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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 55440.x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.x1 | 55440db4 | \([0, 0, 0, -82802523, 290010328778]\) | \(21026497979043461623321/161783881875\) | \(483084082736640000\) | \([2]\) | \(3932160\) | \(2.9857\) | |
| 55440.x2 | 55440db2 | \([0, 0, 0, -5178603, 4525075802]\) | \(5143681768032498601/14238434358225\) | \(42515737178710118400\) | \([2, 2]\) | \(1966080\) | \(2.6392\) | |
| 55440.x3 | 55440db3 | \([0, 0, 0, -3137403, 8128610282]\) | \(-1143792273008057401/8897444448004035\) | \(-26567626762628880445440\) | \([2]\) | \(3932160\) | \(2.9857\) | |
| 55440.x4 | 55440db1 | \([0, 0, 0, -454683, 8063498]\) | \(3481467828171481/2005331497785\) | \(5987887767082045440\) | \([2]\) | \(983040\) | \(2.2926\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 55440.x have rank \(1\).
Complex multiplication
The elliptic curves in class 55440.x do not have complex multiplication.Modular form 55440.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
