Properties

Label 324870.bw
Number of curves 4
Conductor 324870
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("324870.bw1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 324870.bw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
324870.bw1 324870bw3 [1, 0, 1, -36260614, 84039825272] [2] 23592960  
324870.bw2 324870bw4 [1, 0, 1, -4414534, -1540724104] [2] 23592960  
324870.bw3 324870bw2 [1, 0, 1, -2274214, 1303333112] [2, 2] 11796480  
324870.bw4 324870bw1 [1, 0, 1, -16294, 55154936] [2] 5898240 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 324870.bw have rank \(1\).

Modular form 324870.2.a.bw

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} + q^{10} + q^{12} + q^{13} - q^{15} + q^{16} + q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.