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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 320064.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
320064.e1 | 320064e1 | \([0, -1, 0, -6497, 203265]\) | \(115714886617/320064\) | \(83902857216\) | \([2]\) | \(294912\) | \(0.96891\) | \(\Gamma_0(N)\)-optimal |
320064.e2 | 320064e2 | \([0, -1, 0, -3937, 362497]\) | \(-25750777177/200080008\) | \(-52449773617152\) | \([2]\) | \(589824\) | \(1.3155\) |
Rank
sage: E.rank()
The elliptic curves in class 320064.e have rank \(1\).
Complex multiplication
The elliptic curves in class 320064.e do not have complex multiplication.Modular form 320064.2.a.e
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.