Show commands:
SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 12600cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12600.cl3 | 12600cd1 | \([0, 0, 0, -16050, 750125]\) | \(2508888064/118125\) | \(21528281250000\) | \([2]\) | \(36864\) | \(1.3192\) | \(\Gamma_0(N)\)-optimal |
| 12600.cl2 | 12600cd2 | \([0, 0, 0, -44175, -2596750]\) | \(3269383504/893025\) | \(2604060900000000\) | \([2, 2]\) | \(73728\) | \(1.6657\) | |
| 12600.cl1 | 12600cd3 | \([0, 0, 0, -651675, -202464250]\) | \(2624033547076/324135\) | \(3780710640000000\) | \([2]\) | \(147456\) | \(2.0123\) | |
| 12600.cl4 | 12600cd4 | \([0, 0, 0, 113325, -16929250]\) | \(13799183324/18600435\) | \(-216955473840000000\) | \([2]\) | \(147456\) | \(2.0123\) |
Rank
sage: E.rank()
The elliptic curves in class 12600cd have rank \(0\).
Complex multiplication
The elliptic curves in class 12600cd do not have complex multiplication.Modular form 12600.2.a.cd
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.
