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SageMath
E = EllipticCurve("cx1")
E.isogeny_class()
Elliptic curves in class 212160cx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.dl3 | 212160cx1 | \([0, -1, 0, -1665, -24543]\) | \(1948441249/89505\) | \(23463198720\) | \([2]\) | \(229376\) | \(0.75153\) | \(\Gamma_0(N)\)-optimal |
| 212160.dl2 | 212160cx2 | \([0, -1, 0, -4545, 86625]\) | \(39616946929/10989225\) | \(2880759398400\) | \([2, 2]\) | \(458752\) | \(1.0981\) | |
| 212160.dl1 | 212160cx3 | \([0, -1, 0, -66945, 6688545]\) | \(126574061279329/16286595\) | \(4269433159680\) | \([2]\) | \(917504\) | \(1.4447\) | |
| 212160.dl4 | 212160cx4 | \([0, -1, 0, 11775, 553377]\) | \(688699320191/910381875\) | \(-238651146240000\) | \([2]\) | \(917504\) | \(1.4447\) |
Rank
sage: E.rank()
The elliptic curves in class 212160cx have rank \(1\).
Complex multiplication
The elliptic curves in class 212160cx do not have complex multiplication.Modular form 212160.2.a.cx
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
