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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 448.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 448.d1 | 448a4 | \([0, 0, 0, -1196, 15920]\) | \(1443468546/7\) | \(917504\) | \([4]\) | \(128\) | \(0.34454\) | |
| 448.d2 | 448a3 | \([0, 0, 0, -236, -1104]\) | \(11090466/2401\) | \(314703872\) | \([2]\) | \(128\) | \(0.34454\) | |
| 448.d3 | 448a2 | \([0, 0, 0, -76, 240]\) | \(740772/49\) | \(3211264\) | \([2, 2]\) | \(64\) | \(-0.0020328\) | |
| 448.d4 | 448a1 | \([0, 0, 0, 4, 16]\) | \(432/7\) | \(-114688\) | \([2]\) | \(32\) | \(-0.34861\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 448.d have rank \(1\).
Complex multiplication
The elliptic curves in class 448.d do not have complex multiplication.Modular form 448.2.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.
