Show commands:
SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 12600.bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12600.bw1 | 12600q3 | \([0, 0, 0, -34275, -2439250]\) | \(381775972/567\) | \(6613488000000\) | \([2]\) | \(32768\) | \(1.3610\) | |
| 12600.bw2 | 12600q2 | \([0, 0, 0, -2775, -13750]\) | \(810448/441\) | \(1285956000000\) | \([2, 2]\) | \(16384\) | \(1.0144\) | |
| 12600.bw3 | 12600q1 | \([0, 0, 0, -1650, 25625]\) | \(2725888/21\) | \(3827250000\) | \([2]\) | \(8192\) | \(0.66784\) | \(\Gamma_0(N)\)-optimal |
| 12600.bw4 | 12600q4 | \([0, 0, 0, 10725, -108250]\) | \(11696828/7203\) | \(-84015792000000\) | \([2]\) | \(32768\) | \(1.3610\) |
Rank
sage: E.rank()
The elliptic curves in class 12600.bw have rank \(1\).
Complex multiplication
The elliptic curves in class 12600.bw do not have complex multiplication.Modular form 12600.2.a.bw
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.
