Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 720c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 720.d5 | 720c1 | \([0, 0, 0, -138, 623]\) | \(24918016/45\) | \(524880\) | \([2]\) | \(128\) | \(-0.010416\) | \(\Gamma_0(N)\)-optimal |
| 720.d4 | 720c2 | \([0, 0, 0, -183, 182]\) | \(3631696/2025\) | \(377913600\) | \([2, 2]\) | \(256\) | \(0.33616\) | |
| 720.d2 | 720c3 | \([0, 0, 0, -1803, -29302]\) | \(868327204/5625\) | \(4199040000\) | \([2, 2]\) | \(512\) | \(0.68273\) | |
| 720.d6 | 720c4 | \([0, 0, 0, 717, 1442]\) | \(54607676/32805\) | \(-24488801280\) | \([2]\) | \(512\) | \(0.68273\) | |
| 720.d1 | 720c5 | \([0, 0, 0, -28803, -1881502]\) | \(1770025017602/75\) | \(111974400\) | \([2]\) | \(1024\) | \(1.0293\) | |
| 720.d3 | 720c6 | \([0, 0, 0, -723, -64078]\) | \(-27995042/1171875\) | \(-1749600000000\) | \([2]\) | \(1024\) | \(1.0293\) |
Rank
sage: E.rank()
The elliptic curves in class 720c have rank \(0\).
Complex multiplication
The elliptic curves in class 720c do not have complex multiplication.Modular form 720.2.a.c
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
