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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 72128bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72128.j2 | 72128bp1 | \([0, 1, 0, -10649, -426329]\) | \(277167808/161\) | \(77584338944\) | \([2]\) | \(110592\) | \(1.0350\) | \(\Gamma_0(N)\)-optimal |
72128.j1 | 72128bp2 | \([0, 1, 0, -12609, -260513]\) | \(57512456/25921\) | \(99928628559872\) | \([2]\) | \(221184\) | \(1.3815\) |
Rank
sage: E.rank()
The elliptic curves in class 72128bp have rank \(1\).
Complex multiplication
The elliptic curves in class 72128bp do not have complex multiplication.Modular form 72128.2.a.bp
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.