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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 19320.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19320.e1 | 19320m3 | \([0, -1, 0, -131696, -7482804]\) | \(123343086124179938/59429226844575\) | \(121711056577689600\) | \([2]\) | \(196608\) | \(1.9721\) | |
| 19320.e2 | 19320m2 | \([0, -1, 0, -108696, -13748004]\) | \(138697437757771876/106292300625\) | \(108843315840000\) | \([2, 2]\) | \(98304\) | \(1.6255\) | |
| 19320.e3 | 19320m1 | \([0, -1, 0, -108676, -13753340]\) | \(554483565352358224/326025\) | \(83462400\) | \([2]\) | \(49152\) | \(1.2789\) | \(\Gamma_0(N)\)-optimal |
| 19320.e4 | 19320m4 | \([0, -1, 0, -86016, -19672020]\) | \(-34366597532983298/61980408984375\) | \(-126935877600000000\) | \([2]\) | \(196608\) | \(1.9721\) |
Rank
sage: E.rank()
The elliptic curves in class 19320.e have rank \(1\).
Complex multiplication
The elliptic curves in class 19320.e do not have complex multiplication.Modular form 19320.2.a.e
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.
