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Properties

Label 458640.ga
Number of curves $2$
Conductor $458640$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ga1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 458640.ga have rank \(0\).

Complex multiplication

The elliptic curves in class 458640.ga do not have complex multiplication.

Modular form 458640.2.a.ga

Copy content sage:E.q_eigenform(10)
 
\(q - q^{5} + 4 q^{11} + q^{13} - 6 q^{17} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.

Elliptic curves in class 458640.ga

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.ga1 458640ga2 \([0, 0, 0, -848043, -35543718]\) \(7111117467/4057690\) \(38487379767514030080\) \([2]\) \(8847360\) \(2.4479\) \(\Gamma_0(N)\)-optimal*
458640.ga2 458640ga1 \([0, 0, 0, 210357, -4426758]\) \(108531333/63700\) \(-604197484576358400\) \([2]\) \(4423680\) \(2.1013\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 458640.ga1.