Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = x^5 + 2x^4 + x^3$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = x^5z + 2x^4z^2 + x^3z^3$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 4x^5 + 10x^4 + 6x^3 + x^2 + 2x + 1$ | (minimize, homogenize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 1, 2, 1]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 1, 2, 1], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([1, 2, 1, 6, 10, 4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(3319\) | \(=\) | \( 3319 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(3319\) | \(=\) | \( 3319 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(68\) | \(=\) | \( 2^{2} \cdot 17 \) |
| \( I_4 \) | \(=\) | \(3673\) | \(=\) | \( 3673 \) |
| \( I_6 \) | \(=\) | \(38093\) | \(=\) | \( 11 \cdot 3463 \) |
| \( I_{10} \) | \(=\) | \(424832\) | \(=\) | \( 2^{7} \cdot 3319 \) |
| \( J_2 \) | \(=\) | \(17\) | \(=\) | \( 17 \) |
| \( J_4 \) | \(=\) | \(-141\) | \(=\) | \( - 3 \cdot 47 \) |
| \( J_6 \) | \(=\) | \(205\) | \(=\) | \( 5 \cdot 41 \) |
| \( J_8 \) | \(=\) | \(-4099\) | \(=\) | \( -4099 \) |
| \( J_{10} \) | \(=\) | \(3319\) | \(=\) | \( 3319 \) |
| \( g_1 \) | \(=\) | \(1419857/3319\) | ||
| \( g_2 \) | \(=\) | \(-692733/3319\) | ||
| \( g_3 \) | \(=\) | \(59245/3319\) |
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 | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 1 : 1)\) |
| \((1 : 1 : 1)\) | \((-2 : 1 : 1)\) | \((-1 : -1 : 2)\) | \((-1 : -2 : 2)\) | \((1 : -4 : 1)\) | \((-2 : 8 : 1)\) |
magma: [C![-2,1,1],C![-2,8,1],C![-1,-2,2],C![-1,-1,2],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,1,1]];
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: \(\Z \times \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((-1 : 0 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.205416\) | \(\infty\) |
| \((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.036927\) | \(\infty\) |
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(2\) |
| Mordell-Weil rank: | \(2\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 0.007585 \) |
| Real period: | \( 25.35935 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.192363 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(3319\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 74 T + 3319 T^{2} )\) |  |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $\mathrm{USp}(4)$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{USp}(4)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
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 \(\Q\).
Additional information
The conductor $3319$ of the Jacobian of this curve is the smallest known for a genus $2$ curve with analytic rank $2$.