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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 51714d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51714.e4 | 51714d1 | \([1, -1, 0, 12897, -1035059]\) | \(67419143/169728\) | \(-597229341067008\) | \([2]\) | \(172032\) | \(1.5191\) | \(\Gamma_0(N)\)-optimal |
| 51714.e3 | 51714d2 | \([1, -1, 0, -108783, -11523875]\) | \(40459583737/7033104\) | \(24747690820464144\) | \([2, 2]\) | \(344064\) | \(1.8657\) | |
| 51714.e2 | 51714d3 | \([1, -1, 0, -504243, 127124401]\) | \(4029546653497/351790452\) | \(1237860458154369972\) | \([2]\) | \(688128\) | \(2.2122\) | |
| 51714.e1 | 51714d4 | \([1, -1, 0, -1660203, -822916535]\) | \(143820170742457/5826444\) | \(20501763473815884\) | \([2]\) | \(688128\) | \(2.2122\) |
Rank
sage: E.rank()
The elliptic curves in class 51714d have rank \(2\).
Complex multiplication
The elliptic curves in class 51714d do not have complex multiplication.Modular form 51714.2.a.d
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.
