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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 53550.cz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53550.cz1 | 53550dj7 | \([1, -1, 1, -2992469855, 62961632256147]\) | \(260174968233082037895439009/223081361502731896500\) | \(2541036133367055508570312500\) | \([2]\) | \(42467328\) | \(4.1852\) | |
| 53550.cz2 | 53550dj8 | \([1, -1, 1, -1965344855, -33179713493853]\) | \(73704237235978088924479009/899277423164136103500\) | \(10243331898228987803929687500\) | \([2]\) | \(42467328\) | \(4.1852\) | |
| 53550.cz3 | 53550dj5 | \([1, -1, 1, -1959553355, -33386998730853]\) | \(73054578035931991395831649/136386452160\) | \(1553526931635000000\) | \([2]\) | \(14155776\) | \(3.6359\) | |
| 53550.cz4 | 53550dj6 | \([1, -1, 1, -228907355, 510646881147]\) | \(116454264690812369959009/57505157319440250000\) | \(655019682591749097656250000\) | \([2, 2]\) | \(21233664\) | \(3.8387\) | |
| 53550.cz5 | 53550dj4 | \([1, -1, 1, -128593355, -466618730853]\) | \(20645800966247918737249/3688936444974392640\) | \(42019291693536441165000000\) | \([2]\) | \(14155776\) | \(3.6359\) | |
| 53550.cz6 | 53550dj2 | \([1, -1, 1, -122473355, -521637530853]\) | \(17836145204788591940449/770635366502400\) | \(8778018471566400000000\) | \([2, 2]\) | \(7077888\) | \(3.2894\) | |
| 53550.cz7 | 53550dj1 | \([1, -1, 1, -7273355, -8997530853]\) | \(-3735772816268612449/909650165760000\) | \(-10361483919360000000000\) | \([2]\) | \(3538944\) | \(2.9428\) | \(\Gamma_0(N)\)-optimal |
| 53550.cz8 | 53550dj3 | \([1, -1, 1, 52342645, 61209381147]\) | \(1392333139184610040991/947901937500000000\) | \(-10797195506835937500000000\) | \([2]\) | \(10616832\) | \(3.4921\) |
Rank
sage: E.rank()
The elliptic curves in class 53550.cz have rank \(1\).
Complex multiplication
The elliptic curves in class 53550.cz do not have complex multiplication.Modular form 53550.2.a.cz
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
