Properties

Label 205350.dx
Number of curves $6$
Conductor $205350$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("205350.dx1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 205350.dx

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
205350.dx1 205350v3 [1, 0, 0, -11670896838, 485292507728292] [2] 226934784  
205350.dx2 205350v6 [1, 0, 0, -10374453838, -404953052500708] [2] 453869568  
205350.dx3 205350v4 [1, 0, 0, -1003648838, 1374423104292] [2, 2] 226934784  
205350.dx4 205350v2 [1, 0, 0, -729848838, 7573528904292] [2, 2] 113467392  
205350.dx5 205350v1 [1, 0, 0, -28920838, 206074696292] [2] 56733696 \(\Gamma_0(N)\)-optimal
205350.dx6 205350v5 [1, 0, 0, 3986356162, 10960222709292] [2] 453869568  

Rank

sage: E.rank()
 

The elliptic curves in class 205350.dx have rank \(1\).

Modular form 205350.2.a.dx

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + 4q^{11} + q^{12} - 2q^{13} + q^{16} + 2q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.