Properties

Label 388.a.776.1
Conductor 388
Discriminant 776
Mordell-Weil group \(\Z/{21}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 2, 0, -1], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 6, 9, 2, -2, 0, 1]))
 

$y^2 + (x^3 + x + 1)y = -x^4 + 2x^2 + x$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 + 2x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = x^6 - 2x^4 + 2x^3 + 9x^2 + 6x + 1$ (minimize, homogenize)

Invariants

\( N \)  =  \(388\) = \( 2^{2} \cdot 97 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(776\) = \( 2^{3} \cdot 97 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(72\) =  \( 2^{3} \cdot 3^{2} \)
\( I_4 \)  = \(6276\) =  \( 2^{2} \cdot 3 \cdot 523 \)
\( I_6 \)  = \(-109944\) =  \( - 2^{3} \cdot 3^{3} \cdot 509 \)
\( I_{10} \)  = \(3178496\) =  \( 2^{15} \cdot 97 \)
\( J_2 \)  = \(9\) =  \( 3^{2} \)
\( J_4 \)  = \(-62\) =  \( - 2 \cdot 31 \)
\( J_6 \)  = \(356\) =  \( 2^{2} \cdot 89 \)
\( J_8 \)  = \(-160\) =  \( - 2^{5} \cdot 5 \)
\( J_{10} \)  = \(776\) =  \( 2^{3} \cdot 97 \)
\( g_1 \)  = \(59049/776\)
\( g_2 \)  = \(-22599/388\)
\( g_3 \)  = \(7209/194\)

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$

Rational points

magma: [C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]];
 

Points: \((0 : 0 : 1),\, (1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 0),\, (-1 : 1 : 1)\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(0\)

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);
 

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z/{21}\Z\)

Generator Height Order
\(x^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0\) \(21\)

2-torsion field: 6.2.49664.1

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 29.13550 \)
Tamagawa product: \( 3 \)
Torsion order:\( 21 \)
Leading coefficient: \( 0.198200 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(3\) \(2\) \(3\) \(1 + T + T^{2}\)
\(97\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 14 T + 97 T^{2} )\)

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 \(\Q\).