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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 89232.cs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 89232.cs1 | 89232co4 | \([0, 1, 0, -396192, -96023628]\) | \(347873904937/395307\) | \(7815460394446848\) | \([2]\) | \(884736\) | \(1.9630\) | |
| 89232.cs2 | 89232co2 | \([0, 1, 0, -31152, -675180]\) | \(169112377/88209\) | \(1743945707851776\) | \([2, 2]\) | \(442368\) | \(1.6164\) | |
| 89232.cs3 | 89232co1 | \([0, 1, 0, -17632, 887732]\) | \(30664297/297\) | \(5871871070208\) | \([2]\) | \(221184\) | \(1.2698\) | \(\Gamma_0(N)\)-optimal |
| 89232.cs4 | 89232co3 | \([0, 1, 0, 117568, -5136780]\) | \(9090072503/5845851\) | \(-115576038274904064\) | \([2]\) | \(884736\) | \(1.9630\) |
Rank
sage: E.rank()
The elliptic curves in class 89232.cs have rank \(0\).
Complex multiplication
The elliptic curves in class 89232.cs do not have complex multiplication.Modular form 89232.2.a.cs
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.
