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SageMath
E = EllipticCurve("de1")
E.isogeny_class()
Elliptic curves in class 254898de
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 254898.de5 | 254898de1 | \([1, -1, 0, -8952530211, 325631283424789]\) | \(38331145780597164097/55468445663232\) | \(114829980988643259739136851968\) | \([2]\) | \(424673280\) | \(4.4783\) | \(\Gamma_0(N)\)-optimal |
| 254898.de4 | 254898de2 | \([1, -1, 0, -11562685731, 120273121639957]\) | \(82582985847542515777/44772582831427584\) | \(92687559095480860975234438434816\) | \([2, 2]\) | \(849346560\) | \(4.8249\) | |
| 254898.de6 | 254898de3 | \([1, -1, 0, 44596441629, 945935843912149]\) | \(4738217997934888496063/2928751705237796928\) | \(-6063059792134022817162450820663872\) | \([2]\) | \(1698693120\) | \(5.1714\) | |
| 254898.de2 | 254898de4 | \([1, -1, 0, -109484301411, -13848343276727723]\) | \(70108386184777836280897/552468975892674624\) | \(1143713353421458698459693230131776\) | \([2, 2]\) | \(1698693120\) | \(5.1714\) | |
| 254898.de3 | 254898de5 | \([1, -1, 0, -37292089851, -31838079298229555]\) | \(-2770540998624539614657/209924951154647363208\) | \(-434583624291257044043654616756155592\) | \([2]\) | \(3397386240\) | \(5.5180\) | |
| 254898.de1 | 254898de6 | \([1, -1, 0, -1748422363851, -889851231810145571]\) | \(285531136548675601769470657/17941034271597192\) | \(37141271937419963029081880824008\) | \([2]\) | \(3397386240\) | \(5.5180\) |
Rank
sage: E.rank()
The elliptic curves in class 254898de have rank \(1\).
Complex multiplication
The elliptic curves in class 254898de do not have complex multiplication.Modular form 254898.2.a.de
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.
