Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -2x^4 + x^2 - x$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -2x^4z^2 + x^2z^4 - xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 - 8x^4 + 2x^3 + 4x^2 - 4x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 1, 0, -2]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 1, 0, -2], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([1, -4, 4, 2, -8, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(2075\) | \(=\) | \( 5^{2} \cdot 83 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-10375\) | \(=\) | \( - 5^{3} \cdot 83 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(148\) | \(=\) | \( 2^{2} \cdot 37 \) |
\( I_4 \) | \(=\) | \(-6215\) | \(=\) | \( - 5 \cdot 11 \cdot 113 \) |
\( I_6 \) | \(=\) | \(-372115\) | \(=\) | \( - 5 \cdot 19 \cdot 3917 \) |
\( I_{10} \) | \(=\) | \(-1328000\) | \(=\) | \( - 2^{7} \cdot 5^{3} \cdot 83 \) |
\( J_2 \) | \(=\) | \(37\) | \(=\) | \( 37 \) |
\( J_4 \) | \(=\) | \(316\) | \(=\) | \( 2^{2} \cdot 79 \) |
\( J_6 \) | \(=\) | \(2624\) | \(=\) | \( 2^{6} \cdot 41 \) |
\( J_8 \) | \(=\) | \(-692\) | \(=\) | \( - 2^{2} \cdot 173 \) |
\( J_{10} \) | \(=\) | \(-10375\) | \(=\) | \( - 5^{3} \cdot 83 \) |
\( g_1 \) | \(=\) | \(-69343957/10375\) | ||
\( g_2 \) | \(=\) | \(-16006348/10375\) | ||
\( g_3 \) | \(=\) | \(-3592256/10375\) |
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);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (1 : -10 : 3),\, (1 : -18 : 3)\)
magma: [C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-18,3],C![1,-10,3],C![1,-1,0],C![1,0,0]]; // minimal model
magma: [C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-8,3],C![1,8,3],C![1,-1,0],C![1,1,0]]; // 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: \(\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.037751\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - 3xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4xz^2 + z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.037751\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - 3xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4xz^2 + z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.037751\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - 3xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 8xz^2 + 3z^3\) | \(0\) | \(2\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.037751 \) |
Real period: | \( 15.29300 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.288670 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(5\) | \(2\) | \(3\) | \(2\) | \(1 + 4 T + 5 T^{2}\) | |
\(83\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 6 T + 83 T^{2} )\) |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.60.1 | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{USp}(4)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{USp}(4)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
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 \(\Q\).
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);