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SageMath
E = EllipticCurve("mr1")
E.isogeny_class()
Elliptic curves in class 348480.mr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 348480.mr1 | 348480mr3 | \([0, 0, 0, -229998252, 1342563749296]\) | \(15897679904620804/2475\) | \(209478170920550400\) | \([2]\) | \(31457280\) | \(3.1702\) | |
| 348480.mr2 | 348480mr6 | \([0, 0, 0, -121969452, -508534829264]\) | \(1185450336504002/26043266205\) | \(4408481429834653715005440\) | \([2]\) | \(62914560\) | \(3.5168\) | |
| 348480.mr3 | 348480mr4 | \([0, 0, 0, -16554252, 14198064496]\) | \(5927735656804/2401490025\) | \(203256459766039132569600\) | \([2, 2]\) | \(31457280\) | \(3.1702\) | |
| 348480.mr4 | 348480mr2 | \([0, 0, 0, -14376252, 20973386896]\) | \(15529488955216/6125625\) | \(129614618257090560000\) | \([2, 2]\) | \(15728640\) | \(2.8236\) | |
| 348480.mr5 | 348480mr1 | \([0, 0, 0, -763752, 429401896]\) | \(-37256083456/38671875\) | \(-51142131572400000000\) | \([2]\) | \(7864320\) | \(2.4771\) | \(\Gamma_0(N)\)-optimal |
| 348480.mr6 | 348480mr5 | \([0, 0, 0, 54012948, 103310324656]\) | \(102949393183198/86815346805\) | \(-14695692974985575012106240\) | \([2]\) | \(62914560\) | \(3.5168\) |
Rank
sage: E.rank()
The elliptic curves in class 348480.mr have rank \(2\).
Complex multiplication
The elliptic curves in class 348480.mr do not have complex multiplication.Modular form 348480.2.a.mr
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
