Properties

Label 69696.bd
Number of curves 4
Conductor 69696
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("69696.bd1")
sage: E.isogeny_class()

Elliptic curves in class 69696.bd

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
69696.bd1 69696cw4 [0, 0, 0, -10211916, 12548220784] 2 2949120  
69696.bd2 69696cw2 [0, 0, 0, -802956, 86994160] 4 1474560  
69696.bd3 69696cw1 [0, 0, 0, -454476, -116936336] 2 737280 \(\Gamma_0(N)\)-optimal
69696.bd4 69696cw3 [0, 0, 0, 3030324, 677319280] 2 2949120  

Rank

sage: E.rank()

The elliptic curves in class 69696.bd have rank \(1\).

Modular form 69696.2.a.bd

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

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.