Minimal equation
Minimal equation
Simplified equation
$y^2 + (x + 1)y = x^5 + 2x^4$ | (homogenize, simplify) |
$y^2 + (xz^2 + z^3)y = x^5z + 2x^4z^2$ | (dehomogenize, simplify) |
$y^2 = 4x^5 + 8x^4 + x^2 + 2x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 0, 2, 1]), R([1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 0, 2, 1], R![1, 1]);
sage: X = HyperellipticCurve(R([1, 2, 1, 0, 8, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(2102\) | \(=\) | \( 2 \cdot 1051 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(33632\) | \(=\) | \( 2^{5} \cdot 1051 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(48\) | \(=\) | \( 2^{4} \cdot 3 \) |
\( I_4 \) | \(=\) | \(1548\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 43 \) |
\( I_6 \) | \(=\) | \(5463\) | \(=\) | \( 3^{2} \cdot 607 \) |
\( I_{10} \) | \(=\) | \(134528\) | \(=\) | \( 2^{7} \cdot 1051 \) |
\( J_2 \) | \(=\) | \(24\) | \(=\) | \( 2^{3} \cdot 3 \) |
\( J_4 \) | \(=\) | \(-234\) | \(=\) | \( - 2 \cdot 3^{2} \cdot 13 \) |
\( J_6 \) | \(=\) | \(1145\) | \(=\) | \( 5 \cdot 229 \) |
\( J_8 \) | \(=\) | \(-6819\) | \(=\) | \( - 3 \cdot 2273 \) |
\( J_{10} \) | \(=\) | \(33632\) | \(=\) | \( 2^{5} \cdot 1051 \) |
\( g_1 \) | \(=\) | \(248832/1051\) | ||
\( g_2 \) | \(=\) | \(-101088/1051\) | ||
\( g_3 \) | \(=\) | \(20610/1051\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((1 : 1 : 1)\) |
\((-2 : 0 : 1)\) | \((-2 : 1 : 1)\) | \((1 : -3 : 1)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((1 : 1 : 1)\) |
\((-2 : 0 : 1)\) | \((-2 : 1 : 1)\) | \((1 : -3 : 1)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-2 : -1 : 1)\) | \((-2 : 1 : 1)\) | \((-1 : -2 : 1)\) |
\((-1 : 2 : 1)\) | \((1 : -4 : 1)\) | \((1 : 4 : 1)\) |
magma: [C![-2,0,1],C![-2,1,1],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,0,0],C![1,1,1]]; // minimal model
magma: [C![-2,-1,1],C![-2,1,1],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-4,1],C![1,0,0],C![1,4,1]]; // 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: \(\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.003479\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.003479\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -2 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - z^3\) | \(0.003479\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 0.003479 \) |
Real period: | \( 18.67781 \) |
Tamagawa product: | \( 5 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.324934 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(5\) | \(5\) | \(( 1 - T )( 1 + 2 T + 2 T^{2} )\) | |
\(1051\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 27 T + 1051 T^{2} )\) |
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);