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SageMath
E = EllipticCurve("eh1")
E.isogeny_class()
Elliptic curves in class 69696eh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69696.dd2 | 69696eh1 | \([0, 0, 0, -65340, -5174928]\) | \(54000/11\) | \(6284345127616512\) | \([2]\) | \(368640\) | \(1.7470\) | \(\Gamma_0(N)\)-optimal |
69696.dd1 | 69696eh2 | \([0, 0, 0, -326700, 67274064]\) | \(1687500/121\) | \(276511185615126528\) | \([2]\) | \(737280\) | \(2.0936\) |
Rank
sage: E.rank()
The elliptic curves in class 69696eh have rank \(1\).
Complex multiplication
The elliptic curves in class 69696eh do not have complex multiplication.Modular form 69696.2.a.eh
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.