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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 150696bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 150696.x3 | 150696bf1 | \([0, 0, 0, -81639, -8978310]\) | \(322440248841552/27209\) | \(5077852416\) | \([2]\) | \(286720\) | \(1.3044\) | \(\Gamma_0(N)\)-optimal |
| 150696.x2 | 150696bf2 | \([0, 0, 0, -81819, -8936730]\) | \(81144432781668/740329681\) | \(552653145547776\) | \([2, 2]\) | \(573440\) | \(1.6510\) | |
| 150696.x1 | 150696bf3 | \([0, 0, 0, -142659, 6139422]\) | \(215062038362754/113550802729\) | \(169530440067975168\) | \([2]\) | \(1146880\) | \(1.9975\) | |
| 150696.x4 | 150696bf4 | \([0, 0, 0, -23859, -21351762]\) | \(-1006057824354/131332646081\) | \(-196078589937764352\) | \([2]\) | \(1146880\) | \(1.9975\) |
Rank
sage: E.rank()
The elliptic curves in class 150696bf have rank \(1\).
Complex multiplication
The elliptic curves in class 150696bf do not have complex multiplication.Modular form 150696.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
