Minimal equation
Minimal equation
Simplified equation
$y^2 + x^3y = 4x^2 - 4x + 1$ | (homogenize, simplify) |
$y^2 + x^3y = 4x^2z^4 - 4xz^5 + z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 16x^2 - 16x + 4$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -4, 4]), R([0, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -4, 4], R![0, 0, 0, 1]);
sage: X = HyperellipticCurve(R([4, -16, 16, 0, 0, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(3860\) | \(=\) | \( 2^{2} \cdot 5 \cdot 193 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-77200\) | \(=\) | \( - 2^{4} \cdot 5^{2} \cdot 193 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(120\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \) |
\( I_4 \) | \(=\) | \(3477\) | \(=\) | \( 3 \cdot 19 \cdot 61 \) |
\( I_6 \) | \(=\) | \(133257\) | \(=\) | \( 3 \cdot 43 \cdot 1033 \) |
\( I_{10} \) | \(=\) | \(9650\) | \(=\) | \( 2 \cdot 5^{2} \cdot 193 \) |
\( J_2 \) | \(=\) | \(120\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \) |
\( J_4 \) | \(=\) | \(-1718\) | \(=\) | \( - 2 \cdot 859 \) |
\( J_6 \) | \(=\) | \(-37184\) | \(=\) | \( - 2^{6} \cdot 7 \cdot 83 \) |
\( J_8 \) | \(=\) | \(-1853401\) | \(=\) | \( - 11 \cdot 168491 \) |
\( J_{10} \) | \(=\) | \(77200\) | \(=\) | \( 2^{4} \cdot 5^{2} \cdot 193 \) |
\( g_1 \) | \(=\) | \(62208000/193\) | ||
\( g_2 \) | \(=\) | \(-7421760/193\) | ||
\( g_3 \) | \(=\) | \(-1338624/193\) |
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
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : 0 : 2)\) | \((1 : -1 : 2)\) |
\((2 : 1 : 1)\) | \((2 : -9 : 1)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : 0 : 2)\) | \((1 : -1 : 2)\) |
\((2 : 1 : 1)\) | \((2 : -9 : 1)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -2 : 1)\) | \((0 : 2 : 1)\) | \((1 : -1 : 2)\) | \((1 : 1 : 2)\) |
\((2 : -10 : 1)\) | \((2 : 10 : 1)\) |
magma: [C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,-1,2],C![1,0,0],C![1,0,2],C![2,-9,1],C![2,1,1]]; // minimal model
magma: [C![0,-2,1],C![0,2,1],C![1,-1,0],C![1,-1,2],C![1,1,0],C![1,1,2],C![2,-10,1],C![2,10,1]]; // 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: \(\Z \oplus \Z/{3}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 2) + (2 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - 2z) (2x - z)\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(3xz^2 - 2z^3\) | \(0.043869\) | \(\infty\) |
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2 - z^3\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 2) + (2 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - 2z) (2x - z)\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(3xz^2 - 2z^3\) | \(0.043869\) | \(\infty\) |
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2 - z^3\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - 2z) (2x - z)\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(x^3 + 6xz^2 - 4z^3\) | \(0.043869\) | \(\infty\) |
\(2 \cdot(0 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + 4xz^2 - 2z^3\) | \(0\) | \(3\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 0.043869 \) |
Real period: | \( 18.56682 \) |
Tamagawa product: | \( 6 \) |
Torsion order: | \( 3 \) |
Leading coefficient: | \( 0.543012 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(4\) | \(3\) | \(1 + 2 T^{2}\) | |
\(5\) | \(1\) | \(2\) | \(2\) | \(( 1 - T )( 1 + 3 T + 5 T^{2} )\) | |
\(193\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 5 T + 193 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.10.1 | no |
\(3\) | 3.80.1 | 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);