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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 3042.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3042.a1 | 3042f3 | \([1, -1, 0, -698931, 225080181]\) | \(-10730978619193/6656\) | \(-23420758473216\) | \([]\) | \(30240\) | \(1.8862\) | |
| 3042.a2 | 3042f2 | \([1, -1, 0, -6876, 439128]\) | \(-10218313/17576\) | \(-61845440343336\) | \([]\) | \(10080\) | \(1.3369\) | |
| 3042.a3 | 3042f1 | \([1, -1, 0, 729, -12609]\) | \(12167/26\) | \(-91487337786\) | \([]\) | \(3360\) | \(0.78755\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3042.a have rank \(0\).
Complex multiplication
The elliptic curves in class 3042.a do not have complex multiplication.Modular form 3042.2.a.a
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
