Show commands:
SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 3600.x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
|---|---|---|---|---|---|---|---|---|---|
| 3600.x1 | 3600bd1 | \([0, 0, 0, 0, -100]\) | \(0\) | \(-4320000\) | \([]\) | \(576\) | \(-0.047887\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
| 3600.x2 | 3600bd2 | \([0, 0, 0, 0, 2700]\) | \(0\) | \(-3149280000\) | \([]\) | \(1728\) | \(0.50142\) | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 3600.x have rank \(1\).
Complex multiplication
Each elliptic curve in class 3600.x has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 3600.2.a.x
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
