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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 91200.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
91200.e1 | 91200cj2 | \([0, -1, 0, -572833, -166666463]\) | \(81202348906/9747\) | \(2495232000000000\) | \([2]\) | \(1351680\) | \(1.9792\) | |
91200.e2 | 91200cj1 | \([0, -1, 0, -32833, -3046463]\) | \(-30581492/13851\) | \(-1772928000000000\) | \([2]\) | \(675840\) | \(1.6326\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 91200.e have rank \(1\).
Complex multiplication
The elliptic curves in class 91200.e do not have complex multiplication.Modular form 91200.2.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.