 \) | \(=\) | \(45\) | \(=\) | \( 3^{2} \cdot 5 \) |
\( J_4 \) | \(=\) | \(-1246\) | \(=\) | \( - 2 \cdot 7 \cdot 89 \) |
\( J_6 \) | \(=\) | \(-432\) | \(=\) | \( - 2^{4} \cdot 3^{3} \) |
\( J_8 \) | \(=\) | \(-392989\) | \(=\) | \( - 17 \cdot 23117 \) |
\( J_{10} \) | \(=\) | \(-158308\) | \(=\) | \( - 2^{2} \cdot 19 \cdot 2083 \) |
\( g_1 \) | \(=\) | \(-184528125/158308\) | ||
\( g_2 \) | \(=\) | \(56770875/79154\) | ||
\( g_3 \) | \(=\) | \(218700/39577\) |
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);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : 1 : 1)\) | \((1 : -1 : 2)\) | \((-2 : -2 : 1)\) | \((1 : -3 : 1)\) | \((1 : -8 : 2)\) | \((-2 : 9 : 1)\) |
\((1 : -9 : 3)\) | \((1 : -19 : 3)\) | \((-3 : 32 : 4)\) | \((-3 : -69 : 4)\) | \((-5 : 477 : 12)\) | \((-5 : -2080 : 12)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : 1 : 1)\) | \((1 : -1 : 2)\) | \((-2 : -2 : 1)\) | \((1 : -3 : 1)\) | \((1 : -8 : 2)\) | \((-2 : 9 : 1)\) |
\((1 : -9 : 3)\) | \((1 : -19 : 3)\) | \((-3 : 32 : 4)\) | \((-3 : -69 : 4)\) | \((-5 : 477 : 12)\) | \((-5 : -2080 : 12)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -2 : 1)\) | \((-1 : 2 : 1)\) |
\((1 : -4 : 1)\) | \((1 : 4 : 1)\) | \((1 : -7 : 2)\) | \((1 : 7 : 2)\) | \((1 : -10 : 3)\) | \((1 : 10 : 3)\) |
\((-2 : -11 : 1)\) | \((-2 : 11 : 1)\) | \((-3 : -101 : 4)\) | \((-3 : 101 : 4)\) | \((-5 : -2557 : 12)\) | \((-5 : 2557 : 12)\) |
magma: [C![-5,-2080,12],C![-5,477,12],C![-3,-69,4],C![-3,32,4],C![-2,-2,1],C![-2,9,1],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-19,3],C![1,-9,3],C![1,-8,2],C![1,-3,1],C![1,-1,0],C![1,-1,2],C![1,0,0],C![1,1,1]]; // minimal model
magma: [C![-5,-2557,12],C![-5,2557,12],C![-3,-101,4],C![-3,101,4],C![-2,-11,1],C![-2,11,1],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-10,3],C![1,10,3],C![1,-7,2],C![1,-4,1],C![1,-1,0],C![1,7,2],C![1,1,0],C![1,4,1]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.303488\) | \(\infty\) |
\((0 : 0 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.341639\) | \(\infty\) |
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.288905\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.303488\) | \(\infty\) |
\((0 : 0 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.341639\) | \(\infty\) |
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.288905\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - z^3\) | \(0.303488\) | \(\infty\) |
\((0 : 1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + z^3\) | \(0.341639\) | \(\infty\) |
\((-1 : -2 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - z^3\) | \(0.288905\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.021340 \) |
Real period: | \( 18.35420 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.783381 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(19\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 19 T^{2} )\) | |
\(2083\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 26 T + 2083 T^{2} )\) |
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);