Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 213616e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 213616.e3 | 213616e1 | \([0, -1, 0, -125792, -17129984]\) | \(11134383337/316\) | \(6247512653824\) | \([]\) | \(656640\) | \(1.5569\) | \(\Gamma_0(N)\)-optimal |
| 213616.e2 | 213616e2 | \([0, -1, 0, -220432, 12019136]\) | \(59914169497/31554496\) | \(623851623560249344\) | \([]\) | \(1969920\) | \(2.1062\) | |
| 213616.e1 | 213616e3 | \([0, -1, 0, -14105472, 20395257856]\) | \(15698803397448457/20709376\) | \(409436989281009664\) | \([]\) | \(5909760\) | \(2.6555\) |
Rank
sage: E.rank()
The elliptic curves in class 213616e have rank \(0\).
Complex multiplication
The elliptic curves in class 213616e do not have complex multiplication.Modular form 213616.2.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 Cremona numbering.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
