Show commands:
SageMath
E = EllipticCurve("dc1")
E.isogeny_class()
Elliptic curves in class 33600dc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33600.fk5 | 33600dc1 | \([0, 1, 0, -1533, 94563]\) | \(-24918016/229635\) | \(-3674160000000\) | \([2]\) | \(49152\) | \(1.0935\) | \(\Gamma_0(N)\)-optimal |
| 33600.fk4 | 33600dc2 | \([0, 1, 0, -42033, 3294063]\) | \(32082281296/99225\) | \(25401600000000\) | \([2, 2]\) | \(98304\) | \(1.4401\) | |
| 33600.fk3 | 33600dc3 | \([0, 1, 0, -60033, 180063]\) | \(23366901604/13505625\) | \(13829760000000000\) | \([2, 2]\) | \(196608\) | \(1.7866\) | |
| 33600.fk1 | 33600dc4 | \([0, 1, 0, -672033, 211824063]\) | \(32779037733124/315\) | \(322560000000\) | \([2]\) | \(196608\) | \(1.7866\) | |
| 33600.fk6 | 33600dc5 | \([0, 1, 0, 239967, 1680063]\) | \(746185003198/432360075\) | \(-885473433600000000\) | \([2]\) | \(393216\) | \(2.1332\) | |
| 33600.fk2 | 33600dc6 | \([0, 1, 0, -648033, -200327937]\) | \(14695548366242/57421875\) | \(117600000000000000\) | \([2]\) | \(393216\) | \(2.1332\) |
Rank
sage: E.rank()
The elliptic curves in class 33600dc have rank \(1\).
Complex multiplication
The elliptic curves in class 33600dc do not have complex multiplication.Modular form 33600.2.a.dc
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.
