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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 208208i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 208208.o1 | 208208i1 | \([0, -1, 0, -241557, -45615779]\) | \(-78843215872/539\) | \(-10656358608896\) | \([]\) | \(1036800\) | \(1.6806\) | \(\Gamma_0(N)\)-optimal |
| 208208.o2 | 208208i2 | \([0, -1, 0, -133397, -86662499]\) | \(-13278380032/156590819\) | \(-3095895959415074816\) | \([]\) | \(3110400\) | \(2.2299\) | |
| 208208.o3 | 208208i3 | \([0, -1, 0, 1191563, 2243942141]\) | \(9463555063808/115539436859\) | \(-2284285106937663205376\) | \([]\) | \(9331200\) | \(2.7792\) |
Rank
sage: E.rank()
The elliptic curves in class 208208i have rank \(1\).
Complex multiplication
The elliptic curves in class 208208i do not have complex multiplication.Modular form 208208.2.a.i
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.
