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Properties

Label 492765e
Number of curves $4$
Conductor $492765$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("e1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 492765e have rank \(1\).

Complex multiplication

The elliptic curves in class 492765e do not have complex multiplication.

Modular form 492765.2.a.e

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} + q^{7} + 3 q^{8} + q^{9} + q^{10} + q^{12} - q^{13} - q^{14} + q^{15} - q^{16} + 2 q^{17} - q^{18} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 492765e

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
492765.e4 492765e1 \([1, 1, 1, 2339, -45046]\) \(30080231/36855\) \(-1733875944255\) \([2]\) \(663552\) \(1.0346\) \(\Gamma_0(N)\)-optimal*
492765.e3 492765e2 \([1, 1, 1, -13906, -447922]\) \(6321363049/1863225\) \(87657061626225\) \([2, 2]\) \(1327104\) \(1.3811\) \(\Gamma_0(N)\)-optimal*
492765.e2 492765e3 \([1, 1, 1, -84301, 9041324]\) \(1408317602329/58524375\) \(2753330781849375\) \([2]\) \(2654208\) \(1.7277\) \(\Gamma_0(N)\)-optimal*
492765.e1 492765e4 \([1, 1, 1, -203431, -35396332]\) \(19790357598649/2998905\) \(141086127760305\) \([2]\) \(2654208\) \(1.7277\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 492765e1.