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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 57498i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
57498.j2 | 57498i1 | \([1, 0, 1, -65741, -7096840]\) | \(-12246522625/1379952\) | \(-3540579289552368\) | \([2]\) | \(437760\) | \(1.7206\) | \(\Gamma_0(N)\)-optimal |
57498.j1 | 57498i2 | \([1, 0, 1, -1078801, -431366368]\) | \(54117385890625/587412\) | \(1507138481363508\) | \([2]\) | \(875520\) | \(2.0672\) |
Rank
sage: E.rank()
The elliptic curves in class 57498i have rank \(1\).
Complex multiplication
The elliptic curves in class 57498i do not have complex multiplication.Modular form 57498.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.