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SageMath
E = EllipticCurve("gm1")
E.isogeny_class()
Elliptic curves in class 33600.gm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33600.gm1 | 33600gl4 | \([0, 1, 0, -403233, -98690337]\) | \(7080974546692/189\) | \(193536000000\) | \([2]\) | \(196608\) | \(1.6795\) | |
| 33600.gm2 | 33600gl3 | \([0, 1, 0, -39233, 345663]\) | \(6522128932/3720087\) | \(3809369088000000\) | \([2]\) | \(196608\) | \(1.6795\) | |
| 33600.gm3 | 33600gl2 | \([0, 1, 0, -25233, -1544337]\) | \(6940769488/35721\) | \(9144576000000\) | \([2, 2]\) | \(98304\) | \(1.3329\) | |
| 33600.gm4 | 33600gl1 | \([0, 1, 0, -733, -49837]\) | \(-2725888/64827\) | \(-1037232000000\) | \([2]\) | \(49152\) | \(0.98633\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33600.gm have rank \(0\).
Complex multiplication
The elliptic curves in class 33600.gm do not have complex multiplication.Modular form 33600.2.a.gm
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
