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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 22050be
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.bj8 | 22050be1 | \([1, -1, 0, 16308, 2457216]\) | \(357911/2160\) | \(-2894606583750000\) | \([2]\) | \(110592\) | \(1.6487\) | \(\Gamma_0(N)\)-optimal |
| 22050.bj6 | 22050be2 | \([1, -1, 0, -204192, 32224716]\) | \(702595369/72900\) | \(97692972201562500\) | \([2, 2]\) | \(221184\) | \(1.9953\) | |
| 22050.bj7 | 22050be3 | \([1, -1, 0, -149067, -72457659]\) | \(-273359449/1536000\) | \(-2058386904000000000\) | \([2]\) | \(331776\) | \(2.1980\) | |
| 22050.bj5 | 22050be4 | \([1, -1, 0, -755442, -217491534]\) | \(35578826569/5314410\) | \(7121817673493906250\) | \([2]\) | \(442368\) | \(2.3418\) | |
| 22050.bj4 | 22050be5 | \([1, -1, 0, -3180942, 2184414966]\) | \(2656166199049/33750\) | \(45228227871093750\) | \([2]\) | \(442368\) | \(2.3418\) | |
| 22050.bj3 | 22050be6 | \([1, -1, 0, -3677067, -2707873659]\) | \(4102915888729/9000000\) | \(12060860765625000000\) | \([2, 2]\) | \(663552\) | \(2.5446\) | |
| 22050.bj1 | 22050be7 | \([1, -1, 0, -58802067, -173540248659]\) | \(16778985534208729/81000\) | \(108547746890625000\) | \([2]\) | \(1327104\) | \(2.8911\) | |
| 22050.bj2 | 22050be8 | \([1, -1, 0, -5000067, -584458659]\) | \(10316097499609/5859375000\) | \(7852122894287109375000\) | \([2]\) | \(1327104\) | \(2.8911\) |
Rank
sage: E.rank()
The elliptic curves in class 22050be have rank \(1\).
Complex multiplication
The elliptic curves in class 22050be do not have complex multiplication.Modular form 22050.2.a.be
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
