Properties

Label 67760.ba
Number of curves $4$
Conductor $67760$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ba1")
 
E.isogeny_class()
 

Elliptic curves in class 67760.ba

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
67760.ba1 67760ba4 \([0, 0, 0, -518243, -143590942]\) \(2121328796049/120050\) \(871120478412800\) \([2]\) \(491520\) \(1.9311\)  
67760.ba2 67760ba3 \([0, 0, 0, -169763, 25158562]\) \(74565301329/5468750\) \(39682966400000000\) \([2]\) \(491520\) \(1.9311\)  
67760.ba3 67760ba2 \([0, 0, 0, -34243, -1972542]\) \(611960049/122500\) \(888898447360000\) \([2, 2]\) \(245760\) \(1.5845\)  
67760.ba4 67760ba1 \([0, 0, 0, 4477, -183678]\) \(1367631/2800\) \(-20317678796800\) \([2]\) \(122880\) \(1.2379\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 67760.ba have rank \(1\).

Complex multiplication

The elliptic curves in class 67760.ba do not have complex multiplication.

Modular form 67760.2.a.ba

sage: E.q_eigenform(10)
 
\(q - q^{5} - q^{7} - 3 q^{9} + 6 q^{13} - 2 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.