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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1386.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1386.a1 | 1386b3 | \([1, -1, 0, -362223, -83819205]\) | \(7209828390823479793/49509306\) | \(36092284074\) | \([2]\) | \(6144\) | \(1.6255\) | |
| 1386.a2 | 1386b4 | \([1, -1, 0, -31563, -175689]\) | \(4770223741048753/2740574865798\) | \(1997879077166742\) | \([2]\) | \(6144\) | \(1.6255\) | |
| 1386.a3 | 1386b2 | \([1, -1, 0, -22653, -1303695]\) | \(1763535241378513/4612311396\) | \(3362375007684\) | \([2, 2]\) | \(3072\) | \(1.2790\) | |
| 1386.a4 | 1386b1 | \([1, -1, 0, -873, -36099]\) | \(-100999381393/723148272\) | \(-527175090288\) | \([2]\) | \(1536\) | \(0.93239\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1386.a have rank \(0\).
Complex multiplication
The elliptic curves in class 1386.a do not have complex multiplication.Modular form 1386.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
