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Properties

Label 333270ef
Number of curves $2$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ef1")
 
E.isogeny_class()
 

Elliptic curves in class 333270ef

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.ef2 333270ef1 \([1, -1, 1, -2416307, -1440763761]\) \(14457238157881/49990500\) \(5394882931500730500\) \([2]\) \(9732096\) \(2.4573\) \(\Gamma_0(N)\)-optimal
333270.ef1 333270ef2 \([1, -1, 1, -3511337, -3646389]\) \(44365623586201/25674468750\) \(2770741505580538218750\) \([2]\) \(19464192\) \(2.8039\)  

Rank

sage: E.rank()
 

The elliptic curves in class 333270ef have rank \(1\).

Complex multiplication

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

Modular form 333270.2.a.ef

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