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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 2112.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2112.d1 | 2112w3 | \([0, -1, 0, -1889, -30975]\) | \(5690357426/891\) | \(116785152\) | \([2]\) | \(1024\) | \(0.55899\) | |
| 2112.d2 | 2112w2 | \([0, -1, 0, -129, -351]\) | \(3650692/1089\) | \(71368704\) | \([2, 2]\) | \(512\) | \(0.21241\) | |
| 2112.d3 | 2112w1 | \([0, -1, 0, -49, 145]\) | \(810448/33\) | \(540672\) | \([2]\) | \(256\) | \(-0.13416\) | \(\Gamma_0(N)\)-optimal |
| 2112.d4 | 2112w4 | \([0, -1, 0, 351, -2751]\) | \(36382894/43923\) | \(-5757075456\) | \([4]\) | \(1024\) | \(0.55899\) |
Rank
sage: E.rank()
The elliptic curves in class 2112.d have rank \(1\).
Complex multiplication
The elliptic curves in class 2112.d do not have complex multiplication.Modular form 2112.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
