Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^2 + x)y = x^5 - 2x^4 + 2x^3 - 2x^2 + x$ | (homogenize, simplify) |
| $y^2 + (x^2z + xz^2)y = x^5z - 2x^4z^2 + 2x^3z^3 - 2x^2z^4 + xz^5$ | (dehomogenize, simplify) |
| $y^2 = 4x^5 - 7x^4 + 10x^3 - 7x^2 + 4x$ | (minimize, homogenize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -2, 2, -2, 1]), R([0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -2, 2, -2, 1], R![0, 1, 1]);
sage: X = HyperellipticCurve(R([0, 4, -7, 10, -7, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | = | \(578\) | = | \( 2 \cdot 17^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | = | \(2312\) | = | \( 2^{3} \cdot 17^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | = | \(456\) | = | \( 2^{3} \cdot 3 \cdot 19 \) |
| \( I_4 \) | = | \(2820\) | = | \( 2^{2} \cdot 3 \cdot 5 \cdot 47 \) |
| \( I_6 \) | = | \(1086216\) | = | \( 2^{3} \cdot 3 \cdot 45259 \) |
| \( I_{10} \) | = | \(9469952\) | = | \( 2^{15} \cdot 17^{2} \) |
| \( J_2 \) | = | \(57\) | = | \( 3 \cdot 19 \) |
| \( J_4 \) | = | \(106\) | = | \( 2 \cdot 53 \) |
| \( J_6 \) | = | \(-992\) | = | \( - 2^{5} \cdot 31 \) |
| \( J_8 \) | = | \(-16945\) | = | \( - 5 \cdot 3389 \) |
| \( J_{10} \) | = | \(2312\) | = | \( 2^{3} \cdot 17^{2} \) |
| \( g_1 \) | = | \(601692057/2312\) | ||
| \( g_2 \) | = | \(9815229/1156\) | ||
| \( g_3 \) | = | \(-402876/289\) |
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
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 0 : 1),\, (1 : -2 : 1)\)
magma: [C![0,0,1],C![1,-2,1],C![1,0,0],C![1,0,1]];
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: \(\Z/{12}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \((-x + z) x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(12\) |
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(1\) |
| Regulator: | \( 1 \) |
| Real period: | \( 13.91029 \) |
| Tamagawa product: | \( 3 \) |
| Torsion order: | \( 12 \) |
| Leading coefficient: | \( 0.289797 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor |
|---|---|---|---|---|
| \(2\) | \(3\) | \(1\) | \(3\) | \(( 1 - T )( 1 + T + 2 T^{2} )\) |
| \(17\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) |
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 34.a4
Elliptic curve 17.a4
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 \(\Q\).