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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 8400d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8400.b3 | 8400d1 | \([0, -1, 0, -7908, 273312]\) | \(13674725584/945\) | \(3780000000\) | \([2]\) | \(9216\) | \(0.89147\) | \(\Gamma_0(N)\)-optimal |
| 8400.b2 | 8400d2 | \([0, -1, 0, -8408, 237312]\) | \(4108974916/893025\) | \(14288400000000\) | \([2, 2]\) | \(18432\) | \(1.2380\) | |
| 8400.b1 | 8400d3 | \([0, -1, 0, -43408, -3262688]\) | \(282678688658/18600435\) | \(595213920000000\) | \([2]\) | \(36864\) | \(1.5846\) | |
| 8400.b4 | 8400d4 | \([0, -1, 0, 18592, 1425312]\) | \(22208984782/40516875\) | \(-1296540000000000\) | \([2]\) | \(36864\) | \(1.5846\) |
Rank
sage: E.rank()
The elliptic curves in class 8400d have rank \(1\).
Complex multiplication
The elliptic curves in class 8400d do not have complex multiplication.Modular form 8400.2.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 Cremona 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 Cremona labels.
