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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 121968.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121968.bf1 | 121968eg4 | \([0, 0, 0, -701264091, -7147769062390]\) | \(7209828390823479793/49509306\) | \(261896941020854329344\) | \([2]\) | \(17694720\) | \(3.5176\) | |
| 121968.bf2 | 121968eg3 | \([0, 0, 0, -61106331, -15638060086]\) | \(4770223741048753/2740574865798\) | \(14497237630257523196977152\) | \([2]\) | \(17694720\) | \(3.5176\) | |
| 121968.bf3 | 121968eg2 | \([0, 0, 0, -43856571, -111536375830]\) | \(1763535241378513/4612311396\) | \(24398448357325515669504\) | \([2, 2]\) | \(8847360\) | \(3.1711\) | |
| 121968.bf4 | 121968eg1 | \([0, 0, 0, -1690491, -3093651286]\) | \(-100999381393/723148272\) | \(-3825347912194865430528\) | \([2]\) | \(4423680\) | \(2.8245\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121968.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 121968.bf do not have complex multiplication.Modular form 121968.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
