Show commands:
SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 12675.ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12675.ba1 | 12675w3 | \([1, 0, 1, -293726, -61294777]\) | \(37159393753/1053\) | \(79416091828125\) | \([2]\) | \(86016\) | \(1.7688\) | |
| 12675.ba2 | 12675w4 | \([1, 0, 1, -82476, 8248723]\) | \(822656953/85683\) | \(6462116805421875\) | \([2]\) | \(86016\) | \(1.7688\) | |
| 12675.ba3 | 12675w2 | \([1, 0, 1, -19101, -877277]\) | \(10218313/1521\) | \(114712132640625\) | \([2, 2]\) | \(43008\) | \(1.4223\) | |
| 12675.ba4 | 12675w1 | \([1, 0, 1, 2024, -74527]\) | \(12167/39\) | \(-2941336734375\) | \([2]\) | \(21504\) | \(1.0757\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12675.ba have rank \(0\).
Complex multiplication
The elliptic curves in class 12675.ba do not have complex multiplication.Modular form 12675.2.a.ba
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.
