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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)
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.