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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 86640.cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 86640.cf1 | 86640dp4 | \([0, 1, 0, -2675622376, 34014727813940]\) | \(10993009831928446009969/3767761230468750000\) | \(726047320002750000000000000000\) | \([2]\) | \(149299200\) | \(4.4317\) | |
| 86640.cf2 | 86640dp2 | \([0, 1, 0, -2396988136, 45168868986164]\) | \(7903870428425797297009/886464000000\) | \(170821549485195264000000\) | \([2]\) | \(49766400\) | \(3.8824\) | |
| 86640.cf3 | 86640dp1 | \([0, 1, 0, -149431016, 709492084020]\) | \(-1914980734749238129/20440940544000\) | \(-3938967782855062585344000\) | \([2]\) | \(24883200\) | \(3.5358\) | \(\Gamma_0(N)\)-optimal |
| 86640.cf4 | 86640dp3 | \([0, 1, 0, 493784344, 3693647605044]\) | \(69096190760262356111/70568821500000000\) | \(-13598606862742493184000000000\) | \([2]\) | \(74649600\) | \(4.0852\) |
Rank
sage: E.rank()
The elliptic curves in class 86640.cf have rank \(0\).
Complex multiplication
The elliptic curves in class 86640.cf do not have complex multiplication.Modular form 86640.2.a.cf
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.
