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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 6930.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6930.bc1 | 6930be3 | \([1, -1, 1, -26916647, -53548383009]\) | \(2958414657792917260183849/12401051653985258880\) | \(9040366655755253723520\) | \([2]\) | \(802816\) | \(3.0677\) | |
| 6930.bc2 | 6930be2 | \([1, -1, 1, -2523047, 88264671]\) | \(2436531580079063806249/1405478914998681600\) | \(1024594129034038886400\) | \([2, 2]\) | \(401408\) | \(2.7211\) | |
| 6930.bc3 | 6930be1 | \([1, -1, 1, -1785767, 916672479]\) | \(863913648706111516969/2486234429521920\) | \(1812464899121479680\) | \([4]\) | \(200704\) | \(2.3745\) | \(\Gamma_0(N)\)-optimal |
| 6930.bc4 | 6930be4 | \([1, -1, 1, 10074073, 697965279]\) | \(155099895405729262880471/90047655797243760000\) | \(-65644741076190701040000\) | \([2]\) | \(802816\) | \(3.0677\) |
Rank
sage: E.rank()
The elliptic curves in class 6930.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 6930.bc do not have complex multiplication.Modular form 6930.2.a.bc
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.
