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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 55440cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.ca1 | 55440cd1 | \([0, 0, 0, -4203, 104858]\) | \(74246873427/16940\) | \(1873428480\) | \([2]\) | \(49152\) | \(0.77104\) | \(\Gamma_0(N)\)-optimal |
| 55440.ca2 | 55440cd2 | \([0, 0, 0, -3723, 129722]\) | \(-51603494067/35870450\) | \(-3966984806400\) | \([2]\) | \(98304\) | \(1.1176\) |
Rank
sage: E.rank()
The elliptic curves in class 55440cd have rank \(2\).
Complex multiplication
The elliptic curves in class 55440cd do not have complex multiplication.Modular form 55440.2.a.cd
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
