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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 121680dd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121680.bj3 | 121680dd1 | \([0, 0, 0, -329043, 70466578]\) | \(273359449/9360\) | \(134903568805724160\) | \([2]\) | \(1032192\) | \(2.0588\) | \(\Gamma_0(N)\)-optimal |
| 121680.bj2 | 121680dd2 | \([0, 0, 0, -815763, -186424238]\) | \(4165509529/1368900\) | \(19729646937837158400\) | \([2, 2]\) | \(2064384\) | \(2.4054\) | |
| 121680.bj4 | 121680dd3 | \([0, 0, 0, 2347917, -1280424782]\) | \(99317171591/106616250\) | \(-1536635963427701760000\) | \([2]\) | \(4128768\) | \(2.7520\) | |
| 121680.bj1 | 121680dd4 | \([0, 0, 0, -11766963, -15533435918]\) | \(12501706118329/2570490\) | \(37047892583271997440\) | \([2]\) | \(4128768\) | \(2.7520\) |
Rank
sage: E.rank()
The elliptic curves in class 121680dd have rank \(1\).
Complex multiplication
The elliptic curves in class 121680dd do not have complex multiplication.Modular form 121680.2.a.dd
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.
