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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 109820i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
109820.c1 | 109820i1 | \([0, 1, 0, -266265, 49232488]\) | \(5405726654464/407253125\) | \(157281606482450000\) | \([2]\) | \(1228800\) | \(2.0450\) | \(\Gamma_0(N)\)-optimal |
109820.c2 | 109820i2 | \([0, 1, 0, 255380, 219080100]\) | \(298091207216/3525390625\) | \(-21784156022500000000\) | \([2]\) | \(2457600\) | \(2.3916\) |
Rank
sage: E.rank()
The elliptic curves in class 109820i have rank \(1\).
Complex multiplication
The elliptic curves in class 109820i do not have complex multiplication.Modular form 109820.2.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.