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SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 69360.cw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69360.cw1 | 69360du7 | \([0, 1, 0, -24662200, 47132459348]\) | \(16778985534208729/81000\) | \(8008266092544000\) | \([2]\) | \(2654208\) | \(2.6739\) | |
| 69360.cw2 | 69360du8 | \([0, 1, 0, -2097080, 158390100]\) | \(10316097499609/5859375000\) | \(579301656000000000000\) | \([2]\) | \(2654208\) | \(2.6739\) | |
| 69360.cw3 | 69360du6 | \([0, 1, 0, -1542200, 735243348]\) | \(4102915888729/9000000\) | \(889807343616000000\) | \([2, 2]\) | \(1327104\) | \(2.3273\) | |
| 69360.cw4 | 69360du5 | \([0, 1, 0, -1334120, -593555532]\) | \(2656166199049/33750\) | \(3336777538560000\) | \([2]\) | \(884736\) | \(2.1246\) | |
| 69360.cw5 | 69360du4 | \([0, 1, 0, -316840, 59020340]\) | \(35578826569/5314410\) | \(525422338331811840\) | \([2]\) | \(884736\) | \(2.1246\) | |
| 69360.cw6 | 69360du2 | \([0, 1, 0, -85640, -8767500]\) | \(702595369/72900\) | \(7207439483289600\) | \([2, 2]\) | \(442368\) | \(1.7780\) | |
| 69360.cw7 | 69360du3 | \([0, 1, 0, -62520, 19670100]\) | \(-273359449/1536000\) | \(-151860453310464000\) | \([2]\) | \(663552\) | \(1.9808\) | |
| 69360.cw8 | 69360du1 | \([0, 1, 0, 6840, -666252]\) | \(357911/2160\) | \(-213553762467840\) | \([2]\) | \(221184\) | \(1.4315\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 69360.cw have rank \(0\).
Complex multiplication
The elliptic curves in class 69360.cw do not have complex multiplication.Modular form 69360.2.a.cw
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
