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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 162624by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162624.a2 | 162624by1 | \([0, -1, 0, -645, -25419]\) | \(-16384/147\) | \(-266669534208\) | \([2]\) | \(268800\) | \(0.87496\) | \(\Gamma_0(N)\)-optimal |
162624.a1 | 162624by2 | \([0, -1, 0, -17585, -889359]\) | \(20720464/63\) | \(1828591091712\) | \([2]\) | \(537600\) | \(1.2215\) |
Rank
sage: E.rank()
The elliptic curves in class 162624by have rank \(1\).
Complex multiplication
The elliptic curves in class 162624by do not have complex multiplication.Modular form 162624.2.a.by
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.