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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 117600.ex
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 117600.ex1 | 117600hd4 | \([0, 1, 0, -80331008, 277083662988]\) | \(60910917333827912/3255076125\) | \(3063651608241000000000\) | \([2]\) | \(10616832\) | \(3.1902\) | |
| 117600.ex2 | 117600hd3 | \([0, 1, 0, -25971633, -47508415137]\) | \(257307998572864/19456203375\) | \(146496183735384000000000\) | \([2]\) | \(10616832\) | \(3.1902\) | |
| 117600.ex3 | 117600hd1 | \([0, 1, 0, -5299758, 3819850488]\) | \(139927692143296/27348890625\) | \(3217569633140625000000\) | \([2, 2]\) | \(5308416\) | \(2.8437\) | \(\Gamma_0(N)\)-optimal |
| 117600.ex4 | 117600hd2 | \([0, 1, 0, 10906992, 22619680488]\) | \(152461584507448/322998046875\) | \(-304003177734375000000000\) | \([2]\) | \(10616832\) | \(3.1902\) |
Rank
sage: E.rank()
The elliptic curves in class 117600.ex have rank \(0\).
Complex multiplication
The elliptic curves in class 117600.ex do not have complex multiplication.Modular form 117600.2.a.ex
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
