Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x^2 + x + 1)y = -x^6 + 2x^4 + 4x^3 + 2x^2 - 1$ | (homogenize, simplify) |
| $y^2 + (x^3 + x^2z + xz^2 + z^3)y = -x^6 + 2x^4z^2 + 4x^3z^3 + 2x^2z^4 - z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 + 2x^5 + 11x^4 + 20x^3 + 11x^2 + 2x - 3$ | (minimize, homogenize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 0, 2, 4, 2, 0, -1]), R([1, 1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 0, 2, 4, 2, 0, -1], R![1, 1, 1, 1]);
sage: X = HyperellipticCurve(R([-3, 2, 11, 20, 11, 2, -3]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(800\) | \(=\) | \( 2^{5} \cdot 5^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(8000\) | \(=\) | \( 2^{6} \cdot 5^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(-1536\) | \(=\) | \( - 2^{9} \cdot 3 \) |
| \( I_4 \) | \(=\) | \(742656\) | \(=\) | \( 2^{8} \cdot 3 \cdot 967 \) |
| \( I_6 \) | \(=\) | \(-165064704\) | \(=\) | \( - 2^{12} \cdot 3 \cdot 7 \cdot 19 \cdot 101 \) |
| \( I_{10} \) | \(=\) | \(32768000\) | \(=\) | \( 2^{18} \cdot 5^{3} \) |
| \( J_2 \) | \(=\) | \(-192\) | \(=\) | \( - 2^{6} \cdot 3 \) |
| \( J_4 \) | \(=\) | \(-6200\) | \(=\) | \( - 2^{3} \cdot 5^{2} \cdot 31 \) |
| \( J_6 \) | \(=\) | \(-142400\) | \(=\) | \( - 2^{6} \cdot 5^{2} \cdot 89 \) |
| \( J_8 \) | \(=\) | \(-2774800\) | \(=\) | \( - 2^{4} \cdot 5^{2} \cdot 7 \cdot 991 \) |
| \( J_{10} \) | \(=\) | \(8000\) | \(=\) | \( 2^{6} \cdot 5^{3} \) |
| \( g_1 \) | \(=\) | \(-4076863488/125\) | ||
| \( g_2 \) | \(=\) | \(27426816/5\) | ||
| \( g_3 \) | \(=\) | \(-3280896/5\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
This curve has no rational points.
magma: [];
Number of rational Weierstrass points: \(0\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable except over $\Q_{2}$ and $\Q_{5}$.
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: \(\Z/{4}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(4\) |
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(1\) |
| Regulator: | \( 1 \) |
| Real period: | \( 5.590050 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 4 \) |
| Leading coefficient: | \( 0.349378 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(5\) | \(6\) | \(1\) | \(1\) | |
| \(5\) | \(2\) | \(3\) | \(1\) | \(( 1 - T )( 1 + T )\) |  |
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 20.a1
Elliptic curve 40.a3
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 \(\Q\).