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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 18240.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18240.bf1 | 18240q4 | \([0, -1, 0, -29646785, 39684026817]\) | \(10993009831928446009969/3767761230468750000\) | \(987696000000000000000000\) | \([2]\) | \(3317760\) | \(3.3061\) | |
| 18240.bf2 | 18240q2 | \([0, -1, 0, -26559425, 52692530625]\) | \(7903870428425797297009/886464000000\) | \(232381218816000000\) | \([2]\) | \(1105920\) | \(2.7568\) | |
| 18240.bf3 | 18240q1 | \([0, -1, 0, -1655745, 828126657]\) | \(-1914980734749238129/20440940544000\) | \(-5358469917966336000\) | \([2]\) | \(552960\) | \(2.4102\) | \(\Gamma_0(N)\)-optimal |
| 18240.bf4 | 18240q3 | \([0, -1, 0, 5471295, 4306073025]\) | \(69096190760262356111/70568821500000000\) | \(-18499193143296000000000\) | \([2]\) | \(1658880\) | \(2.9595\) |
Rank
sage: E.rank()
The elliptic curves in class 18240.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 18240.bf do not have complex multiplication.Modular form 18240.2.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.
