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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 51842c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51842.e2 | 51842c1 | \([1, -1, 0, -212228, 175200444]\) | \(-60698457/725788\) | \(-12640522895249034268\) | \([2]\) | \(1216512\) | \(2.3471\) | \(\Gamma_0(N)\)-optimal |
51842.e1 | 51842c2 | \([1, -1, 0, -6174058, 5887825950]\) | \(1494447319737/5411854\) | \(94254333762400407694\) | \([2]\) | \(2433024\) | \(2.6937\) |
Rank
sage: E.rank()
The elliptic curves in class 51842c have rank \(1\).
Complex multiplication
The elliptic curves in class 51842c do not have complex multiplication.Modular form 51842.2.a.c
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.