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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 1610.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1610.f1 | 1610c3 | \([1, -1, 1, -36598, 2703961]\) | \(5421065386069310769/1919709260\) | \(1919709260\) | \([2]\) | \(3072\) | \(1.1357\) | |
| 1610.f2 | 1610c2 | \([1, -1, 1, -2298, 42281]\) | \(1341518286067569/24894528400\) | \(24894528400\) | \([2, 2]\) | \(1536\) | \(0.78916\) | |
| 1610.f3 | 1610c1 | \([1, -1, 1, -298, -919]\) | \(2917464019569/1262240000\) | \(1262240000\) | \([2]\) | \(768\) | \(0.44258\) | \(\Gamma_0(N)\)-optimal |
| 1610.f4 | 1610c4 | \([1, -1, 1, 2, 121401]\) | \(1367631/6366992112460\) | \(-6366992112460\) | \([2]\) | \(3072\) | \(1.1357\) |
Rank
sage: E.rank()
The elliptic curves in class 1610.f have rank \(1\).
Complex multiplication
The elliptic curves in class 1610.f do not have complex multiplication.Modular form 1610.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
