Show commands:
SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 463680bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 463680.bt3 | 463680bt1 | \([0, 0, 0, -86988, 9588112]\) | \(380920459249/12622400\) | \(2412178204262400\) | \([2]\) | \(2654208\) | \(1.7246\) | \(\Gamma_0(N)\)-optimal |
| 463680.bt4 | 463680bt2 | \([0, 0, 0, 28212, 33134992]\) | \(12994449551/2489452840\) | \(-475741846335651840\) | \([2]\) | \(5308416\) | \(2.0712\) | |
| 463680.bt1 | 463680bt3 | \([0, 0, 0, -974028, -366807152]\) | \(534774372149809/5323062500\) | \(1017253085184000000\) | \([2]\) | \(7962624\) | \(2.2739\) | |
| 463680.bt2 | 463680bt4 | \([0, 0, 0, -254028, -897015152]\) | \(-9486391169809/1813439640250\) | \(-346553712048144384000\) | \([2]\) | \(15925248\) | \(2.6205\) |
Rank
sage: E.rank()
The elliptic curves in class 463680bt have rank \(1\).
Complex multiplication
The elliptic curves in class 463680bt do not have complex multiplication.Modular form 463680.2.a.bt
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
