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SageMath
E = EllipticCurve("dj1")
E.isogeny_class()
Elliptic curves in class 107712.dj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 107712.dj1 | 107712dm4 | \([0, 0, 0, -1115724, -420411760]\) | \(803760366578833/65593817586\) | \(12535173747885735936\) | \([2]\) | \(2359296\) | \(2.4073\) | |
| 107712.dj2 | 107712dm2 | \([0, 0, 0, -234444, 36091280]\) | \(7457162887153/1370924676\) | \(261987785455435776\) | \([2, 2]\) | \(1179648\) | \(2.0607\) | |
| 107712.dj3 | 107712dm1 | \([0, 0, 0, -222924, 40510352]\) | \(6411014266033/296208\) | \(56606230315008\) | \([2]\) | \(589824\) | \(1.7141\) | \(\Gamma_0(N)\)-optimal |
| 107712.dj4 | 107712dm3 | \([0, 0, 0, 462516, 209773712]\) | \(57258048889007/132611470002\) | \(-25342446569116925952\) | \([2]\) | \(2359296\) | \(2.4073\) |
Rank
sage: E.rank()
The elliptic curves in class 107712.dj have rank \(1\).
Complex multiplication
The elliptic curves in class 107712.dj do not have complex multiplication.Modular form 107712.2.a.dj
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
