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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 51842.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51842.i1 | 51842h2 | \([1, 1, 0, -2968494, 1967687156]\) | \(-313994137/64\) | \(-589645382868759616\) | \([]\) | \(1987200\) | \(2.4074\) | |
51842.i2 | 51842h1 | \([1, 1, 0, 12421, 9226001]\) | \(23/4\) | \(-36852836429297476\) | \([]\) | \(662400\) | \(1.8581\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 51842.i have rank \(1\).
Complex multiplication
The elliptic curves in class 51842.i do not have complex multiplication.Modular form 51842.2.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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.