Properties

Label 722.e
Number of curves $3$
Conductor $722$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 722.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
722.e1 722e3 \([1, 1, 1, -30873, 16782247]\) \(-69173457625/2550136832\) \(-119973433931988992\) \([]\) \(6480\) \(1.9574\)  
722.e2 722e1 \([1, 1, 1, -5603, -163815]\) \(-413493625/152\) \(-7150973912\) \([]\) \(720\) \(0.85878\) \(\Gamma_0(N)\)-optimal
722.e3 722e2 \([1, 1, 1, 3422, -612177]\) \(94196375/3511808\) \(-165216101262848\) \([]\) \(2160\) \(1.4081\)  

Rank

sage: E.rank()
 

The elliptic curves in class 722.e have rank \(1\).

Complex multiplication

The elliptic curves in class 722.e do not have complex multiplication.

Modular form 722.2.a.e

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} - 2q^{9} - 6q^{11} - q^{12} - 5q^{13} - q^{14} + q^{16} + 3q^{17} - 2q^{18} + O(q^{20})\)  Toggle raw display

Isogeny matrix

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.

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.