Minimal equation
Minimal equation
Simplified equation
$y^2 + y = x^5 - 2x^3 + x^2$ | (homogenize, simplify) |
$y^2 + z^3y = x^5z - 2x^3z^3 + x^2z^4$ | (dehomogenize, simplify) |
$y^2 = 4x^5 - 8x^3 + 4x^2 + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 1, -2, 0, 1]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 1, -2, 0, 1], R![1]);
sage: X = HyperellipticCurve(R([1, 0, 4, -8, 0, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(9783\) | \(=\) | \( 3^{2} \cdot 1087 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(29349\) | \(=\) | \( 3^{3} \cdot 1087 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(96\) | \(=\) | \( 2^{5} \cdot 3 \) |
\( I_4 \) | \(=\) | \(1584\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 11 \) |
\( I_6 \) | \(=\) | \(13104\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 7 \cdot 13 \) |
\( I_{10} \) | \(=\) | \(117396\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 1087 \) |
\( J_2 \) | \(=\) | \(48\) | \(=\) | \( 2^{4} \cdot 3 \) |
\( J_4 \) | \(=\) | \(-168\) | \(=\) | \( - 2^{3} \cdot 3 \cdot 7 \) |
\( J_6 \) | \(=\) | \(2320\) | \(=\) | \( 2^{4} \cdot 5 \cdot 29 \) |
\( J_8 \) | \(=\) | \(20784\) | \(=\) | \( 2^{4} \cdot 3 \cdot 433 \) |
\( J_{10} \) | \(=\) | \(29349\) | \(=\) | \( 3^{3} \cdot 1087 \) |
\( g_1 \) | \(=\) | \(9437184/1087\) | ||
\( g_2 \) | \(=\) | \(-688128/1087\) | ||
\( g_3 \) | \(=\) | \(593920/3261\) |
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
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) |
\((-1 : -2 : 1)\) | \((1 : 2 : 4)\) | \((2 : 4 : 1)\) | \((2 : -5 : 1)\) | \((1 : -66 : 4)\) | \((12 : 495 : 1)\) |
\((12 : -496 : 1)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) |
\((-1 : -2 : 1)\) | \((1 : 2 : 4)\) | \((2 : 4 : 1)\) | \((2 : -5 : 1)\) | \((1 : -66 : 4)\) | \((12 : 495 : 1)\) |
\((12 : -496 : 1)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) | \((-1 : -3 : 1)\) |
\((-1 : 3 : 1)\) | \((2 : -9 : 1)\) | \((2 : 9 : 1)\) | \((1 : -68 : 4)\) | \((1 : 68 : 4)\) | \((12 : -991 : 1)\) |
\((12 : 991 : 1)\) |
magma: [C![-1,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-66,4],C![1,-1,1],C![1,0,0],C![1,0,1],C![1,2,4],C![2,-5,1],C![2,4,1],C![12,-496,1],C![12,495,1]]; // minimal model
magma: [C![-1,-3,1],C![-1,3,1],C![0,-1,1],C![0,1,1],C![1,-68,4],C![1,-1,1],C![1,0,0],C![1,1,1],C![1,68,4],C![2,-9,1],C![2,9,1],C![12,-991,1],C![12,991,1]]; // 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: \(\Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -2 : 1) + (0 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - z^3\) | \(0.308770\) | \(\infty\) |
\((-1 : 1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.035307\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -2 : 1) + (0 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - z^3\) | \(0.308770\) | \(\infty\) |
\((-1 : 1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.035307\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -3 : 1) + (0 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2 - z^3\) | \(0.308770\) | \(\infty\) |
\((-1 : 3 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 + z^3\) | \(0.035307\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.010663 \) |
Real period: | \( 20.29496 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.432833 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(2\) | \(3\) | \(2\) | \(1 + 2 T + 3 T^{2}\) | |
\(1087\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 8 T + 1087 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.6.1 | no |
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);