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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 3978j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3978.j4 | 3978j1 | \([1, -1, 1, 76, -489]\) | \(67419143/169728\) | \(-123731712\) | \([2]\) | \(1024\) | \(0.23661\) | \(\Gamma_0(N)\)-optimal |
| 3978.j3 | 3978j2 | \([1, -1, 1, -644, -5097]\) | \(40459583737/7033104\) | \(5127132816\) | \([2, 2]\) | \(2048\) | \(0.58318\) | |
| 3978.j1 | 3978j3 | \([1, -1, 1, -9824, -372297]\) | \(143820170742457/5826444\) | \(4247477676\) | \([2]\) | \(4096\) | \(0.92975\) | |
| 3978.j2 | 3978j4 | \([1, -1, 1, -2984, 58551]\) | \(4029546653497/351790452\) | \(256455239508\) | \([2]\) | \(4096\) | \(0.92975\) |
Rank
sage: E.rank()
The elliptic curves in class 3978j have rank \(0\).
Complex multiplication
The elliptic curves in class 3978j do not have complex multiplication.Modular form 3978.2.a.j
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.
