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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 1755.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1755.c1 | 1755e1 | \([0, 0, 1, -438, 3528]\) | \(-344177344512/65\) | \(-1755\) | \([3]\) | \(216\) | \(0.015108\) | \(\Gamma_0(N)\)-optimal |
| 1755.c2 | 1755e2 | \([0, 0, 1, -378, 4529]\) | \(-303464448/274625\) | \(-5405443875\) | \([3]\) | \(648\) | \(0.56441\) | |
| 1755.c3 | 1755e3 | \([0, 0, 1, 3132, -76552]\) | \(19180290048/25390625\) | \(-4497873046875\) | \([]\) | \(1944\) | \(1.1137\) |
Rank
sage: E.rank()
The elliptic curves in class 1755.c have rank \(1\).
Complex multiplication
The elliptic curves in class 1755.c do not have complex multiplication.Modular form 1755.2.a.c
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.
