Show commands:
SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 75712.bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75712.bw1 | 75712z4 | \([0, 0, 0, -202124, 34976240]\) | \(1443468546/7\) | \(4428616564736\) | \([2]\) | \(294912\) | \(1.6270\) | |
| 75712.bw2 | 75712z3 | \([0, 0, 0, -39884, -2425488]\) | \(11090466/2401\) | \(1519015481704448\) | \([2]\) | \(294912\) | \(1.6270\) | |
| 75712.bw3 | 75712z2 | \([0, 0, 0, -12844, 527280]\) | \(740772/49\) | \(15500157976576\) | \([2, 2]\) | \(147456\) | \(1.2804\) | |
| 75712.bw4 | 75712z1 | \([0, 0, 0, 676, 35152]\) | \(432/7\) | \(-553577070592\) | \([2]\) | \(73728\) | \(0.93387\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 75712.bw have rank \(0\).
Complex multiplication
The elliptic curves in class 75712.bw do not have complex multiplication.Modular form 75712.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
