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SageMath
E = EllipticCurve("ga1")
E.isogeny_class()
Elliptic curves in class 52416ga
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 52416.bx3 | 52416ga1 | \([0, 0, 0, -32556, -2198576]\) | \(19968681097/628992\) | \(120202243080192\) | \([2]\) | \(147456\) | \(1.4765\) | \(\Gamma_0(N)\)-optimal |
| 52416.bx2 | 52416ga2 | \([0, 0, 0, -78636, 5413840]\) | \(281397674377/96589584\) | \(18458556953001984\) | \([2, 2]\) | \(294912\) | \(1.8231\) | |
| 52416.bx4 | 52416ga3 | \([0, 0, 0, 232404, 37637584]\) | \(7264187703863/7406095788\) | \(-1415326945627865088\) | \([2]\) | \(589824\) | \(2.1697\) | |
| 52416.bx1 | 52416ga4 | \([0, 0, 0, -1126956, 460384720]\) | \(828279937799497/193444524\) | \(36967824227303424\) | \([2]\) | \(589824\) | \(2.1697\) |
Rank
sage: E.rank()
The elliptic curves in class 52416ga have rank \(0\).
Complex multiplication
The elliptic curves in class 52416ga do not have complex multiplication.Modular form 52416.2.a.ga
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
