Minimal equation
Minimal equation
Simplified equation
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-306, 272, 10, -39, -1, 1], R![1]);
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: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1223, 1088, 40, -156, -4, 4]))
| $y^2 + y = x^5 - x^4 - 39x^3 + 10x^2 + 272x - 306$ | (homogenize, simplify) |
| $y^2 + z^3y = x^5z - x^4z^2 - 39x^3z^3 + 10x^2z^4 + 272xz^5 - 306z^6$ | (dehomogenize, simplify) |
| $y^2 = 4x^5 - 4x^4 - 156x^3 + 40x^2 + 1088x - 1223$ | (minimize, homogenize) |
Invariants
| \( N \) | = | \(461\) | = | \( 461 \) | magma: Conductor(LSeries(C)); Factorization($1);
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| \( \Delta \) | = | \(461\) | = | \( 461 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
Igusa invariants
magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
G2 invariants
magma: G2Invariants(C);
| \( I_2 \) | = | \(322656\) | = | \( 2^{5} \cdot 3 \cdot 3361 \) |
| \( I_4 \) | = | \(2657873664\) | = | \( 2^{8} \cdot 3^{2} \cdot 29 \cdot 39779 \) |
| \( I_6 \) | = | \(240174460988928\) | = | \( 2^{9} \cdot 3^{2} \cdot 71 \cdot 1319 \cdot 556559 \) |
| \( I_{10} \) | = | \(1888256\) | = | \( 2^{12} \cdot 461 \) |
| \( J_2 \) | = | \(40332\) | = | \( 2^{2} \cdot 3 \cdot 3361 \) |
| \( J_4 \) | = | \(40091742\) | = | \( 2 \cdot 3^{2} \cdot 31 \cdot 71849 \) |
| \( J_6 \) | = | \(45075737276\) | = | \( 2^{2} \cdot 83 \cdot 135770293 \) |
| \( J_8 \) | = | \(52661714805267\) | = | \( 3 \cdot 613 \cdot 8819 \cdot 3247087 \) |
| \( J_{10} \) | = | \(461\) | = | \( 461 \) |
| \( g_1 \) | = | \(106720731303787612818432/461\) | ||
| \( g_2 \) | = | \(2630293443843585469056/461\) | ||
| \( g_3 \) | = | \(73323359651716069824/461\) |
Automorphism group
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magma: AutomorphismGroup(C); IdentifyGroup($1);
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| \(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | ||
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magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | ||
Rational points
magma: [C![1,0,0]];
Points: \((1 : 0 : 0)\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
Number of rational Weierstrass points: \(1\)
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);
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian:
magma: MordellWeilGroupGenus2(Jacobian(C));
Group structure: trivial
BSD invariants
| 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 |
|---|---|---|---|---|
| \(461\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 461 T^{2} )\) |
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 \(\Q\).