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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 78033b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 78033.a3 | 78033b1 | \([1, 0, 0, -2082, -31773]\) | \(389017/57\) | \(146246405313\) | \([2]\) | \(72576\) | \(0.86730\) | \(\Gamma_0(N)\)-optimal |
| 78033.a2 | 78033b2 | \([1, 0, 0, -8927, 292680]\) | \(30664297/3249\) | \(8336045102841\) | \([2, 2]\) | \(145152\) | \(1.2139\) | |
| 78033.a4 | 78033b3 | \([1, 0, 0, 11608, 1446747]\) | \(67419143/390963\) | \(-1003104094041867\) | \([2]\) | \(290304\) | \(1.5604\) | |
| 78033.a1 | 78033b4 | \([1, 0, 0, -138982, 19930985]\) | \(115714886617/1539\) | \(3948652943451\) | \([2]\) | \(290304\) | \(1.5604\) |
Rank
sage: E.rank()
The elliptic curves in class 78033b have rank \(1\).
Complex multiplication
The elliptic curves in class 78033b do not have complex multiplication.Modular form 78033.2.a.b
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.
