Properties

Label 85680.bf
Number of curves $4$
Conductor $85680$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bf1")
 
E.isogeny_class()
 

Elliptic curves in class 85680.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
85680.bf1 85680dp4 \([0, 0, 0, -146163, 16597618]\) \(115650783909361/27072079335\) \(80836795741040640\) \([2]\) \(786432\) \(1.9561\)  
85680.bf2 85680dp2 \([0, 0, 0, -48963, -3950462]\) \(4347507044161/258084225\) \(770635366502400\) \([2, 2]\) \(393216\) \(1.6095\)  
85680.bf3 85680dp1 \([0, 0, 0, -48243, -4078478]\) \(4158523459441/16065\) \(47969832960\) \([2]\) \(196608\) \(1.2630\) \(\Gamma_0(N)\)-optimal
85680.bf4 85680dp3 \([0, 0, 0, 36717, -16305518]\) \(1833318007919/39525924375\) \(-118023777768960000\) \([2]\) \(786432\) \(1.9561\)  

Rank

sage: E.rank()
 

The elliptic curves in class 85680.bf have rank \(1\).

Complex multiplication

The elliptic curves in class 85680.bf do not have complex multiplication.

Modular form 85680.2.a.bf

sage: E.q_eigenform(10)
 
\(q - q^{5} - q^{7} + 4 q^{11} + 2 q^{13} - q^{17} - 8 q^{19} + O(q^{20})\) 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.