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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 74529.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
|---|---|---|---|---|---|---|---|---|---|
| 74529.w1 | 74529d1 | \([0, 0, 1, 0, -296]\) | \(0\) | \(-37786203\) | \([]\) | \(9504\) | \(0.13284\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
| 74529.w2 | 74529d2 | \([0, 0, 1, 0, 7985]\) | \(0\) | \(-27546141987\) | \([]\) | \(28512\) | \(0.68214\) | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 74529.w have rank \(0\).
Complex multiplication
Each elliptic curve in class 74529.w has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 74529.2.a.w
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.
