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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 77616.bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 77616.bz1 | 77616gn4 | \([0, 0, 0, -1033851, -404211094]\) | \(347873904937/395307\) | \(138870570984026112\) | \([2]\) | \(884736\) | \(2.2028\) | |
| 77616.bz2 | 77616gn2 | \([0, 0, 0, -81291, -2802310]\) | \(169112377/88209\) | \(30987648070815744\) | \([2, 2]\) | \(442368\) | \(1.8562\) | |
| 77616.bz3 | 77616gn1 | \([0, 0, 0, -46011, 3766826]\) | \(30664297/297\) | \(104335515389952\) | \([2]\) | \(221184\) | \(1.5096\) | \(\Gamma_0(N)\)-optimal |
| 77616.bz4 | 77616gn3 | \([0, 0, 0, 306789, -21818230]\) | \(9090072503/5845851\) | \(-2053635949420425216\) | \([2]\) | \(884736\) | \(2.2028\) |
Rank
sage: E.rank()
The elliptic curves in class 77616.bz have rank \(1\).
Complex multiplication
The elliptic curves in class 77616.bz do not have complex multiplication.Modular form 77616.2.a.bz
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
