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SageMath
E = EllipticCurve("ff1")
E.isogeny_class()
Elliptic curves in class 288990ff
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 288990.ff4 | 288990ff1 | \([1, -1, 1, -15242, -1232071]\) | \(-111284641/123120\) | \(-433227731854320\) | \([2]\) | \(1769472\) | \(1.5031\) | \(\Gamma_0(N)\)-optimal |
| 288990.ff3 | 288990ff2 | \([1, -1, 1, -289022, -59711479]\) | \(758800078561/324900\) | \(1143239847948900\) | \([2, 2]\) | \(3538944\) | \(1.8497\) | |
| 288990.ff2 | 288990ff3 | \([1, -1, 1, -334652, -39561271]\) | \(1177918188481/488703750\) | \(1719623271289803750\) | \([2]\) | \(7077888\) | \(2.1963\) | |
| 288990.ff1 | 288990ff4 | \([1, -1, 1, -4623872, -3825829159]\) | \(3107086841064961/570\) | \(2005683943770\) | \([2]\) | \(7077888\) | \(2.1963\) |
Rank
sage: E.rank()
The elliptic curves in class 288990ff have rank \(1\).
Complex multiplication
The elliptic curves in class 288990ff do not have complex multiplication.Modular form 288990.2.a.ff
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.
