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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 6090.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6090.bc1 | 6090bc3 | \([1, 0, 0, -4690, -123970]\) | \(11409011759446561/5015376870\) | \(5015376870\) | \([2]\) | \(8192\) | \(0.82035\) | |
| 6090.bc2 | 6090bc2 | \([1, 0, 0, -340, -1300]\) | \(4347507044161/1817316900\) | \(1817316900\) | \([2, 2]\) | \(4096\) | \(0.47378\) | |
| 6090.bc3 | 6090bc1 | \([1, 0, 0, -160, 752]\) | \(453161802241/9208080\) | \(9208080\) | \([4]\) | \(2048\) | \(0.12721\) | \(\Gamma_0(N)\)-optimal |
| 6090.bc4 | 6090bc4 | \([1, 0, 0, 1130, -9238]\) | \(159564039253919/129962883750\) | \(-129962883750\) | \([2]\) | \(8192\) | \(0.82035\) |
Rank
sage: E.rank()
The elliptic curves in class 6090.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 6090.bc do not have complex multiplication.Modular form 6090.2.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.
