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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 61710o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 61710.m4 | 61710o1 | \([1, 1, 0, -884512, -320555264]\) | \(43199583152847841/89760000\) | \(159015315360000\) | \([2]\) | \(737280\) | \(1.9744\) | \(\Gamma_0(N)\)-optimal |
| 61710.m3 | 61710o2 | \([1, 1, 0, -894192, -313192656]\) | \(44633474953947361/1967006250000\) | \(3484671559256250000\) | \([2, 2]\) | \(1474560\) | \(2.3210\) | |
| 61710.m5 | 61710o3 | \([1, 1, 0, 463428, -1178539644]\) | \(6213165856218719/342407226562500\) | \(-606595288696289062500\) | \([2]\) | \(2949120\) | \(2.6675\) | |
| 61710.m2 | 61710o4 | \([1, 1, 0, -2406692, 1023554844]\) | \(870220733067747361/247623269602500\) | \(438679727120274502500\) | \([2, 2]\) | \(2949120\) | \(2.6675\) | |
| 61710.m6 | 61710o5 | \([1, 1, 0, 6335558, 6760219294]\) | \(15875306080318016639/20322604533582450\) | \(-36002733610117858704450\) | \([2]\) | \(5898240\) | \(3.0141\) | |
| 61710.m1 | 61710o6 | \([1, 1, 0, -35348942, 80868980394]\) | \(2757381641970898311361/379829992662450\) | \(672892001631082584450\) | \([2]\) | \(5898240\) | \(3.0141\) |
Rank
sage: E.rank()
The elliptic curves in class 61710o have rank \(1\).
Complex multiplication
The elliptic curves in class 61710o do not have complex multiplication.Modular form 61710.2.a.o
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
