Properties

Label 277.a.277.1
Conductor 277
Discriminant 277
Mordell-Weil group \(\Z/{15}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Show commands for: SageMath / Magma

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)
$y^2 = x^6 + 2x^5 + 3x^4 + 4x^3 - x^2 - 2x + 1$ (minimize, homogenize)

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

\( I_2 \)  = \(-256\) =  \( - 2^{8} \)
\( I_4 \)  = \(5632\) =  \( 2^{9} \cdot 11 \)
\( I_6 \)  = \(-611328\) =  \( - 2^{10} \cdot 3 \cdot 199 \)
\( I_{10} \)  = \(1134592\) =  \( 2^{12} \cdot 277 \)
\( J_2 \)  = \(-32\) =  \( - 2^{5} \)
\( J_4 \)  = \(-16\) =  \( - 2^{4} \)
\( J_6 \)  = \(464\) =  \( 2^{4} \cdot 29 \)
\( J_8 \)  = \(-3776\) =  \( - 2^{6} \cdot 59 \)
\( J_{10} \)  = \(277\) =  \( 277 \)
\( g_1 \)  = \(-33554432/277\)
\( g_2 \)  = \(524288/277\)
\( g_3 \)  = \(475136/277\)

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);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$
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]];
 

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
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(15\)

2-torsion field: 5.1.4432.1

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
\(277\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 8 T + 277 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

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 \(\Q\).