Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("dm1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 333270.dm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
333270.dm1 333270dm3 [1, -1, 1, -1778333, 913226991] [2] 5767168  
333270.dm2 333270dm2 [1, -1, 1, -111983, 14064531] [2, 2] 2883584  
333270.dm3 333270dm1 [1, -1, 1, -16763, -523173] [2] 1441792 \(\Gamma_0(N)\)-optimal
333270.dm4 333270dm4 [1, -1, 1, 30847, 47372487] [2] 5767168  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 333270.2.a.dm

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.