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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 121242o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121242.p4 | 121242o1 | \([1, 0, 1, 37870, -804652]\) | \(3390548817167/2121250032\) | \(-3757923827939952\) | \([2]\) | \(921600\) | \(1.6776\) | \(\Gamma_0(N)\)-optimal |
| 121242.p3 | 121242o2 | \([1, 0, 1, -158150, -6606844]\) | \(246928513749553/132296603076\) | \(234371502441921636\) | \([2, 2]\) | \(1843200\) | \(2.0242\) | |
| 121242.p2 | 121242o3 | \([1, 0, 1, -1475840, 684916868]\) | \(200671182100438993/1694040387138\) | \(3001095882278582418\) | \([2]\) | \(3686400\) | \(2.3708\) | |
| 121242.p1 | 121242o4 | \([1, 0, 1, -1976780, -1068686764]\) | \(482216760027353233/644362796286\) | \(1141527999751222446\) | \([2]\) | \(3686400\) | \(2.3708\) |
Rank
sage: E.rank()
The elliptic curves in class 121242o have rank \(0\).
Complex multiplication
The elliptic curves in class 121242o do not have complex multiplication.Modular form 121242.2.a.o
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.
