Show commands:
SageMath
E = EllipticCurve("fm1")
E.isogeny_class()
Elliptic curves in class 262080fm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 262080.fm7 | 262080fm1 | \([0, 0, 0, -1971948, -1574104912]\) | \(-4437543642183289/3033210136320\) | \(-579655483884117688320\) | \([2]\) | \(10616832\) | \(2.6845\) | \(\Gamma_0(N)\)-optimal |
| 262080.fm6 | 262080fm2 | \([0, 0, 0, -35564268, -81617885008]\) | \(26031421522845051769/5797789779600\) | \(1107974881103944089600\) | \([2, 2]\) | \(21233664\) | \(3.0311\) | |
| 262080.fm8 | 262080fm3 | \([0, 0, 0, 15990612, 23450027312]\) | \(2366200373628880151/2612420149248000\) | \(-499241265083656962048000\) | \([2]\) | \(31850496\) | \(3.2338\) | |
| 262080.fm5 | 262080fm4 | \([0, 0, 0, -39607788, -61906533712]\) | \(35958207000163259449/12145729518877500\) | \(2321085056748538429440000\) | \([2]\) | \(42467328\) | \(3.3776\) | |
| 262080.fm3 | 262080fm5 | \([0, 0, 0, -568997868, -5224131162448]\) | \(106607603143751752938169/5290068420\) | \(1010947818305617920\) | \([2]\) | \(42467328\) | \(3.3776\) | |
| 262080.fm4 | 262080fm6 | \([0, 0, 0, -90177708, 219946353968]\) | \(424378956393532177129/136231857216000000\) | \(26034313339984674816000000\) | \([2, 2]\) | \(63700992\) | \(3.5804\) | |
| 262080.fm1 | 262080fm7 | \([0, 0, 0, -1305030828, 18142917403952]\) | \(1286229821345376481036009/247265484375000000\) | \(47253169926144000000000000\) | \([2]\) | \(127401984\) | \(3.9270\) | |
| 262080.fm2 | 262080fm8 | \([0, 0, 0, -574017708, -5127259790032]\) | \(109454124781830273937129/3914078300576808000\) | \(747992011537250525380608000\) | \([2]\) | \(127401984\) | \(3.9270\) |
Rank
sage: E.rank()
The elliptic curves in class 262080fm have rank \(1\).
Complex multiplication
The elliptic curves in class 262080fm do not have complex multiplication.Modular form 262080.2.a.fm
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
