Properties

Label 69696.cb
Number of curves $4$
Conductor $69696$
CM no
Rank $0$
Graph

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

Elliptic curves in class 69696.cb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
69696.cb1 69696gp4 \([0, 0, 0, -10211916, -12548220784]\) \(347873904937/395307\) \(133831413837721239552\) \([2]\) \(2949120\) \(2.7753\)  
69696.cb2 69696gp2 \([0, 0, 0, -802956, -86994160]\) \(169112377/88209\) \(29863208046433665024\) \([2, 2]\) \(1474560\) \(2.4288\)  
69696.cb3 69696gp1 \([0, 0, 0, -454476, 116936336]\) \(30664297/297\) \(100549522041864192\) \([2]\) \(737280\) \(2.0822\) \(\Gamma_0(N)\)-optimal
69696.cb4 69696gp3 \([0, 0, 0, 3030324, -677319280]\) \(9090072503/5845851\) \(-1979116242350012891136\) \([2]\) \(2949120\) \(2.7753\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 69696.cb do not have complex multiplication.

Modular form 69696.2.a.cb

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

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.