Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = x^6 + 2x^3 - x$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = x^6 + 2x^3z^3 - xz^5$ | (dehomogenize, simplify) |
$y^2 = 5x^6 + 10x^3 - 4x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 2, 0, 0, 1]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 2, 0, 0, 1], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([1, -4, 0, 10, 0, 0, 5]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(12500\) | \(=\) | \( 2^{2} \cdot 5^{5} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(12500\) | \(=\) | \( 2^{2} \cdot 5^{5} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(60\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \) |
\( I_4 \) | \(=\) | \(2025\) | \(=\) | \( 3^{4} \cdot 5^{2} \) |
\( I_6 \) | \(=\) | \(67995\) | \(=\) | \( 3^{2} \cdot 5 \cdot 1511 \) |
\( I_{10} \) | \(=\) | \(-512\) | \(=\) | \( - 2^{9} \) |
\( J_2 \) | \(=\) | \(75\) | \(=\) | \( 3 \cdot 5^{2} \) |
\( J_4 \) | \(=\) | \(-1875\) | \(=\) | \( - 3 \cdot 5^{4} \) |
\( J_6 \) | \(=\) | \(-73125\) | \(=\) | \( - 3^{2} \cdot 5^{4} \cdot 13 \) |
\( J_8 \) | \(=\) | \(-2250000\) | \(=\) | \( - 2^{4} \cdot 3^{2} \cdot 5^{6} \) |
\( J_{10} \) | \(=\) | \(-12500\) | \(=\) | \( - 2^{2} \cdot 5^{5} \) |
\( g_1 \) | \(=\) | \(-759375/4\) | ||
\( g_2 \) | \(=\) | \(253125/4\) | ||
\( g_3 \) | \(=\) | \(131625/4\) |
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: \((0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1)\)
magma: [C![-1,0,1],C![0,-1,1],C![0,0,1]]; // minimal model
magma: [C![-1,0,1],C![0,-1,1],C![0,1,1]]; // 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: \(\Z/{5}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (0 : 0 : 1) - D_\infty\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(5\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (0 : 0 : 1) - D_\infty\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(5\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (0 : 1 : 1) - D_\infty\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + z^3\) | \(0\) | \(5\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(0\) |
Regulator: | \( 1 \) |
Real period: | \( 25.01138 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 5 \) |
Leading coefficient: | \( 1.000455 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(2\) | \(1\) | \(1 + T^{2}\) | |
\(5\) | \(5\) | \(5\) | \(1\) | \(1\) |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.36.1 | no |
\(3\) | 3.36.1 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
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 \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{5}) \) with defining polynomial \(x^{2} - x - 1\)
Of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{5}}{2}]\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{5}) \) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
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);