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SageMath
E = EllipticCurve("ff1")
E.isogeny_class()
Elliptic curves in class 221760ff
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.d3 | 221760ff1 | \([0, 0, 0, -220863, 39951088]\) | \(25537895799171904/269028375\) | \(12551787864000\) | \([2]\) | \(1179648\) | \(1.6699\) | \(\Gamma_0(N)\)-optimal |
| 221760.d2 | 221760ff2 | \([0, 0, 0, -226308, 37877632]\) | \(429275354481856/40854515625\) | \(121990929984000000\) | \([2, 2]\) | \(2359296\) | \(2.0164\) | |
| 221760.d4 | 221760ff3 | \([0, 0, 0, 268692, 180635632]\) | \(89807055379768/642054711375\) | \(-15337320762322944000\) | \([2]\) | \(4718592\) | \(2.3630\) | |
| 221760.d1 | 221760ff4 | \([0, 0, 0, -808428, -237581552]\) | \(2446077932835272/394775390625\) | \(9430344000000000000\) | \([2]\) | \(4718592\) | \(2.3630\) |
Rank
sage: E.rank()
The elliptic curves in class 221760ff have rank \(1\).
Complex multiplication
The elliptic curves in class 221760ff do not have complex multiplication.Modular form 221760.2.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.
