Properties

Label 333270de
Number of curves $4$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("333270.de1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 333270de

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
333270.de3 333270de1 [1, -1, 1, -24698, -1302919] [2] 1216512 \(\Gamma_0(N)\)-optimal
333270.de4 333270de2 [1, -1, 1, 38782, -6939943] [2] 2433024  
333270.de1 333270de3 [1, -1, 1, -500798, 136364041] [2] 3649536  
333270.de2 333270de4 [1, -1, 1, -357968, 215663257] [2] 7299072  

Rank

sage: E.rank()
 

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

Modular form 333270.2.a.de

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

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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.