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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 121968fk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121968.dj4 | 121968fk1 | \([0, 0, 0, 7260, 81191]\) | \(2048000/1323\) | \(-27337793967792\) | \([2]\) | \(207360\) | \(1.2672\) | \(\Gamma_0(N)\)-optimal |
| 121968.dj3 | 121968fk2 | \([0, 0, 0, -30855, 668162]\) | \(9826000/5103\) | \(1687132427726592\) | \([2]\) | \(414720\) | \(1.6138\) | |
| 121968.dj2 | 121968fk3 | \([0, 0, 0, -123420, 17187203]\) | \(-10061824000/352947\) | \(-7293115924074288\) | \([2]\) | \(622080\) | \(1.8165\) | |
| 121968.dj1 | 121968fk4 | \([0, 0, 0, -1991055, 1081365626]\) | \(2640279346000/3087\) | \(1020610974797568\) | \([2]\) | \(1244160\) | \(2.1631\) |
Rank
sage: E.rank()
The elliptic curves in class 121968fk have rank \(1\).
Complex multiplication
The elliptic curves in class 121968fk do not have complex multiplication.Modular form 121968.2.a.fk
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
