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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 2205.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2205.f1 | 2205i2 | \([0, 0, 1, -90552, 10509777]\) | \(-19539165184/46875\) | \(-196994059171875\) | \([3]\) | \(8064\) | \(1.6222\) | |
| 2205.f2 | 2205i1 | \([0, 0, 1, 2058, 72630]\) | \(229376/675\) | \(-2836714452075\) | \([]\) | \(2688\) | \(1.0729\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2205.f have rank \(1\).
Complex multiplication
The elliptic curves in class 2205.f do not have complex multiplication.Modular form 2205.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
