Show commands:
SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 14400.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 14400.cm1 | 14400z5 | \([0, 0, 0, -2880300, 1881502000]\) | \(1770025017602/75\) | \(111974400000000\) | \([2]\) | \(196608\) | \(2.1806\) | |
| 14400.cm2 | 14400z3 | \([0, 0, 0, -180300, 29302000]\) | \(868327204/5625\) | \(4199040000000000\) | \([2, 2]\) | \(98304\) | \(1.8340\) | |
| 14400.cm3 | 14400z6 | \([0, 0, 0, -72300, 64078000]\) | \(-27995042/1171875\) | \(-1749600000000000000\) | \([2]\) | \(196608\) | \(2.1806\) | |
| 14400.cm4 | 14400z2 | \([0, 0, 0, -18300, -182000]\) | \(3631696/2025\) | \(377913600000000\) | \([2, 2]\) | \(49152\) | \(1.4874\) | |
| 14400.cm5 | 14400z1 | \([0, 0, 0, -13800, -623000]\) | \(24918016/45\) | \(524880000000\) | \([2]\) | \(24576\) | \(1.1409\) | \(\Gamma_0(N)\)-optimal |
| 14400.cm6 | 14400z4 | \([0, 0, 0, 71700, -1442000]\) | \(54607676/32805\) | \(-24488801280000000\) | \([2]\) | \(98304\) | \(1.8340\) |
Rank
sage: E.rank()
The elliptic curves in class 14400.cm have rank \(0\).
Complex multiplication
The elliptic curves in class 14400.cm do not have complex multiplication.Modular form 14400.2.a.cm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
