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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 68450d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68450.f3 | 68450d1 | \([1, 1, 0, -4135, 121685]\) | \(-121945/32\) | \(-2052581127200\) | \([]\) | \(103680\) | \(1.0794\) | \(\Gamma_0(N)\)-optimal |
| 68450.f4 | 68450d2 | \([1, 1, 0, 30090, -898220]\) | \(46969655/32768\) | \(-2101843074252800\) | \([]\) | \(311040\) | \(1.6287\) | |
| 68450.f2 | 68450d3 | \([1, 1, 0, -17825, -10816625]\) | \(-25/2\) | \(-50111843925781250\) | \([]\) | \(518400\) | \(1.8841\) | |
| 68450.f1 | 68450d4 | \([1, 1, 0, -4295950, -3429038500]\) | \(-349938025/8\) | \(-200447375703125000\) | \([]\) | \(1555200\) | \(2.4334\) |
Rank
sage: E.rank()
The elliptic curves in class 68450d have rank \(1\).
Complex multiplication
The elliptic curves in class 68450d do not have complex multiplication.Modular form 68450.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 5 & 15 \\ 3 & 1 & 15 & 5 \\ 5 & 15 & 1 & 3 \\ 15 & 5 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
