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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 8400.bl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8400.bl1 | 8400w3 | \([0, 1, 0, -168008, -26562012]\) | \(32779037733124/315\) | \(5040000000\) | \([2]\) | \(24576\) | \(1.4401\) | |
| 8400.bl2 | 8400w5 | \([0, 1, 0, -162008, 24959988]\) | \(14695548366242/57421875\) | \(1837500000000000\) | \([2]\) | \(49152\) | \(1.7866\) | |
| 8400.bl3 | 8400w4 | \([0, 1, 0, -15008, -30012]\) | \(23366901604/13505625\) | \(216090000000000\) | \([2, 2]\) | \(24576\) | \(1.4401\) | |
| 8400.bl4 | 8400w2 | \([0, 1, 0, -10508, -417012]\) | \(32082281296/99225\) | \(396900000000\) | \([2, 2]\) | \(12288\) | \(1.0935\) | |
| 8400.bl5 | 8400w1 | \([0, 1, 0, -383, -12012]\) | \(-24918016/229635\) | \(-57408750000\) | \([2]\) | \(6144\) | \(0.74691\) | \(\Gamma_0(N)\)-optimal |
| 8400.bl6 | 8400w6 | \([0, 1, 0, 59992, -180012]\) | \(746185003198/432360075\) | \(-13835522400000000\) | \([2]\) | \(49152\) | \(1.7866\) |
Rank
sage: E.rank()
The elliptic curves in class 8400.bl have rank \(0\).
Complex multiplication
The elliptic curves in class 8400.bl do not have complex multiplication.Modular form 8400.2.a.bl
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
