Properties

Label 11449.a.11449.1
Conductor $11449$
Discriminant $11449$
Mordell-Weil group \(\Z \oplus \Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{RM}\)
\(\End(J) \otimes \Q\) \(\mathsf{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

This is a model for the quotient of the modular curve $X_0(107)$ by its Fricke involution $w_{107}$; this quotient is also denoted $X_0^+(107)$.

Minimal equation

Minimal equation

Simplified equation

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -1, -1, 0, 1]), R([1, 0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -1, -1, 0, 1], R![1, 0, 1, 1]);
 
sage: X = HyperellipticCurve(R([-3, -4, -2, 2, 5, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(11449\) \(=\) \( 107^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(11449\) \(=\) \( 107^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(292\) \(=\)  \( 2^{2} \cdot 73 \)
\( I_4 \)  \(=\) \(1\) \(=\)  \( 1 \)
\( I_6 \)  \(=\) \(154177\) \(=\)  \( 53 \cdot 2909 \)
\( I_{10} \)  \(=\) \(1465472\) \(=\)  \( 2^{7} \cdot 107^{2} \)
\( J_2 \)  \(=\) \(73\) \(=\)  \( 73 \)
\( J_4 \)  \(=\) \(222\) \(=\)  \( 2 \cdot 3 \cdot 37 \)
\( J_6 \)  \(=\) \(-1240\) \(=\)  \( - 2^{3} \cdot 5 \cdot 31 \)
\( J_8 \)  \(=\) \(-34951\) \(=\)  \( - 7 \cdot 4993 \)
\( J_{10} \)  \(=\) \(11449\) \(=\)  \( 107^{2} \)
\( g_1 \)  \(=\) \(2073071593/11449\)
\( g_2 \)  \(=\) \(86361774/11449\)
\( g_3 \)  \(=\) \(-6607960/11449\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (-1 : -1 : 1),\, (1 : -1 : 1),\, (1 : -2 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (-1 : -1 : 1),\, (1 : -1 : 1),\, (1 : -2 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (1 : -1 : 1),\, (1 : 1 : 1)\)

magma: [C![-1,-1,1],C![-1,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0]]; // minimal model
 
magma: [C![-1,-1,1],C![-1,1,1],C![1,-1,1],C![1,-1,0],C![1,1,1],C![1,1,0]]; // simplified model
 

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: \(\Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.266345\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.190461\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.266345\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.190461\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z - z^3\) \(0.266345\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + z^3\) \(0.190461\) \(\infty\)

2-torsion field: 5.1.732736.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.044970 \)
Real period: \( 11.88300 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.534380 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(107\) \(2\) \(2\) \(1\) \(( 1 + T )^{2}\)

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.12.2 no
\(3\) 3.432.4 no

Sato-Tate group

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

Decomposition of the Jacobian

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

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{5}}{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{5}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);
 

Additional information

The set of rational points on this genus 2 curve with rank 2 Jacobian was computed by Balakrishnan--Dogra--Müller--Tuitman--Vonk using quadratic Chabauty and the Mordell--Weil sieve.