Properties

Label 676.a.5408.1
Conductor 676
Discriminant -5408
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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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2 + x)y = x^3 + 3x^2 + 3x + 1$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2)y = x^3z^3 + 3x^2z^4 + 3xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 3x^4 + 6x^3 + 13x^2 + 12x + 4$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 3, 3, 1]), R([0, 1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 3, 3, 1], R![0, 1, 1, 1]);
 
sage: X = HyperellipticCurve(R([4, 12, 13, 6, 3, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(676\) = \( 2^{2} \cdot 13^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-5408\) = \( - 2^{5} \cdot 13^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-408\) =  \( - 2^{3} \cdot 3 \cdot 17 \)
\( I_4 \)  = \(13092\) =  \( 2^{2} \cdot 3 \cdot 1091 \)
\( I_6 \)  = \(-1289688\) =  \( - 2^{3} \cdot 3 \cdot 17 \cdot 29 \cdot 109 \)
\( I_{10} \)  = \(-22151168\) =  \( - 2^{17} \cdot 13^{2} \)
\( J_2 \)  = \(-51\) =  \( - 3 \cdot 17 \)
\( J_4 \)  = \(-28\) =  \( - 2^{2} \cdot 7 \)
\( J_6 \)  = \(0\) =  \( 0 \)
\( J_8 \)  = \(-196\) =  \( - 2^{2} \cdot 7^{2} \)
\( J_{10} \)  = \(-5408\) =  \( - 2^{5} \cdot 13^{2} \)
\( g_1 \)  = \(345025251/5408\)
\( g_2 \)  = \(-928557/1352\)
\( g_3 \)  = \(0\)

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

magma: [C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,1,1],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
\((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0\) \(21\)

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: \( 20.16977 \)
Tamagawa product: \( 7 \)
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
\(2\) \(5\) \(2\) \(7\) \(( 1 - T )( 1 + T )\)
\(13\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

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.a3
  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 \(\Q\).