Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -x^4 + x^3 - x^2 - 2x + 2$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -x^4z^2 + x^3z^3 - x^2z^4 - 2xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 4x^4 + 6x^3 - 4x^2 - 8x + 9$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -2, -1, 1, -1]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -2, -1, 1, -1], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([9, -8, -4, 6, -4, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(241706\) | \(=\) | \( 2 \cdot 17 \cdot 7109 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-483412\) | \(=\) | \( - 2^{2} \cdot 17 \cdot 7109 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1100\) | \(=\) | \( 2^{2} \cdot 5^{2} \cdot 11 \) |
\( I_4 \) | \(=\) | \(7945\) | \(=\) | \( 5 \cdot 7 \cdot 227 \) |
\( I_6 \) | \(=\) | \(5431163\) | \(=\) | \( 547 \cdot 9929 \) |
\( I_{10} \) | \(=\) | \(61876736\) | \(=\) | \( 2^{9} \cdot 17 \cdot 7109 \) |
\( J_2 \) | \(=\) | \(275\) | \(=\) | \( 5^{2} \cdot 11 \) |
\( J_4 \) | \(=\) | \(2820\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 47 \) |
\( J_6 \) | \(=\) | \(-2004\) | \(=\) | \( - 2^{2} \cdot 3 \cdot 167 \) |
\( J_8 \) | \(=\) | \(-2125875\) | \(=\) | \( - 3 \cdot 5^{3} \cdot 5669 \) |
\( J_{10} \) | \(=\) | \(483412\) | \(=\) | \( 2^{2} \cdot 17 \cdot 7109 \) |
\( g_1 \) | \(=\) | \(1572763671875/483412\) | ||
\( g_2 \) | \(=\) | \(14661796875/120853\) | ||
\( g_3 \) | \(=\) | \(-37888125/120853\) |
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);
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Rational points
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) |
\((0 : -2 : 1)\) | \((2 : -2 : 1)\) | \((1 : 4 : 2)\) | \((2 : -7 : 1)\) | \((1 : -13 : 2)\) | \((-7 : 43 : 2)\) |
\((5 : -53 : 3)\) | \((5 : -99 : 3)\) | \((-7 : 292 : 2)\) | \((62 : -199422 : 63)\) | \((62 : -288953 : 63)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) |
\((0 : -2 : 1)\) | \((2 : -2 : 1)\) | \((1 : 4 : 2)\) | \((2 : -7 : 1)\) | \((1 : -13 : 2)\) | \((-7 : 43 : 2)\) |
\((5 : -53 : 3)\) | \((5 : -99 : 3)\) | \((-7 : 292 : 2)\) | \((62 : -199422 : 63)\) | \((62 : -288953 : 63)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((1 : 0 : 1)\) | \((-1 : -2 : 1)\) | \((-1 : 2 : 1)\) | \((0 : -3 : 1)\) |
\((0 : 3 : 1)\) | \((2 : -5 : 1)\) | \((2 : 5 : 1)\) | \((1 : -17 : 2)\) | \((1 : 17 : 2)\) | \((5 : -46 : 3)\) |
\((5 : 46 : 3)\) | \((-7 : -249 : 2)\) | \((-7 : 249 : 2)\) | \((62 : -89531 : 63)\) | \((62 : 89531 : 63)\) |
magma: [C![-7,43,2],C![-7,292,2],C![-1,-1,1],C![-1,1,1],C![0,-2,1],C![0,1,1],C![1,-13,2],C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,4,2],C![2,-7,1],C![2,-2,1],C![5,-99,3],C![5,-53,3],C![62,-288953,63],C![62,-199422,63]]; // minimal model
magma: [C![-7,-249,2],C![-7,249,2],C![-1,-2,1],C![-1,2,1],C![0,-3,1],C![0,3,1],C![1,-17,2],C![1,-1,0],C![1,0,1],C![1,1,0],C![1,17,2],C![2,-5,1],C![2,5,1],C![5,-46,3],C![5,46,3],C![62,-89531,63],C![62,89531,63]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 2z^3\) | \(0.716894\) | \(\infty\) |
\((0 : -2 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 2z^3\) | \(0.804321\) | \(\infty\) |
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.102770\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 2z^3\) | \(0.716894\) | \(\infty\) |
\((0 : -2 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 2z^3\) | \(0.804321\) | \(\infty\) |
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.102770\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -2 : 1) - (1 : 1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 3z^3\) | \(0.716894\) | \(\infty\) |
\((0 : -3 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 3z^3\) | \(0.804321\) | \(\infty\) |
\((1 : 0 : 1) - (1 : 1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + z^3\) | \(0.102770\) | \(\infty\) |
2-torsion field: 5.3.1933648.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.050151 \) |
Real period: | \( 14.54123 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.458518 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(17\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 17 T^{2} )\) | |
\(7109\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 3 T + 7109 T^{2} )\) |
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);