Properties

Label 78650.bf
Number of curves $4$
Conductor $78650$
CM no
Rank $1$
Graph

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

Elliptic curves in class 78650.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
78650.bf1 78650u3 \([1, 1, 0, -627750, 183492500]\) \(988345570681/44994560\) \(1245478245440000000\) \([2]\) \(1866240\) \(2.2341\)  
78650.bf2 78650u1 \([1, 1, 0, -98375, -11846875]\) \(3803721481/26000\) \(719696656250000\) \([2]\) \(622080\) \(1.6848\) \(\Gamma_0(N)\)-optimal
78650.bf3 78650u2 \([1, 1, 0, -37875, -26185375]\) \(-217081801/10562500\) \(-292376766601562500\) \([2]\) \(1244160\) \(2.0314\)  
78650.bf4 78650u4 \([1, 1, 0, 340250, 699436500]\) \(157376536199/7722894400\) \(-213774664471225000000\) \([2]\) \(3732480\) \(2.5807\)  

Rank

sage: E.rank()
 

The elliptic curves in class 78650.bf have rank \(1\).

Complex multiplication

The elliptic curves in class 78650.bf do not have complex multiplication.

Modular form 78650.2.a.bf

sage: E.q_eigenform(10)
 
\(q - q^{2} + 2 q^{3} + q^{4} - 2 q^{6} - 4 q^{7} - q^{8} + q^{9} + 2 q^{12} + q^{13} + 4 q^{14} + q^{16} - 6 q^{17} - q^{18} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.