Properties

Label 69696gp
Number of curves 4
Conductor 69696
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("69696.cb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 69696gp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
69696.cb3 69696gp1 [0, 0, 0, -454476, 116936336] [2] 737280 \(\Gamma_0(N)\)-optimal
69696.cb2 69696gp2 [0, 0, 0, -802956, -86994160] [2, 2] 1474560  
69696.cb4 69696gp3 [0, 0, 0, 3030324, -677319280] [2] 2949120  
69696.cb1 69696gp4 [0, 0, 0, -10211916, -12548220784] [2] 2949120  

Rank

sage: E.rank()
 

The elliptic curves in class 69696gp have rank \(0\).

Modular form 69696.2.a.cb

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 Cremona 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 Cremona labels.