Minimal equation
Minimal equation
Simplified equation
| $y^2 + y = x^5 - x^4 - 39x^3 + 10x^2 + 272x - 306$ | (homogenize, simplify) |
| $y^2 + z^3y = x^5z - x^4z^2 - 39x^3z^3 + 10x^2z^4 + 272xz^5 - 306z^6$ | (dehomogenize, simplify) |
| $y^2 = 4x^5 - 4x^4 - 156x^3 + 40x^2 + 1088x - 1223$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-306, 272, 10, -39, -1, 1]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-306, 272, 10, -39, -1, 1], R![1]);
sage: X = HyperellipticCurve(R([-1223, 1088, 40, -156, -4, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(461\) | \(=\) | \( 461 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(461\) | \(=\) | \( 461 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(80664\) | \(=\) | \( 2^{3} \cdot 3 \cdot 3361 \) |
| \( I_4 \) | \(=\) | \(166117104\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 29 \cdot 39779 \) |
| \( I_6 \) | \(=\) | \(3752725952952\) | \(=\) | \( 2^{3} \cdot 3^{2} \cdot 71 \cdot 1319 \cdot 556559 \) |
| \( I_{10} \) | \(=\) | \(1844\) | \(=\) | \( 2^{2} \cdot 461 \) |
| \( J_2 \) | \(=\) | \(40332\) | \(=\) | \( 2^{2} \cdot 3 \cdot 3361 \) |
| \( J_4 \) | \(=\) | \(40091742\) | \(=\) | \( 2 \cdot 3^{2} \cdot 31 \cdot 71849 \) |
| \( J_6 \) | \(=\) | \(45075737276\) | \(=\) | \( 2^{2} \cdot 83 \cdot 135770293 \) |
| \( J_8 \) | \(=\) | \(52661714805267\) | \(=\) | \( 3 \cdot 613 \cdot 8819 \cdot 3247087 \) |
| \( J_{10} \) | \(=\) | \(461\) | \(=\) | \( 461 \) |
| \( g_1 \) | \(=\) | \(106720731303787612818432/461\) | ||
| \( g_2 \) | \(=\) | \(2630293443843585469056/461\) | ||
| \( g_3 \) | \(=\) | \(73323359651716069824/461\) |
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: \((1 : 0 : 0)\)
magma: [C![1,0,0]]; // minimal model
magma: [C![1,0,0]]; // simplified model
Number of rational Weierstrass points: \(1\)
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: trivial
magma: MordellWeilGroupGenus2(Jacobian(C));
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(0\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.245886 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.245886 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(461\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 461 T^{2} )\) |  |
Galois representations
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not the maximal image $\GSp(4,\F_\ell)$
| Prime \(\ell\) | mod-\(\ell\) image |
|---|---|
| \(2\) | 2.6.1 |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $\mathrm{USp}(4)$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{USp}(4)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).