Properties

Label 8281.a.8281.1
Conductor $8281$
Discriminant $8281$
Mordell-Weil group \(\Z \times \Z \times \Z/{3}\Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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This curve is isomorphic to the quotient of the modular curve $X_0(91)$ by its Fricke involution $w_{91}$.

Minimal equation

Minimal equation

Simplified equation

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -1, -3, -1]), R([1, 1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -1, -3, -1], R![1, 1, 1, 1]);
 
sage: X = HyperellipticCurve(R([1, 2, -1, -8, -1, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(8281\) \(=\) \( 7^{2} \cdot 13^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(8281\) \(=\) \( 7^{2} \cdot 13^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(72\) \(=\)  \( 2^{3} \cdot 3^{2} \)
\( I_4 \)  \(=\) \(1236\) \(=\)  \( 2^{2} \cdot 3 \cdot 103 \)
\( I_6 \)  \(=\) \(15984\) \(=\)  \( 2^{4} \cdot 3^{3} \cdot 37 \)
\( I_{10} \)  \(=\) \(33124\) \(=\)  \( 2^{2} \cdot 7^{2} \cdot 13^{2} \)
\( J_2 \)  \(=\) \(36\) \(=\)  \( 2^{2} \cdot 3^{2} \)
\( J_4 \)  \(=\) \(-152\) \(=\)  \( - 2^{3} \cdot 19 \)
\( J_6 \)  \(=\) \(392\) \(=\)  \( 2^{3} \cdot 7^{2} \)
\( J_8 \)  \(=\) \(-2248\) \(=\)  \( - 2^{3} \cdot 281 \)
\( J_{10} \)  \(=\) \(8281\) \(=\)  \( 7^{2} \cdot 13^{2} \)
\( g_1 \)  \(=\) \(60466176/8281\)
\( g_2 \)  \(=\) \(-7091712/8281\)
\( g_3 \)  \(=\) \(10368/169\)

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

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : -4 : 2)\) \((2 : -4 : 1)\) \((1 : -11 : 2)\) \((2 : -11 : 1)\)

magma: [C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-11,2],C![1,-4,2],C![1,-1,0],C![1,0,0],C![2,-11,1],C![2,-4,1]];
 

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: \(\Z \times \Z \times \Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.600818\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.284784\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(3\)

2-torsion field: 4.2.1456.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.150828 \)
Real period: \( 23.53747 \)
Tamagawa product: \( 1 \)
Torsion order:\( 3 \)
Leading coefficient: \( 0.394457 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

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 91.b2
  Elliptic curve 91.a1

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 \(\Q\).