Properties

Label 377706.cl
Number of curves $4$
Conductor $377706$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("cl1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 377706.cl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
377706.cl1 377706cl4 \([1, 1, 1, -2686802, -1696243417]\) \(14489843500598257/6246072\) \(924642821278008\) \([2]\) \(8650752\) \(2.2143\)  
377706.cl2 377706cl3 \([1, 1, 1, -359202, 43650279]\) \(34623662831857/14438442312\) \(2137407643432135368\) \([2]\) \(8650752\) \(2.2143\)  
377706.cl3 377706cl2 \([1, 1, 1, -168762, -26279289]\) \(3590714269297/73410624\) \(10867406985884736\) \([2, 2]\) \(4325376\) \(1.8677\)  
377706.cl4 377706cl1 \([1, 1, 1, 518, -1225849]\) \(103823/4386816\) \(-649406206439424\) \([2]\) \(2162688\) \(1.5212\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 377706.cl have rank \(2\).

Complex multiplication

The elliptic curves in class 377706.cl do not have complex multiplication.

Modular form 377706.2.a.cl

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + 2q^{10} - q^{12} - 6q^{13} + q^{14} - 2q^{15} + q^{16} - q^{17} + q^{18} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.