Properties

Label 85680fi
Number of curves 4
Conductor 85680
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("85680.dp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 85680fi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
85680.dp4 85680fi1 [0, 0, 0, 2013, 58466] [2] 131072 \(\Gamma_0(N)\)-optimal
85680.dp3 85680fi2 [0, 0, 0, -15987, 638066] [2, 2] 262144  
85680.dp2 85680fi3 [0, 0, 0, -77187, -7672894] [2] 524288  
85680.dp1 85680fi4 [0, 0, 0, -242787, 46043426] [2] 524288  

Rank

sage: E.rank()
 

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

Modular form 85680.2.a.dp

sage: E.q_eigenform(10)
 
\( q + q^{5} - q^{7} - 6q^{13} + q^{17} - 4q^{19} + O(q^{20}) \)

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.