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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 36822i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 36822.e5 | 36822i1 | \([1, 1, 0, -12281, -490971]\) | \(4354703137/352512\) | \(16584237603072\) | \([2]\) | \(110592\) | \(1.2796\) | \(\Gamma_0(N)\)-optimal |
| 36822.e4 | 36822i2 | \([1, 1, 0, -41161, 2633845]\) | \(163936758817/30338064\) | \(1427280948714384\) | \([2, 2]\) | \(221184\) | \(1.6262\) | |
| 36822.e6 | 36822i3 | \([1, 1, 0, 81579, 15472449]\) | \(1276229915423/2927177028\) | \(-137711622125221668\) | \([2]\) | \(442368\) | \(1.9727\) | |
| 36822.e2 | 36822i4 | \([1, 1, 0, -625981, 190361065]\) | \(576615941610337/27060804\) | \(1273099364748324\) | \([2, 2]\) | \(442368\) | \(1.9727\) | |
| 36822.e3 | 36822i5 | \([1, 1, 0, -593491, 211044199]\) | \(-491411892194497/125563633938\) | \(-5907251780174709378\) | \([2]\) | \(884736\) | \(2.3193\) | |
| 36822.e1 | 36822i6 | \([1, 1, 0, -10015591, 12195916411]\) | \(2361739090258884097/5202\) | \(244732672962\) | \([2]\) | \(884736\) | \(2.3193\) |
Rank
sage: E.rank()
The elliptic curves in class 36822i have rank \(1\).
Complex multiplication
The elliptic curves in class 36822i do not have complex multiplication.Modular form 36822.2.a.i
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
