Show commands:
SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 225225.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 225225.bf1 | 225225be3 | \([1, -1, 1, -2231030, 1283201722]\) | \(107818231938348177/4463459\) | \(50841587671875\) | \([2]\) | \(2490368\) | \(2.1163\) | |
| 225225.bf2 | 225225be4 | \([1, -1, 1, -226280, -7708778]\) | \(112489728522417/62811265517\) | \(715459571279578125\) | \([2]\) | \(2490368\) | \(2.1163\) | |
| 225225.bf3 | 225225be2 | \([1, -1, 1, -139655, 20011222]\) | \(26444947540257/169338169\) | \(1928867581265625\) | \([2, 2]\) | \(1245184\) | \(1.7697\) | |
| 225225.bf4 | 225225be1 | \([1, -1, 1, -3530, 681472]\) | \(-426957777/17320303\) | \(-197289076359375\) | \([2]\) | \(622592\) | \(1.4231\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 225225.bf have rank \(2\).
Complex multiplication
The elliptic curves in class 225225.bf do not have complex multiplication.Modular form 225225.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
