Properties

Label 333270.ev
Number of curves $2$
Conductor $333270$
CM no
Rank $2$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("ev1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 333270.ev

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.ev1 333270ev2 \([1, -1, 1, -178292, 25468759]\) \(70665260180607/9411920000\) \(83481311536560000\) \([2]\) \(4128768\) \(1.9747\)  
333270.ev2 333270ev1 \([1, -1, 1, -45812, -3358889]\) \(1198785674367/140492800\) \(1246135029350400\) \([2]\) \(2064384\) \(1.6281\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 333270.ev have rank \(2\).

Complex multiplication

The elliptic curves in class 333270.ev do not have complex multiplication.

Modular form 333270.2.a.ev

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} - 2q^{11} - 4q^{13} + q^{14} + q^{16} - 6q^{17} - 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.