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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 53550ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53550.o4 | 53550ba1 | \([1, -1, 0, 120708, 1935616]\) | \(17075848639751/10028415600\) | \(-114229921443750000\) | \([2]\) | \(442368\) | \(1.9625\) | \(\Gamma_0(N)\)-optimal |
| 53550.o3 | 53550ba2 | \([1, -1, 0, -486792, 15908116]\) | \(1119971462469049/638680075740\) | \(7274965237725937500\) | \([2]\) | \(884736\) | \(2.3090\) | |
| 53550.o2 | 53550ba3 | \([1, -1, 0, -1772667, 956368741]\) | \(-54082626581000809/3358656000000\) | \(-38257191000000000000\) | \([2]\) | \(1327104\) | \(2.5118\) | |
| 53550.o1 | 53550ba4 | \([1, -1, 0, -28772667, 59411368741]\) | \(231268521845235080809/816013464000\) | \(9294903363375000000\) | \([2]\) | \(2654208\) | \(2.8584\) |
Rank
sage: E.rank()
The elliptic curves in class 53550ba have rank \(1\).
Complex multiplication
The elliptic curves in class 53550ba do not have complex multiplication.Modular form 53550.2.a.ba
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
