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SageMath
E = EllipticCurve("il1")
E.isogeny_class()
Elliptic curves in class 242550il
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
242550.il1 | 242550il1 | \([1, -1, 1, -230716730, 1348745899897]\) | \(37537160298467283/5519360000\) | \(199704697426080000000000\) | \([2]\) | \(49545216\) | \(3.4852\) | \(\Gamma_0(N)\)-optimal |
242550.il2 | 242550il2 | \([1, -1, 1, -209548730, 1606233451897]\) | \(-28124139978713043/14526050000000\) | \(-525589999573521093750000000\) | \([2]\) | \(99090432\) | \(3.8317\) |
Rank
sage: E.rank()
The elliptic curves in class 242550il have rank \(1\).
Complex multiplication
The elliptic curves in class 242550il do not have complex multiplication.Modular form 242550.2.a.il
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.