Properties

Label 353.a.353.1
Conductor $353$
Discriminant $-353$
Mordell-Weil group \(\Z/{11}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = x^2$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = x^2z^4$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^4 + 2x^3 + 5x^2 + 2x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 2, 5, 2, 2, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(353\) \(=\) \( 353 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-353\) \(=\) \( -353 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(188\) \(=\)  \( 2^{2} \cdot 47 \)
\( I_4 \)  \(=\) \(817\) \(=\)  \( 19 \cdot 43 \)
\( I_6 \)  \(=\) \(30871\) \(=\)  \( 30871 \)
\( I_{10} \)  \(=\) \(45184\) \(=\)  \( 2^{7} \cdot 353 \)
\( J_2 \)  \(=\) \(47\) \(=\)  \( 47 \)
\( J_4 \)  \(=\) \(58\) \(=\)  \( 2 \cdot 29 \)
\( J_6 \)  \(=\) \(256\) \(=\)  \( 2^{8} \)
\( J_8 \)  \(=\) \(2167\) \(=\)  \( 11 \cdot 197 \)
\( J_{10} \)  \(=\) \(353\) \(=\)  \( 353 \)
\( g_1 \)  \(=\) \(229345007/353\)
\( g_2 \)  \(=\) \(6021734/353\)
\( g_3 \)  \(=\) \(565504/353\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1)\)

magma: [C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [C![0,-1,1],C![0,1,1],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: \(\Z/{11}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(11\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(11\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 - z^3\) \(0\) \(11\)

2-torsion field: 6.0.22592.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 22.49549 \)
Tamagawa product: \( 1 \)
Torsion order:\( 11 \)
Leading coefficient: \( 0.185913 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(353\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 9 T + 353 T^{2} )\)

Galois representations

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not the maximal image $\GSp(4,\F_\ell)$

Prime \(\ell\) mod-\(\ell\) image
\(2\) 2.10.1

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 \(\Q\).