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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1755.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1755.d1 | 1755c2 | \([0, 0, 1, -3942, -95263]\) | \(-344177344512/65\) | \(-1279395\) | \([]\) | \(648\) | \(0.56441\) | |
| 1755.d2 | 1755c1 | \([0, 0, 1, -42, -168]\) | \(-303464448/274625\) | \(-7414875\) | \([3]\) | \(216\) | \(0.015108\) | \(\Gamma_0(N)\)-optimal |
| 1755.d3 | 1755c3 | \([0, 0, 1, 348, 2835]\) | \(19180290048/25390625\) | \(-6169921875\) | \([3]\) | \(648\) | \(0.56441\) |
Rank
sage: E.rank()
The elliptic curves in class 1755.d have rank \(1\).
Complex multiplication
The elliptic curves in class 1755.d do not have complex multiplication.Modular form 1755.2.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 LMFDB numbering.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
