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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 1710.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1710.f1 | 1710d3 | \([1, -1, 0, -27360, -1735074]\) | \(3107086841064961/570\) | \(415530\) | \([2]\) | \(3072\) | \(0.91382\) | |
| 1710.f2 | 1710d4 | \([1, -1, 0, -1980, -17550]\) | \(1177918188481/488703750\) | \(356265033750\) | \([2]\) | \(3072\) | \(0.91382\) | |
| 1710.f3 | 1710d2 | \([1, -1, 0, -1710, -26784]\) | \(758800078561/324900\) | \(236852100\) | \([2, 2]\) | \(1536\) | \(0.56725\) | |
| 1710.f4 | 1710d1 | \([1, -1, 0, -90, -540]\) | \(-111284641/123120\) | \(-89754480\) | \([2]\) | \(768\) | \(0.22067\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1710.f have rank \(0\).
Complex multiplication
The elliptic curves in class 1710.f do not have complex multiplication.Modular form 1710.2.a.f
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.
