Properties

Label 1225.a.6125.1
Conductor 1225
Discriminant 6125
Mordell-Weil group \(\Z/{8}\Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

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The Jacobian of this curve is isogenous to that of the quotient of the modular curve $X_0(35)$ by the involution $W_7$ (see [MR:1373390]), which has discriminant $5^77^3$.

Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(1225\) \(=\) \( 5^{2} \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(6125\) \(=\) \( 5^{3} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-1280\) \(=\)  \( - 2^{8} \cdot 5 \)
\( I_4 \)  \(=\) \(229504\) \(=\)  \( 2^{7} \cdot 11 \cdot 163 \)
\( I_6 \)  \(=\) \(-61598784\) \(=\)  \( - 2^{6} \cdot 3 \cdot 13 \cdot 23 \cdot 29 \cdot 37 \)
\( I_{10} \)  \(=\) \(25088000\) \(=\)  \( 2^{12} \cdot 5^{3} \cdot 7^{2} \)
\( J_2 \)  \(=\) \(-160\) \(=\)  \( - 2^{5} \cdot 5 \)
\( J_4 \)  \(=\) \(-1324\) \(=\)  \( - 2^{2} \cdot 331 \)
\( J_6 \)  \(=\) \(-8791\) \(=\)  \( - 59 \cdot 149 \)
\( J_8 \)  \(=\) \(-86604\) \(=\)  \( - 2^{2} \cdot 3 \cdot 7 \cdot 1031 \)
\( J_{10} \)  \(=\) \(6125\) \(=\)  \( 5^{3} \cdot 7^{2} \)
\( g_1 \)  \(=\) \(-838860800/49\)
\( g_2 \)  \(=\) \(43384832/49\)
\( g_3 \)  \(=\) \(-9001984/245\)

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

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

Number of rational Weierstrass points: \(1\)

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: \(\Z/{8}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 - xz + 2z^2\) \(=\) \(0,\) \(4y\) \(=\) \(xz^2 + 6z^3\) \(0\) \(8\)

2-torsion field: 6.2.98000.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 11.92789 \)
Tamagawa product: \( 2 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.372746 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(5\) \(2\) \(3\) \(2\) \(( 1 - T )^{2}\)
\(7\) \(2\) \(2\) \(1\) \(( 1 + T )^{2}\)

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 \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{17}}{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{17}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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