Minimal equation
Minimal equation
Simplified equation
$y^2 + (x + 1)y = 2x^5 - 2x^4 - x^2$ | (homogenize, simplify) |
$y^2 + (xz^2 + z^3)y = 2x^5z - 2x^4z^2 - x^2z^4$ | (dehomogenize, simplify) |
$y^2 = 8x^5 - 8x^4 - 3x^2 + 2x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -1, 0, -2, 2]), R([1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -1, 0, -2, 2], R![1, 1]);
sage: X = HyperellipticCurve(R([1, 2, -3, 0, -8, 8]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(3080\) | \(=\) | \( 2^{3} \cdot 5 \cdot 7 \cdot 11 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-67760\) | \(=\) | \( - 2^{4} \cdot 5 \cdot 7 \cdot 11^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(64\) | \(=\) | \( 2^{6} \) |
\( I_4 \) | \(=\) | \(-6176\) | \(=\) | \( - 2^{5} \cdot 193 \) |
\( I_6 \) | \(=\) | \(-164068\) | \(=\) | \( - 2^{2} \cdot 41017 \) |
\( I_{10} \) | \(=\) | \(-271040\) | \(=\) | \( - 2^{6} \cdot 5 \cdot 7 \cdot 11^{2} \) |
\( J_2 \) | \(=\) | \(32\) | \(=\) | \( 2^{5} \) |
\( J_4 \) | \(=\) | \(1072\) | \(=\) | \( 2^{4} \cdot 67 \) |
\( J_6 \) | \(=\) | \(9156\) | \(=\) | \( 2^{2} \cdot 3 \cdot 7 \cdot 109 \) |
\( J_8 \) | \(=\) | \(-214048\) | \(=\) | \( - 2^{5} \cdot 6689 \) |
\( J_{10} \) | \(=\) | \(-67760\) | \(=\) | \( - 2^{4} \cdot 5 \cdot 7 \cdot 11^{2} \) |
\( g_1 \) | \(=\) | \(-2097152/4235\) | ||
\( g_2 \) | \(=\) | \(-2195456/4235\) | ||
\( g_3 \) | \(=\) | \(-83712/605\) |
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
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : -1 : 1)\) | \((1 : -2 : 2)\) | \((2 : 4 : 1)\) |
\((2 : -7 : 1)\) | \((1 : -10 : 2)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : -1 : 1)\) | \((1 : -2 : 2)\) | \((2 : 4 : 1)\) |
\((2 : -7 : 1)\) | \((1 : -10 : 2)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -8 : 2)\) | \((1 : 8 : 2)\) |
\((2 : -11 : 1)\) | \((2 : 11 : 1)\) |
magma: [C![0,-1,1],C![0,0,1],C![1,-10,2],C![1,-2,2],C![1,-1,1],C![1,0,0],C![2,-7,1],C![2,4,1]]; // minimal model
magma: [C![0,-1,1],C![0,1,1],C![1,-8,2],C![1,8,2],C![1,0,1],C![1,0,0],C![2,-11,1],C![2,11,1]]; // simplified model
Number of rational Weierstrass points: \(2\)
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: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.121980\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 - 2xz - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(2x^2 + xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.121980\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 - 2xz - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(2x^2 + xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - z^3\) | \(0.121980\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 - 2xz - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(2x^2 + xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
2-torsion field: \(\Q(\sqrt{5}, \sqrt{-7})\)
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.121980 \) |
Real period: | \( 13.60185 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 0.414790 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(3\) | \(4\) | \(2\) | \(1 + T\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 5 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 4 T + 7 T^{2} )\) | |
\(11\) | \(1\) | \(2\) | \(2\) | \(( 1 - T )( 1 + 4 T + 11 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.180.3 | yes |
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);