The Jacobian of this curve is isogenous to that of the quotient of the modular curve $X_0(33)$ by the Atkin-Lehner involution $w_3$, which has discriminant $3\cdot 11^9$.
Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^2 + 1)y = x^5 + 2x^3 + 4x^2 + 2x$ | (homogenize, simplify) |
| $y^2 + (x^2z + z^3)y = x^5z + 2x^3z^3 + 4x^2z^4 + 2xz^5$ | (dehomogenize, simplify) |
| $y^2 = 4x^5 + x^4 + 8x^3 + 18x^2 + 8x + 1$ | (minimize, homogenize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 4, 2, 0, 1]), R([1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 4, 2, 0, 1], R![1, 0, 1]);
sage: X = HyperellipticCurve(R([1, 8, 18, 8, 1, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(363\) | \(=\) | \( 3 \cdot 11^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-11979\) | \(=\) | \( - 3^{2} \cdot 11^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(344\) | \(=\) | \( 2^{3} \cdot 43 \) |
| \( I_4 \) | \(=\) | \(-3068\) | \(=\) | \( - 2^{2} \cdot 13 \cdot 59 \) |
| \( I_6 \) | \(=\) | \(-526433\) | \(=\) | \( - 19 \cdot 103 \cdot 269 \) |
| \( I_{10} \) | \(=\) | \(-47916\) | \(=\) | \( - 2^{2} \cdot 3^{2} \cdot 11^{3} \) |
| \( J_2 \) | \(=\) | \(172\) | \(=\) | \( 2^{2} \cdot 43 \) |
| \( J_4 \) | \(=\) | \(1744\) | \(=\) | \( 2^{4} \cdot 109 \) |
| \( J_6 \) | \(=\) | \(45841\) | \(=\) | \( 45841 \) |
| \( J_8 \) | \(=\) | \(1210779\) | \(=\) | \( 3^{2} \cdot 29 \cdot 4639 \) |
| \( J_{10} \) | \(=\) | \(-11979\) | \(=\) | \( - 3^{2} \cdot 11^{3} \) |
| \( g_1 \) | \(=\) | \(-150536645632/11979\) | ||
| \( g_2 \) | \(=\) | \(-8874253312/11979\) | ||
| \( g_3 \) | \(=\) | \(-1356160144/11979\) |
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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -1 : 1),\, (-1 : -34 : 4)\)
magma: [C![-1,-34,4],C![-1,-1,1],C![0,-1,1],C![0,0,1],C![1,0,0]];
Number of rational Weierstrass points: \(3\)
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: \(\Z/{2}\Z \times \Z/{10}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((-1 : -1 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 5xz - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-15xz^2 - 6z^3\) | \(0\) | \(10\) |
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1 \) |
| Real period: | \( 18.97059 \) |
| Tamagawa product: | \( 4 \) |
| Torsion order: | \( 20 \) |
| Leading coefficient: | \( 0.189705 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(3\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 3 T^{2} )\) |  |
| \(11\) | \(2\) | \(3\) | \(2\) | \(( 1 - T )^{2}\) |  |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $G_{3,3}$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 11.a3
Elliptic curve 33.a2
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | an order of index \(3\) in \(\Z \times \Z\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).