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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 68544bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68544.dn1 | 68544bm1 | \([0, 0, 0, -12144, 514600]\) | \(265327034368/297381\) | \(221993726976\) | \([2]\) | \(92160\) | \(1.0912\) | \(\Gamma_0(N)\)-optimal |
68544.dn2 | 68544bm2 | \([0, 0, 0, -9084, 780208]\) | \(-6940769488/18000297\) | \(-214994395348992\) | \([2]\) | \(184320\) | \(1.4378\) |
Rank
sage: E.rank()
The elliptic curves in class 68544bm have rank \(1\).
Complex multiplication
The elliptic curves in class 68544bm do not have complex multiplication.Modular form 68544.2.a.bm
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.