Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 41616.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 41616.e1 | 41616cp2 | \([0, 0, 0, -18737859, -31219652414]\) | \(-843137281012581793/216\) | \(-186397065216\) | \([]\) | \(1306368\) | \(2.4450\) | |
| 41616.e2 | 41616cp1 | \([0, 0, 0, -230979, -42962366]\) | \(-1579268174113/10077696\) | \(-8696541474717696\) | \([]\) | \(435456\) | \(1.8957\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 41616.e have rank \(1\).
Complex multiplication
The elliptic curves in class 41616.e do not have complex multiplication.Modular form 41616.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
