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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 429429cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 429429.cm2 | 429429cm1 | \([0, 1, 1, -149959, 6800239]\) | \(360448/189\) | \(195552512071697781\) | \([]\) | \(4561920\) | \(2.0096\) | \(\Gamma_0(N)\)-optimal |
| 429429.cm1 | 429429cm2 | \([0, 1, 1, -6898129, -6975531260]\) | \(35084566528/1029\) | \(1064674787945910141\) | \([]\) | \(13685760\) | \(2.5589\) |
Rank
sage: E.rank()
The elliptic curves in class 429429cm have rank \(1\).
Complex multiplication
The elliptic curves in class 429429cm do not have complex multiplication.Modular form 429429.2.a.cm
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.
