Show commands:
SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 6450.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6450.bc1 | 6450bb2 | \([1, 1, 1, -5938, -172969]\) | \(1481933914201/53916840\) | \(842450625000\) | \([2]\) | \(13824\) | \(1.0580\) | |
6450.bc2 | 6450bb1 | \([1, 1, 1, -938, 7031]\) | \(5841725401/1857600\) | \(29025000000\) | \([2]\) | \(6912\) | \(0.71145\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6450.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 6450.bc do not have complex multiplication.Modular form 6450.2.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.