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SageMath
E = EllipticCurve("ey1")
E.isogeny_class()
Elliptic curves in class 136710ey
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
136710.x2 | 136710ey1 | \([1, -1, 0, -18090, -956124]\) | \(-7633736209/230640\) | \(-19781098147440\) | \([2]\) | \(460800\) | \(1.3291\) | \(\Gamma_0(N)\)-optimal |
136710.x1 | 136710ey2 | \([1, -1, 0, -291510, -60507000]\) | \(31942518433489/27900\) | \(2392874775900\) | \([2]\) | \(921600\) | \(1.6757\) |
Rank
sage: E.rank()
The elliptic curves in class 136710ey have rank \(0\).
Complex multiplication
The elliptic curves in class 136710ey do not have complex multiplication.Modular form 136710.2.a.ey
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.