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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 369600.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 369600.bg1 | 369600bg6 | \([0, -1, 0, -205336033, -1132451516063]\) | \(467508233804095622882/315748125\) | \(646652160000000000\) | \([2]\) | \(37748736\) | \(3.1674\) | |
| 369600.bg2 | 369600bg4 | \([0, -1, 0, -12836033, -17684016063]\) | \(228410605013945764/187597265625\) | \(192099600000000000000\) | \([2, 2]\) | \(18874368\) | \(2.8208\) | |
| 369600.bg3 | 369600bg5 | \([0, -1, 0, -10064033, -25537092063]\) | \(-55043996611705922/105743408203125\) | \(-216562500000000000000000\) | \([2]\) | \(37748736\) | \(3.1674\) | |
| 369600.bg4 | 369600bg3 | \([0, -1, 0, -8328033, 9151715937]\) | \(62380825826921284/787768887675\) | \(806675340979200000000\) | \([2]\) | \(18874368\) | \(2.8208\) | |
| 369600.bg5 | 369600bg2 | \([0, -1, 0, -978033, -146034063]\) | \(404151985581136/197735855625\) | \(50620379040000000000\) | \([2, 2]\) | \(9437184\) | \(2.4742\) | |
| 369600.bg6 | 369600bg1 | \([0, -1, 0, 222467, -17580563]\) | \(76102438406144/52315569075\) | \(-837049105200000000\) | \([2]\) | \(4718592\) | \(2.1277\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 369600.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 369600.bg do not have complex multiplication.Modular form 369600.2.a.bg
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
