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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 222180.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
222180.i1 | 222180t1 | \([0, -1, 0, -1936845, 1037692650]\) | \(339251313639424/173578125\) | \(411132672725250000\) | \([2]\) | \(3649536\) | \(2.3314\) | \(\Gamma_0(N)\)-optimal |
222180.i2 | 222180t2 | \([0, -1, 0, -1606220, 1403099400]\) | \(-12092945312464/15426235125\) | \(-584610926247014688000\) | \([2]\) | \(7299072\) | \(2.6780\) |
Rank
sage: E.rank()
The elliptic curves in class 222180.i have rank \(1\).
Complex multiplication
The elliptic curves in class 222180.i do not have complex multiplication.Modular form 222180.2.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 LMFDB 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 LMFDB labels.