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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 129360i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129360.r2 | 129360i1 | \([0, -1, 0, -9718816, 11665032016]\) | \(289047861148528498972/2210462409375\) | \(776385132969600000\) | \([2]\) | \(4055040\) | \(2.6073\) | \(\Gamma_0(N)\)-optimal |
129360.r1 | 129360i2 | \([0, -1, 0, -9922936, 11149669840]\) | \(153822637773009613406/12611991884765625\) | \(8859470267340000000000\) | \([2]\) | \(8110080\) | \(2.9539\) |
Rank
sage: E.rank()
The elliptic curves in class 129360i have rank \(0\).
Complex multiplication
The elliptic curves in class 129360i do not have complex multiplication.Modular form 129360.2.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.