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Properties

Label 496860a
Number of curves $4$
Conductor $496860$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("a1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 496860a have rank \(1\).

Complex multiplication

The elliptic curves in class 496860a do not have complex multiplication.

Modular form 496860.2.a.a

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} - 6 q^{11} + q^{15} - 6 q^{17} + 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 496860a

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
496860.a2 496860a1 \([0, -1, 0, -507901, -138794774]\) \(1594753024/4725\) \(42930915454299600\) \([2]\) \(7464960\) \(2.0613\) \(\Gamma_0(N)\)-optimal*
496860.a3 496860a2 \([0, -1, 0, -300876, -253155384]\) \(-20720464/178605\) \(-25964617666760398080\) \([2]\) \(14929920\) \(2.4079\)  
496860.a1 496860a3 \([0, -1, 0, -2495341, 1402166830]\) \(189123395584/16078125\) \(146084365087547250000\) \([2]\) \(22394880\) \(2.6106\) \(\Gamma_0(N)\)-optimal*
496860.a4 496860a4 \([0, -1, 0, 2680284, 6455647080]\) \(14647977776/132355125\) \(-19241063894411023392000\) \([2]\) \(44789760\) \(2.9572\)  
*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 496860a1.