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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 38640q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 38640.bx3 | 38640q1 | \([0, 1, 0, -108676, 13753340]\) | \(554483565352358224/326025\) | \(83462400\) | \([2]\) | \(98304\) | \(1.2789\) | \(\Gamma_0(N)\)-optimal |
| 38640.bx2 | 38640q2 | \([0, 1, 0, -108696, 13748004]\) | \(138697437757771876/106292300625\) | \(108843315840000\) | \([2, 2]\) | \(196608\) | \(1.6255\) | |
| 38640.bx4 | 38640q3 | \([0, 1, 0, -86016, 19672020]\) | \(-34366597532983298/61980408984375\) | \(-126935877600000000\) | \([2]\) | \(393216\) | \(1.9721\) | |
| 38640.bx1 | 38640q4 | \([0, 1, 0, -131696, 7482804]\) | \(123343086124179938/59429226844575\) | \(121711056577689600\) | \([2]\) | \(393216\) | \(1.9721\) |
Rank
sage: E.rank()
The elliptic curves in class 38640q have rank \(1\).
Complex multiplication
The elliptic curves in class 38640q do not have complex multiplication.Modular form 38640.2.a.q
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.
