Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = x^5 + 4x^4 + 6x^3 + 10x^2 + 3x + 1$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = x^5z + 4x^4z^2 + 6x^3z^3 + 10x^2z^4 + 3xz^5 + z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 + 16x^4 + 26x^3 + 40x^2 + 12x + 5$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 3, 10, 6, 4, 1]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 3, 10, 6, 4, 1], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([5, 12, 40, 26, 16, 4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(4225\) | \(=\) | \( 5^{2} \cdot 13^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-274625\) | \(=\) | \( - 5^{3} \cdot 13^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(2732\) | \(=\) | \( 2^{2} \cdot 683 \) |
\( I_4 \) | \(=\) | \(286321\) | \(=\) | \( 7 \cdot 40903 \) |
\( I_6 \) | \(=\) | \(209098567\) | \(=\) | \( 209098567 \) |
\( I_{10} \) | \(=\) | \(35152000\) | \(=\) | \( 2^{7} \cdot 5^{3} \cdot 13^{3} \) |
\( J_2 \) | \(=\) | \(683\) | \(=\) | \( 683 \) |
\( J_4 \) | \(=\) | \(7507\) | \(=\) | \( 7507 \) |
\( J_6 \) | \(=\) | \(96775\) | \(=\) | \( 5^{2} \cdot 7^{2} \cdot 79 \) |
\( J_8 \) | \(=\) | \(2435569\) | \(=\) | \( 2435569 \) |
\( J_{10} \) | \(=\) | \(274625\) | \(=\) | \( 5^{3} \cdot 13^{3} \) |
\( g_1 \) | \(=\) | \(148628987203643/274625\) | ||
\( g_2 \) | \(=\) | \(2391820186409/274625\) | ||
\( g_3 \) | \(=\) | \(1805778919/10985\) |
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);
|
Rational points
All points: \((1 : 0 : 0),\, (1 : -1 : 0)\)
magma: [C![1,-1,0],C![1,0,0]]; // minimal model
magma: [C![1,-1,0],C![1,1,0]]; // simplified model
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: \(\Z/{6}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(2x^2 + 2xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-4xz^2 - z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(2x^2 + 2xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-4xz^2 - z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(2x^2 + 2xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^3 - 8xz^2 - z^3\) | \(0\) | \(6\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 8.210257 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 6 \) |
Leading coefficient: | \( 0.912250 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(5\) | \(2\) | \(3\) | \(2\) | \(( 1 + T )^{2}\) | |
\(13\) | \(2\) | \(3\) | \(2\) | \(( 1 - T )^{2}\) |
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? |
---|---|---|
\(2\) | 2.90.1 | yes |
\(3\) | 3.1920.1 | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z [\sqrt{3}]\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{3}) \) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
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);