Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^2 + x)y = x^6 - x^5 - 2x^4 + 4x^3 - 3x + 1$ | (homogenize, simplify) |
| $y^2 + (x^2z + xz^2)y = x^6 - x^5z - 2x^4z^2 + 4x^3z^3 - 3xz^5 + z^6$ | (dehomogenize, simplify) |
| $y^2 = 4x^6 - 4x^5 - 7x^4 + 18x^3 + x^2 - 12x + 4$ | (minimize, homogenize) |
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -3, 0, 4, -2, -1, 1], R![0, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -3, 0, 4, -2, -1, 1]), R([0, 1, 1]));
magma: X,pi:= SimplifiedModel(C);
sage: X = HyperellipticCurve(R([4, -12, 1, 18, -7, -4, 4]))
Invariants
| Conductor: | \( N \) | = | \(20172\) | = | \( 2^{2} \cdot 3 \cdot 41^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | = | \(-968256\) | = | \( - 2^{6} \cdot 3^{2} \cdot 41^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | = | \(136\) | = | \( 2^{3} \cdot 17 \) |
| \( I_4 \) | = | \(774916\) | = | \( 2^{2} \cdot 23 \cdot 8423 \) |
| \( I_6 \) | = | \(-219321592\) | = | \( - 2^{3} \cdot 7 \cdot 613 \cdot 6389 \) |
| \( I_{10} \) | = | \(-3965976576\) | = | \( - 2^{18} \cdot 3^{2} \cdot 41^{2} \) |
| \( J_2 \) | = | \(17\) | = | \( 17 \) |
| \( J_4 \) | = | \(-8060\) | = | \( - 2^{2} \cdot 5 \cdot 13 \cdot 31 \) |
| \( J_6 \) | = | \(418896\) | = | \( 2^{4} \cdot 3^{2} \cdot 2909 \) |
| \( J_8 \) | = | \(-14460592\) | = | \( - 2^{4} \cdot 83 \cdot 10889 \) |
| \( J_{10} \) | = | \(-968256\) | = | \( - 2^{6} \cdot 3^{2} \cdot 41^{2} \) |
| \( g_1 \) | = | \(-1419857/968256\) | ||
| \( g_2 \) | = | \(9899695/242064\) | ||
| \( g_3 \) | = | \(-840701/6724\) |
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) |
| \((1 : -2 : 1)\) | \((1 : -3 : 2)\) | \((2 : 3 : 1)\) | \((-3 : 5 : 2)\) | \((2 : -9 : 1)\) | \((-3 : -11 : 2)\) |
| \((3 : -52 : 5)\) | \((3 : -68 : 5)\) |
magma: [C![-3,-11,2],C![-3,5,2],C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-3,2],C![1,-2,1],C![1,-1,0],C![1,0,1],C![1,1,0],C![2,-9,1],C![2,3,1],C![3,-68,5],C![3,-52,5]];
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: \(\Z \times \Z \times \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((-1 : 0 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0.158713\) | \(\infty\) |
| \((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.112353\) | \(\infty\) |
| \((-1 : 0 : 1) + (1 : -3 : 2) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x + z) (2x - z)\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(2\) |
| Mordell-Weil rank: | \(2\) |
| 2-Selmer rank: | \(3\) |
| Regulator: | \( 0.017831 \) |
| Real period: | \( 16.48290 \) |
| Tamagawa product: | \( 8 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 0.587845 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor |
|---|---|---|---|---|
| \(2\) | \(6\) | \(2\) | \(4\) | \(( 1 + T )^{2}\) |
| \(3\) | \(2\) | \(1\) | \(2\) | \(( 1 + T )( 1 + 2 T + 3 T^{2} )\) |
| \(41\) | \(2\) | \(2\) | \(1\) | \(( 1 + T )^{2}\) |
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 246.a1
Elliptic curve 82.a2
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | an order of index \(2\) 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 \(\Q\).