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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 21840a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21840.j4 | 21840a1 | \([0, -1, 0, 49, 546]\) | \(796706816/8996715\) | \(-143947440\) | \([2]\) | \(6144\) | \(0.24638\) | \(\Gamma_0(N)\)-optimal |
| 21840.j3 | 21840a2 | \([0, -1, 0, -796, 8320]\) | \(218156637904/16769025\) | \(4292870400\) | \([2, 2]\) | \(12288\) | \(0.59295\) | |
| 21840.j2 | 21840a3 | \([0, -1, 0, -2616, -41184]\) | \(1934207124196/373156875\) | \(382112640000\) | \([2]\) | \(24576\) | \(0.93953\) | |
| 21840.j1 | 21840a4 | \([0, -1, 0, -12496, 541840]\) | \(210751929444676/1404585\) | \(1438295040\) | \([2]\) | \(24576\) | \(0.93953\) |
Rank
sage: E.rank()
The elliptic curves in class 21840a have rank \(1\).
Complex multiplication
The elliptic curves in class 21840a do not have complex multiplication.Modular form 21840.2.a.a
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.
