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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 105393j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 105393.d3 | 105393j1 | \([1, 1, 1, -2812, 48428]\) | \(389017/57\) | \(360317693793\) | \([2]\) | \(120960\) | \(0.94244\) | \(\Gamma_0(N)\)-optimal |
| 105393.d2 | 105393j2 | \([1, 1, 1, -12057, -465594]\) | \(30664297/3249\) | \(20538108546201\) | \([2, 2]\) | \(241920\) | \(1.2890\) | |
| 105393.d4 | 105393j3 | \([1, 1, 1, 15678, -2262822]\) | \(67419143/390963\) | \(-2471419061726187\) | \([2]\) | \(483840\) | \(1.6356\) | |
| 105393.d1 | 105393j4 | \([1, 1, 1, -187712, -31380874]\) | \(115714886617/1539\) | \(9728577732411\) | \([2]\) | \(483840\) | \(1.6356\) |
Rank
sage: E.rank()
The elliptic curves in class 105393j have rank \(1\).
Complex multiplication
The elliptic curves in class 105393j do not have complex multiplication.Modular form 105393.2.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.
