Show commands:
SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 272832.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 272832.bc1 | 272832bc4 | \([0, -1, 0, -271329, -52425855]\) | \(143256979154/5639949\) | \(86970775620943872\) | \([2]\) | \(3932160\) | \(2.0173\) | |
| 272832.bc2 | 272832bc2 | \([0, -1, 0, -43969, 2458849]\) | \(1219284868/370881\) | \(2859583325405184\) | \([2, 2]\) | \(1966080\) | \(1.6708\) | |
| 272832.bc3 | 272832bc1 | \([0, -1, 0, -40049, 3097809]\) | \(3685542352/609\) | \(1173884780544\) | \([2]\) | \(983040\) | \(1.3242\) | \(\Gamma_0(N)\)-optimal |
| 272832.bc4 | 272832bc3 | \([0, -1, 0, 120671, 16387393]\) | \(12601744846/14852901\) | \(-229039007301500928\) | \([2]\) | \(3932160\) | \(2.0173\) |
Rank
sage: E.rank()
The elliptic curves in class 272832.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 272832.bc do not have complex multiplication.Modular form 272832.2.a.bc
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.
