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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 256.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
|---|---|---|---|---|---|---|---|---|---|
| 256.d1 | 256d2 | \([0, -1, 0, -13, 21]\) | \(8000\) | \(32768\) | \([2]\) | \(16\) | \(-0.40397\) | \(-8\) | |
| 256.d2 | 256d1 | \([0, -1, 0, -3, -1]\) | \(8000\) | \(512\) | \([2]\) | \(8\) | \(-0.75055\) | \(\Gamma_0(N)\)-optimal | \(-8\) |
Rank
sage: E.rank()
The elliptic curves in class 256.d have rank \(0\).
Complex multiplication
Each elliptic curve in class 256.d has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-2}) \).Modular form 256.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
