Properties

Label 841.a.841.1
Conductor $841$
Discriminant $-841$
Mordell-Weil group \(\Z/{7}\Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{RM}\)
\(\End(J) \otimes \Q\) \(\mathsf{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

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The Jacobian of this curve is isogenous to that of the modular curve $X_0(29)$ (which has discriminant $29^5$).

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2 + x)y = x^4 + x^3 + 3x^2 + x + 2$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2)y = x^4z^2 + x^3z^3 + 3x^2z^4 + xz^5 + 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 7x^4 + 6x^3 + 13x^2 + 4x + 8$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 1, 3, 1, 1]), R([0, 1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 1, 3, 1, 1], R![0, 1, 1, 1]);
 
sage: X = HyperellipticCurve(R([8, 4, 13, 6, 7, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(841\) \(=\) \( 29^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-841\) \(=\) \( - 29^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1420\) \(=\)  \( 2^{2} \cdot 5 \cdot 71 \)
\( I_4 \)  \(=\) \(4201\) \(=\)  \( 4201 \)
\( I_6 \)  \(=\) \(1973899\) \(=\)  \( 61 \cdot 32359 \)
\( I_{10} \)  \(=\) \(107648\) \(=\)  \( 2^{7} \cdot 29^{2} \)
\( J_2 \)  \(=\) \(355\) \(=\)  \( 5 \cdot 71 \)
\( J_4 \)  \(=\) \(5076\) \(=\)  \( 2^{2} \cdot 3^{3} \cdot 47 \)
\( J_6 \)  \(=\) \(93408\) \(=\)  \( 2^{5} \cdot 3 \cdot 7 \cdot 139 \)
\( J_8 \)  \(=\) \(1848516\) \(=\)  \( 2^{2} \cdot 3 \cdot 154043 \)
\( J_{10} \)  \(=\) \(841\) \(=\)  \( 29^{2} \)
\( g_1 \)  \(=\) \(5638216721875/841\)
\( g_2 \)  \(=\) \(227094529500/841\)
\( g_3 \)  \(=\) \(11771743200/841\)

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

magma: [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: \(\Z/{7}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + z^2\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0\) \(7\)

2-torsion field: 6.0.53824.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 14.28455 \)
Tamagawa product: \( 1 \)
Torsion order:\( 7 \)
Leading coefficient: \( 0.291521 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(29\) \(2\) \(2\) \(1\) \(( 1 - T )^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\sqrt{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{2}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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