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SageMath
E = EllipticCurve("pi1")
E.isogeny_class()
Elliptic curves in class 342720.pi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 342720.pi1 | 342720pi4 | \([0, 0, 0, -584652, -132780944]\) | \(115650783909361/27072079335\) | \(5173554927426600960\) | \([2]\) | \(6291456\) | \(2.3027\) | |
| 342720.pi2 | 342720pi2 | \([0, 0, 0, -195852, 31603696]\) | \(4347507044161/258084225\) | \(49320663456153600\) | \([2, 2]\) | \(3145728\) | \(1.9561\) | |
| 342720.pi3 | 342720pi1 | \([0, 0, 0, -192972, 32627824]\) | \(4158523459441/16065\) | \(3070069309440\) | \([2]\) | \(1572864\) | \(1.6095\) | \(\Gamma_0(N)\)-optimal |
| 342720.pi4 | 342720pi3 | \([0, 0, 0, 146868, 130444144]\) | \(1833318007919/39525924375\) | \(-7553521777213440000\) | \([2]\) | \(6291456\) | \(2.3027\) |
Rank
sage: E.rank()
The elliptic curves in class 342720.pi have rank \(0\).
Complex multiplication
The elliptic curves in class 342720.pi do not have complex multiplication.Modular form 342720.2.a.pi
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 LMFDB 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 LMFDB labels.
