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SageMath
E = EllipticCurve("gm1")
E.isogeny_class()
Elliptic curves in class 121968gm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121968.gm2 | 121968gm1 | \([0, 0, 0, -7057083, -11269696790]\) | \(-7347774183121/6119866368\) | \(-32373192248657373560832\) | \([2]\) | \(15482880\) | \(3.0172\) | \(\Gamma_0(N)\)-optimal |
| 121968.gm1 | 121968gm2 | \([0, 0, 0, -129722043, -568536610070]\) | \(45637459887836881/13417633152\) | \(70977304312219681947648\) | \([2]\) | \(30965760\) | \(3.3637\) |
Rank
sage: E.rank()
The elliptic curves in class 121968gm have rank \(1\).
Complex multiplication
The elliptic curves in class 121968gm do not have complex multiplication.Modular form 121968.2.a.gm
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
