Show commands:
SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 6300.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6300.bc1 | 6300be2 | \([0, 0, 0, -12993375, 18021293750]\) | \(665567485783184/257298363\) | \(93785253313500000000\) | \([2]\) | \(322560\) | \(2.7977\) | |
6300.bc2 | 6300be1 | \([0, 0, 0, -691500, 368103125]\) | \(-1605176213504/1640558367\) | \(-37373970298218750000\) | \([2]\) | \(161280\) | \(2.4511\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6300.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 6300.bc do not have complex multiplication.Modular form 6300.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.