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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 2233a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2233.a4 | 2233a1 | \([1, -1, 1, 584, 6050]\) | \(22062729659823/29354283343\) | \(-29354283343\) | \([4]\) | \(1344\) | \(0.69438\) | \(\Gamma_0(N)\)-optimal |
| 2233.a3 | 2233a2 | \([1, -1, 1, -3621, 61556]\) | \(5249244962308257/1448621666569\) | \(1448621666569\) | \([2, 2]\) | \(2688\) | \(1.0409\) | |
| 2233.a2 | 2233a3 | \([1, -1, 1, -21166, -1131504]\) | \(1048626554636928177/48569076788309\) | \(48569076788309\) | \([2]\) | \(5376\) | \(1.3875\) | |
| 2233.a1 | 2233a4 | \([1, -1, 1, -53356, 4756540]\) | \(16798320881842096017/2132227789307\) | \(2132227789307\) | \([2]\) | \(5376\) | \(1.3875\) |
Rank
sage: E.rank()
The elliptic curves in class 2233a have rank \(0\).
Complex multiplication
The elliptic curves in class 2233a do not have complex multiplication.Modular form 2233.2.a.a
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.
