Minimal equation
Minimal equation
Simplified equation
| $y^2 + x^3y = x^5 - 7x^3 - 16x^2 - 15x - 5$ | (homogenize, simplify) |
| $y^2 + x^3y = x^5z - 7x^3z^3 - 16x^2z^4 - 15xz^5 - 5z^6$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 4x^5 - 28x^3 - 64x^2 - 60x - 20$ | (minimize, homogenize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-5, -15, -16, -7, 0, 1]), R([0, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-5, -15, -16, -7, 0, 1], R![0, 0, 0, 1]);
sage: X = HyperellipticCurve(R([-20, -60, -64, -28, 0, 4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(6845\) | \(=\) | \( 5 \cdot 37^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(6845\) | \(=\) | \( 5 \cdot 37^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(24\) | \(=\) | \( 2^{3} \cdot 3 \) |
| \( I_4 \) | \(=\) | \(852\) | \(=\) | \( 2^{2} \cdot 3 \cdot 71 \) |
| \( I_6 \) | \(=\) | \(-14064\) | \(=\) | \( - 2^{4} \cdot 3 \cdot 293 \) |
| \( I_{10} \) | \(=\) | \(-27380\) | \(=\) | \( - 2^{2} \cdot 5 \cdot 37^{2} \) |
| \( J_2 \) | \(=\) | \(12\) | \(=\) | \( 2^{2} \cdot 3 \) |
| \( J_4 \) | \(=\) | \(-136\) | \(=\) | \( - 2^{3} \cdot 17 \) |
| \( J_6 \) | \(=\) | \(2040\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 17 \) |
| \( J_8 \) | \(=\) | \(1496\) | \(=\) | \( 2^{3} \cdot 11 \cdot 17 \) |
| \( J_{10} \) | \(=\) | \(-6845\) | \(=\) | \( - 5 \cdot 37^{2} \) |
| \( g_1 \) | \(=\) | \(-248832/6845\) | ||
| \( g_2 \) | \(=\) | \(235008/6845\) | ||
| \( g_3 \) | \(=\) | \(-58752/1369\) |
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
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((-2 : 3 : 1)\) | \((-2 : 5 : 1)\) |
| \((3 : -7 : 1)\) | \((-3 : 7 : 4)\) | \((3 : -20 : 1)\) | \((-3 : 20 : 4)\) |
magma: [C![-3,7,4],C![-3,20,4],C![-2,3,1],C![-2,5,1],C![-1,0,1],C![-1,1,1],C![1,-1,0],C![1,0,0],C![3,-20,1],C![3,-7,1]];
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 | |||||
|---|---|---|---|---|---|---|---|---|
| \((-1 : 0 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.220557\) | \(\infty\) |
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 3xz + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2 - 4z^3\) | \(0.102222\) | \(\infty\) |
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(2\) |
| Mordell-Weil rank: | \(2\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 0.022546 \) |
| Real period: | \( 14.66307 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.330594 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 5 T^{2} )\) |  |
| \(37\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) |  |
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 37.a1
Elliptic curve 185.b1
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 \(\Q\).