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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 16320.bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16320.bw1 | 16320bb7 | \([0, 1, 0, -7262081, -7534894881]\) | \(161572377633716256481/914742821250\) | \(239794342133760000\) | \([2]\) | \(393216\) | \(2.5265\) | |
| 16320.bw2 | 16320bb4 | \([0, 1, 0, -1392641, 632103519]\) | \(1139466686381936641/4080\) | \(1069547520\) | \([2]\) | \(98304\) | \(1.8333\) | |
| 16320.bw3 | 16320bb5 | \([0, 1, 0, -462081, -113374881]\) | \(41623544884956481/2962701562500\) | \(776654438400000000\) | \([2, 2]\) | \(196608\) | \(2.1799\) | |
| 16320.bw4 | 16320bb3 | \([0, 1, 0, -92161, 8624735]\) | \(330240275458561/67652010000\) | \(17734568509440000\) | \([2, 2]\) | \(98304\) | \(1.8333\) | |
| 16320.bw5 | 16320bb2 | \([0, 1, 0, -87041, 9854559]\) | \(278202094583041/16646400\) | \(4363753881600\) | \([2, 2]\) | \(49152\) | \(1.4867\) | |
| 16320.bw6 | 16320bb1 | \([0, 1, 0, -5121, 171615]\) | \(-56667352321/16711680\) | \(-4380866641920\) | \([2]\) | \(24576\) | \(1.1402\) | \(\Gamma_0(N)\)-optimal |
| 16320.bw7 | 16320bb6 | \([0, 1, 0, 195839, 51997535]\) | \(3168685387909439/6278181696900\) | \(-1645787662752153600\) | \([2]\) | \(196608\) | \(2.1799\) | |
| 16320.bw8 | 16320bb8 | \([0, 1, 0, 419199, -493911585]\) | \(31077313442863199/420227050781250\) | \(-110160000000000000000\) | \([2]\) | \(393216\) | \(2.5265\) |
Rank
sage: E.rank()
The elliptic curves in class 16320.bw have rank \(1\).
Complex multiplication
The elliptic curves in class 16320.bw do not have complex multiplication.Modular form 16320.2.a.bw
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 16 & 8 & 4 \\ 16 & 1 & 8 & 4 & 2 & 4 & 8 & 16 \\ 2 & 8 & 1 & 2 & 4 & 8 & 4 & 2 \\ 4 & 4 & 2 & 1 & 2 & 4 & 2 & 4 \\ 8 & 2 & 4 & 2 & 1 & 2 & 4 & 8 \\ 16 & 4 & 8 & 4 & 2 & 1 & 8 & 16 \\ 8 & 8 & 4 & 2 & 4 & 8 & 1 & 8 \\ 4 & 16 & 2 & 4 & 8 & 16 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
