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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 226512.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
226512.k1 | 226512ei1 | \([0, 0, 0, -709302, -216014645]\) | \(1909913257984/129730653\) | \(2680687727151260112\) | \([2]\) | \(5376000\) | \(2.2846\) | \(\Gamma_0(N)\)-optimal |
226512.k2 | 226512ei2 | \([0, 0, 0, 613833, -929184410]\) | \(77366117936/1172914587\) | \(-387784094587741133568\) | \([2]\) | \(10752000\) | \(2.6312\) |
Rank
sage: E.rank()
The elliptic curves in class 226512.k have rank \(0\).
Complex multiplication
The elliptic curves in class 226512.k do not have complex multiplication.Modular form 226512.2.a.k
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.