Properties

Label 676.a.562432.1
Conductor 676
Discriminant 562432
Mordell-Weil group \(\Z/{21}\Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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This is a model for the modular curve $X_0(26)$.

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = 2x^5 + 2x^4 + 4x^3 + 2x^2 + 2x$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = 2x^5z + 2x^4z^2 + 4x^3z^3 + 2x^2z^4 + 2xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 8x^5 + 8x^4 + 18x^3 + 8x^2 + 8x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 2, 4, 2, 2]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 2, 4, 2, 2], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 8, 8, 18, 8, 8, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(676\) \(=\) \( 2^{2} \cdot 13^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(562432\) \(=\) \( 2^{8} \cdot 13^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(3240\) \(=\)  \( 2^{3} \cdot 3^{4} \cdot 5 \)
\( I_4 \)  \(=\) \(211812\) \(=\)  \( 2^{2} \cdot 3 \cdot 19 \cdot 929 \)
\( I_6 \)  \(=\) \(236219112\) \(=\)  \( 2^{3} \cdot 3^{3} \cdot 251 \cdot 4357 \)
\( I_{10} \)  \(=\) \(2303721472\) \(=\)  \( 2^{20} \cdot 13^{3} \)
\( J_2 \)  \(=\) \(405\) \(=\)  \( 3^{4} \cdot 5 \)
\( J_4 \)  \(=\) \(4628\) \(=\)  \( 2^{2} \cdot 13 \cdot 89 \)
\( J_6 \)  \(=\) \(-8112\) \(=\)  \( - 2^{4} \cdot 3 \cdot 13^{2} \)
\( J_8 \)  \(=\) \(-6175936\) \(=\)  \( - 2^{6} \cdot 13^{2} \cdot 571 \)
\( J_{10} \)  \(=\) \(562432\) \(=\)  \( 2^{8} \cdot 13^{3} \)
\( g_1 \)  \(=\) \(10896201253125/562432\)
\( g_2 \)  \(=\) \(5912281125/10816\)
\( g_3 \)  \(=\) \(-492075/208\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

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

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 + xz + 3z^2\) \(=\) \(0,\) \(4y\) \(=\) \(xz^2 - 3z^3\) \(0\) \(21\)

2-torsion field: 6.0.86528.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 6.723259 \)
Tamagawa product: \( 21 \)
Torsion order:\( 21 \)
Leading coefficient: \( 0.320155 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(8\) \(7\) \(( 1 - T )( 1 + T )\)
\(13\) \(2\) \(3\) \(3\) \(( 1 - T )( 1 + T )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 26.a2
  Elliptic curve 26.b2

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).