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Properties

Label 462722.i
Number of curves $2$
Conductor $462722$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("i1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 462722.i have rank \(0\).

Complex multiplication

The elliptic curves in class 462722.i do not have complex multiplication.

Modular form 462722.2.a.i

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - 3 q^{3} + q^{4} - q^{5} - 3 q^{6} - q^{7} + q^{8} + 6 q^{9} - q^{10} + 2 q^{11} - 3 q^{12} - q^{14} + 3 q^{15} + q^{16} + 3 q^{17} + 6 q^{18} + 6 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 462722.i

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462722.i1 462722i2 \([1, -1, 1, -49207593, -145760144877]\) \(-1064019559329/125497034\) \(-1554189319224867033818954\) \([]\) \(121504320\) \(3.3777\)  
462722.i2 462722i1 \([1, -1, 1, -621783, 288799983]\) \(-2146689/1664\) \(-20607427480638137984\) \([]\) \(17357760\) \(2.4048\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 462722.i1.