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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 10626.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10626.p1 | 10626r5 | \([1, 0, 0, -68877094, -220024810702]\) | \(36136672427711016379227705697/1011258101510224722\) | \(1011258101510224722\) | \([2]\) | \(819200\) | \(2.9656\) | |
| 10626.p2 | 10626r4 | \([1, 0, 0, -4929124, 4203486548]\) | \(13244420128496241770842177/29965867631164664892\) | \(29965867631164664892\) | \([4]\) | \(409600\) | \(2.6190\) | |
| 10626.p3 | 10626r3 | \([1, 0, 0, -4310284, -3428989876]\) | \(8856076866003496152467137/46664863048067576004\) | \(46664863048067576004\) | \([2, 2]\) | \(409600\) | \(2.6190\) | |
| 10626.p4 | 10626r6 | \([1, 0, 0, -1977394, -7125687370]\) | \(-855073332201294509246497/21439133060285771735058\) | \(-21439133060285771735058\) | \([2]\) | \(819200\) | \(2.9656\) | |
| 10626.p5 | 10626r2 | \([1, 0, 0, -420664, 13323824]\) | \(8232463578739844255617/4687062591766850064\) | \(4687062591766850064\) | \([2, 4]\) | \(204800\) | \(2.2724\) | |
| 10626.p6 | 10626r1 | \([1, 0, 0, 104216, 1671488]\) | \(125177609053596564863/73635189229502208\) | \(-73635189229502208\) | \([8]\) | \(102400\) | \(1.9259\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10626.p have rank \(0\).
Complex multiplication
The elliptic curves in class 10626.p do not have complex multiplication.Modular form 10626.2.a.p
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
