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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 3864.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3864.d1 | 3864e3 | \([0, 1, 0, -18992, -1013472]\) | \(369937818893666/123409881\) | \(252743436288\) | \([2]\) | \(8192\) | \(1.1607\) | |
| 3864.d2 | 3864e4 | \([0, 1, 0, -9632, 352800]\) | \(48260105780546/1193313807\) | \(2443906676736\) | \([2]\) | \(8192\) | \(1.1607\) | |
| 3864.d3 | 3864e2 | \([0, 1, 0, -1352, -11520]\) | \(267100692772/102880449\) | \(105349579776\) | \([2, 2]\) | \(4096\) | \(0.81410\) | |
| 3864.d4 | 3864e1 | \([0, 1, 0, 268, -1152]\) | \(8284506032/7394247\) | \(-1892927232\) | \([4]\) | \(2048\) | \(0.46752\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3864.d have rank \(0\).
Complex multiplication
The elliptic curves in class 3864.d do not have complex multiplication.Modular form 3864.2.a.d
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.
