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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 12240r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12240.br3 | 12240r1 | \([0, 0, 0, -1287, 17766]\) | \(1263257424/425\) | \(79315200\) | \([2]\) | \(4096\) | \(0.48802\) | \(\Gamma_0(N)\)-optimal |
| 12240.br2 | 12240r2 | \([0, 0, 0, -1467, 12474]\) | \(467720676/180625\) | \(134835840000\) | \([2, 2]\) | \(8192\) | \(0.83460\) | |
| 12240.br1 | 12240r3 | \([0, 0, 0, -10467, -403326]\) | \(84944038338/2088025\) | \(3117404620800\) | \([2]\) | \(16384\) | \(1.1812\) | |
| 12240.br4 | 12240r4 | \([0, 0, 0, 4653, 89586]\) | \(7462174302/6640625\) | \(-9914400000000\) | \([2]\) | \(16384\) | \(1.1812\) |
Rank
sage: E.rank()
The elliptic curves in class 12240r have rank \(1\).
Complex multiplication
The elliptic curves in class 12240r do not have complex multiplication.Modular form 12240.2.a.r
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.
