This is the first proven instance of the paramodular conjecture for an abelian surface $A$ with trivial geometric endomorphism ring (meaning $\End(A_{\overline{\mathbb{Q}}})=\mathbb{Z}$); see [arXiv:1805.10873].
Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2 + x + 1)y = -x^2 - x$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = -x^2z^4 - xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^5 + 3x^4 + 4x^3 - x^2 - 2x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -1]), R([1, 1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1], R![1, 1, 1, 1]);
sage: X = HyperellipticCurve(R([1, -2, -1, 4, 3, 2, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(277\) | \(=\) | \( 277 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(277\) | \(=\) | \( 277 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(64\) | \(=\) | \( 2^{6} \) |
\( I_4 \) | \(=\) | \(352\) | \(=\) | \( 2^{5} \cdot 11 \) |
\( I_6 \) | \(=\) | \(9552\) | \(=\) | \( 2^{4} \cdot 3 \cdot 199 \) |
\( I_{10} \) | \(=\) | \(-1108\) | \(=\) | \( - 2^{2} \cdot 277 \) |
\( J_2 \) | \(=\) | \(32\) | \(=\) | \( 2^{5} \) |
\( J_4 \) | \(=\) | \(-16\) | \(=\) | \( - 2^{4} \) |
\( J_6 \) | \(=\) | \(-464\) | \(=\) | \( - 2^{4} \cdot 29 \) |
\( J_8 \) | \(=\) | \(-3776\) | \(=\) | \( - 2^{6} \cdot 59 \) |
\( J_{10} \) | \(=\) | \(-277\) | \(=\) | \( -277 \) |
\( g_1 \) | \(=\) | \(-33554432/277\) | ||
\( g_2 \) | \(=\) | \(524288/277\) | ||
\( g_3 \) | \(=\) | \(475136/277\) |
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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1)\)
magma: [C![-1,0,1],C![0,-1,1],C![0,0,1],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,-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: \(\Z/{15}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(15\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(15\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + xz^2 + z^3\) | \(0\) | \(15\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(0\) |
Regulator: | \( 1 \) |
Real period: | \( 32.20574 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 15 \) |
Leading coefficient: | \( 0.143136 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(277\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 8 T + 277 T^{2} )\) |
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 |
\(3\) | 3.80.1 | yes |
\(5\) | not computed | 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);