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SageMath
E = EllipticCurve("eg1")
E.isogeny_class()
Elliptic curves in class 327184eg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 327184.eg2 | 327184eg1 | \([0, -1, 0, 8056, -108304]\) | \(24167/16\) | \(-38275900309504\) | \([]\) | \(787968\) | \(1.2951\) | \(\Gamma_0(N)\)-optimal |
| 327184.eg1 | 327184eg2 | \([0, -1, 0, -140664, -20810128]\) | \(-128667913/4096\) | \(-9798630479233024\) | \([]\) | \(2363904\) | \(1.8444\) |
Rank
sage: E.rank()
The elliptic curves in class 327184eg have rank \(1\).
Complex multiplication
The elliptic curves in class 327184eg do not have complex multiplication.Modular form 327184.2.a.eg
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 Cremona 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 Cremona labels.
