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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 8450.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8450.d1 | 8450d4 | \([1, 1, 0, -530325, -148872875]\) | \(-349938025/8\) | \(-377094453125000\) | \([]\) | \(64800\) | \(1.9104\) | |
| 8450.d2 | 8450d3 | \([1, 1, 0, -2200, -469750]\) | \(-25/2\) | \(-94273613281250\) | \([]\) | \(21600\) | \(1.3611\) | |
| 8450.d3 | 8450d1 | \([1, 1, 0, -510, 5140]\) | \(-121945/32\) | \(-3861447200\) | \([]\) | \(4320\) | \(0.55638\) | \(\Gamma_0(N)\)-optimal |
| 8450.d4 | 8450d2 | \([1, 1, 0, 3715, -37955]\) | \(46969655/32768\) | \(-3954121932800\) | \([]\) | \(12960\) | \(1.1057\) |
Rank
sage: E.rank()
The elliptic curves in class 8450.d have rank \(1\).
Complex multiplication
The elliptic curves in class 8450.d do not have complex multiplication.Modular form 8450.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 15 & 5 \\ 3 & 1 & 5 & 15 \\ 15 & 5 & 1 & 3 \\ 5 & 15 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
