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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 38640.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 38640.bf1 | 38640k6 | \([0, -1, 0, -1422160, 653260192]\) | \(155324313723954725282/13018359375\) | \(26661600000000\) | \([4]\) | \(458752\) | \(2.0187\) | |
| 38640.bf2 | 38640k4 | \([0, -1, 0, -122400, -16424208]\) | \(198048499826486404/242568272835\) | \(248389911383040\) | \([2]\) | \(229376\) | \(1.6721\) | |
| 38640.bf3 | 38640k3 | \([0, -1, 0, -89080, 10182400]\) | \(76343005935514084/694180580625\) | \(710840914560000\) | \([2, 4]\) | \(229376\) | \(1.6721\) | |
| 38640.bf4 | 38640k5 | \([0, -1, 0, -26080, 24244000]\) | \(-957928673903042/123339801817575\) | \(-252599914122393600\) | \([4]\) | \(458752\) | \(2.0187\) | |
| 38640.bf5 | 38640k2 | \([0, -1, 0, -9700, -105248]\) | \(394315384276816/208332909225\) | \(53333224761600\) | \([2, 2]\) | \(114688\) | \(1.3255\) | |
| 38640.bf6 | 38640k1 | \([0, -1, 0, 2305, -14010]\) | \(84611246065664/53699121315\) | \(-859185941040\) | \([2]\) | \(57344\) | \(0.97893\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 38640.bf have rank \(1\).
Complex multiplication
The elliptic curves in class 38640.bf do not have complex multiplication.Modular form 38640.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
