Show commands:
SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 12075.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12075.r1 | 12075c3 | \([1, 1, 0, -1150025, 474210000]\) | \(10765299591712341649/20708625\) | \(323572265625\) | \([2]\) | \(101376\) | \(1.8902\) | |
| 12075.r2 | 12075c2 | \([1, 1, 0, -71900, 7381875]\) | \(2630872462131649/3645140625\) | \(56955322265625\) | \([2, 2]\) | \(50688\) | \(1.5436\) | |
| 12075.r3 | 12075c4 | \([1, 1, 0, -51775, 11628250]\) | \(-982374577874929/3183837890625\) | \(-49747467041015625\) | \([2]\) | \(101376\) | \(1.8902\) | |
| 12075.r4 | 12075c1 | \([1, 1, 0, -5775, 42000]\) | \(1363569097969/734582625\) | \(11477853515625\) | \([2]\) | \(25344\) | \(1.1971\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12075.r have rank \(1\).
Complex multiplication
The elliptic curves in class 12075.r do not have complex multiplication.Modular form 12075.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
