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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 382200v
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 382200.v5 | 382200v1 | \([0, -1, 0, -900783, -429604188]\) | \(-2748251600896/1124136195\) | \(-33063374801388750000\) | \([2]\) | \(9437184\) | \(2.4538\) | \(\Gamma_0(N)\)-optimal |
| 382200.v4 | 382200v2 | \([0, -1, 0, -15606908, -23724106188]\) | \(893359210685776/91298025\) | \(42964485372900000000\) | \([2, 2]\) | \(18874368\) | \(2.8004\) | |
| 382200.v3 | 382200v3 | \([0, -1, 0, -16807408, -19860897188]\) | \(278944461825124/70849130625\) | \(133365269902410000000000\) | \([2, 2]\) | \(37748736\) | \(3.1469\) | |
| 382200.v1 | 382200v4 | \([0, -1, 0, -249704408, -1518670741188]\) | \(914732517663095044/9555\) | \(17986179120000000\) | \([2]\) | \(37748736\) | \(3.1469\) | |
| 382200.v2 | 382200v5 | \([0, -1, 0, -93982408, 334372352812]\) | \(24385137179326562/1284775885575\) | \(4836883141184421600000000\) | \([2]\) | \(75497472\) | \(3.4935\) | |
| 382200.v6 | 382200v6 | \([0, -1, 0, 41159592, -126867979188]\) | \(2048324060764798/3031899609375\) | \(-11414398628587500000000000\) | \([2]\) | \(75497472\) | \(3.4935\) |
Rank
sage: E.rank()
The elliptic curves in class 382200v have rank \(1\).
Complex multiplication
The elliptic curves in class 382200v do not have complex multiplication.Modular form 382200.2.a.v
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
