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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 49725f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 49725.k3 | 49725f1 | \([1, -1, 1, -5855, -163978]\) | \(1948441249/89505\) | \(1019517890625\) | \([2]\) | \(86016\) | \(1.0658\) | \(\Gamma_0(N)\)-optimal |
| 49725.k2 | 49725f2 | \([1, -1, 1, -15980, 565022]\) | \(39616946929/10989225\) | \(125174141015625\) | \([2, 2]\) | \(172032\) | \(1.4124\) | |
| 49725.k4 | 49725f3 | \([1, -1, 1, 41395, 3663272]\) | \(688699320191/910381875\) | \(-10369818544921875\) | \([2]\) | \(344064\) | \(1.7590\) | |
| 49725.k1 | 49725f4 | \([1, -1, 1, -235355, 44001272]\) | \(126574061279329/16286595\) | \(185514496171875\) | \([2]\) | \(344064\) | \(1.7590\) |
Rank
sage: E.rank()
The elliptic curves in class 49725f have rank \(0\).
Complex multiplication
The elliptic curves in class 49725f do not have complex multiplication.Modular form 49725.2.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 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.
