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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 5712.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5712.o1 | 5712t5 | \([0, 1, 0, -219497664, 1251605773620]\) | \(285531136548675601769470657/17941034271597192\) | \(73486476376462098432\) | \([8]\) | \(737280\) | \(3.2723\) | |
| 5712.o2 | 5712t3 | \([0, 1, 0, -13744704, 19474747956]\) | \(70108386184777836280897/552468975892674624\) | \(2262912925256395259904\) | \([2, 4]\) | \(368640\) | \(2.9257\) | |
| 5712.o3 | 5712t6 | \([0, 1, 0, -4681664, 44782380852]\) | \(-2770540998624539614657/209924951154647363208\) | \(-859852599929435599699968\) | \([4]\) | \(737280\) | \(3.2723\) | |
| 5712.o4 | 5712t2 | \([0, 1, 0, -1451584, -169657804]\) | \(82582985847542515777/44772582831427584\) | \(183388499277527384064\) | \([2, 2]\) | \(184320\) | \(2.5791\) | |
| 5712.o5 | 5712t1 | \([0, 1, 0, -1123904, -458409420]\) | \(38331145780597164097/55468445663232\) | \(227198753436598272\) | \([2]\) | \(92160\) | \(2.2326\) | \(\Gamma_0(N)\)-optimal |
| 5712.o6 | 5712t4 | \([0, 1, 0, 5598656, -1328717260]\) | \(4738217997934888496063/2928751705237796928\) | \(-11996166984654016217088\) | \([2]\) | \(368640\) | \(2.9257\) |
Rank
sage: E.rank()
The elliptic curves in class 5712.o have rank \(0\).
Complex multiplication
The elliptic curves in class 5712.o do not have complex multiplication.Modular form 5712.2.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 LMFDB 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 LMFDB labels.
