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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 51842.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51842.j1 | 51842e6 | \([1, 1, 0, -70777830, -229218601324]\) | \(2251439055699625/25088\) | \(436939489762861568\) | \([2]\) | \(3649536\) | \(2.9538\) | |
| 51842.j2 | 51842e5 | \([1, 1, 0, -4420070, -3588945772]\) | \(-548347731625/1835008\) | \(-31959002679797874688\) | \([2]\) | \(1824768\) | \(2.6072\) | |
| 51842.j3 | 51842e4 | \([1, 1, 0, -920735, -279119203]\) | \(4956477625/941192\) | \(16392058045634853512\) | \([2]\) | \(1216512\) | \(2.4045\) | |
| 51842.j4 | 51842e2 | \([1, 1, 0, -272710, 54665514]\) | \(128787625/98\) | \(1706794881886178\) | \([2]\) | \(405504\) | \(1.8552\) | |
| 51842.j5 | 51842e1 | \([1, 1, 0, -13500, 1216412]\) | \(-15625/28\) | \(-487655680538908\) | \([2]\) | \(202752\) | \(1.5086\) | \(\Gamma_0(N)\)-optimal |
| 51842.j6 | 51842e3 | \([1, 1, 0, 116105, -25508139]\) | \(9938375/21952\) | \(-382322053542503872\) | \([2]\) | \(608256\) | \(2.0579\) |
Rank
sage: E.rank()
The elliptic curves in class 51842.j have rank \(1\).
Complex multiplication
The elliptic curves in class 51842.j do not have complex multiplication.Modular form 51842.2.a.j
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
