Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 112651j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 112651.e1 | 112651j1 | \([0, -1, 1, -162059, 29420082]\) | \(-2258403328/480491\) | \(-100145077831343699\) | \([]\) | \(829440\) | \(1.9835\) | \(\Gamma_0(N)\)-optimal |
| 112651.e2 | 112651j2 | \([0, -1, 1, 1142321, -170215277]\) | \(790939860992/517504691\) | \(-107859559405442497499\) | \([]\) | \(2488320\) | \(2.5328\) |
Rank
sage: E.rank()
The elliptic curves in class 112651j have rank \(0\).
Complex multiplication
The elliptic curves in class 112651j do not have complex multiplication.Modular form 112651.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
