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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 56550w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 56550.v4 | 56550w1 | \([1, 0, 1, -258376, 50517398]\) | \(122083727651299441/32242728960\) | \(503792640000000\) | \([2]\) | \(491520\) | \(1.8055\) | \(\Gamma_0(N)\)-optimal |
| 56550.v3 | 56550w2 | \([1, 0, 1, -290376, 37205398]\) | \(173294065906331761/61964605497600\) | \(968196960900000000\) | \([2, 2]\) | \(983040\) | \(2.1521\) | |
| 56550.v6 | 56550w3 | \([1, 0, 1, 879624, 261845398]\) | \(4817210305461175439/4682306425314960\) | \(-73161037895546250000\) | \([2]\) | \(1966080\) | \(2.4986\) | |
| 56550.v2 | 56550w4 | \([1, 0, 1, -1972376, -1039274602]\) | \(54309086480107021681/1575939143610000\) | \(24624049118906250000\) | \([2, 2]\) | \(1966080\) | \(2.4986\) | |
| 56550.v5 | 56550w5 | \([1, 0, 1, 478124, -3450566602]\) | \(773618103830753999/329643718157812500\) | \(-5150683096215820312500\) | \([2]\) | \(3932160\) | \(2.8452\) | |
| 56550.v1 | 56550w6 | \([1, 0, 1, -31334876, -67515974602]\) | \(217764763259392950709681/191615146362900\) | \(2993986661920312500\) | \([2]\) | \(3932160\) | \(2.8452\) |
Rank
sage: E.rank()
The elliptic curves in class 56550w have rank \(1\).
Complex multiplication
The elliptic curves in class 56550w do not have complex multiplication.Modular form 56550.2.a.w
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.
