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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 1323.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
|---|---|---|---|---|---|---|---|---|---|
| 1323.j1 | 1323m1 | \([0, 0, 1, 0, -2]\) | \(0\) | \(-1323\) | \([]\) | \(36\) | \(-0.72215\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
| 1323.j2 | 1323m2 | \([0, 0, 1, 0, 47]\) | \(0\) | \(-964467\) | \([]\) | \(108\) | \(-0.17284\) | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 1323.j have rank \(1\).
Complex multiplication
Each elliptic curve in class 1323.j has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 1323.2.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 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.
