Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 5712j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5712.x4 | 5712j1 | \([0, 1, 0, 28, 60]\) | \(9148592/9639\) | \(-2467584\) | \([2]\) | \(768\) | \(-0.086725\) | \(\Gamma_0(N)\)-optimal |
| 5712.x3 | 5712j2 | \([0, 1, 0, -152, 420]\) | \(381775972/127449\) | \(130507776\) | \([2, 2]\) | \(1536\) | \(0.25985\) | |
| 5712.x2 | 5712j3 | \([0, 1, 0, -992, -12012]\) | \(52767497666/1753941\) | \(3592071168\) | \([2]\) | \(3072\) | \(0.60642\) | |
| 5712.x1 | 5712j4 | \([0, 1, 0, -2192, 38772]\) | \(569001644066/122451\) | \(250779648\) | \([2]\) | \(3072\) | \(0.60642\) |
Rank
sage: E.rank()
The elliptic curves in class 5712j have rank \(0\).
Complex multiplication
The elliptic curves in class 5712j do not have complex multiplication.Modular form 5712.2.a.j
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 Cremona 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 Cremona labels.
