Show commands:
SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 338130.cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
338130.cm1 | 338130cm1 | \([1, -1, 1, -743, -12189]\) | \(-215038729/189540\) | \(-39932476740\) | \([]\) | \(248832\) | \(0.73138\) | \(\Gamma_0(N)\)-optimal |
338130.cm2 | 338130cm2 | \([1, -1, 1, 6142, 205377]\) | \(121644944711/158184000\) | \(-33326363304000\) | \([]\) | \(746496\) | \(1.2807\) |
Rank
sage: E.rank()
The elliptic curves in class 338130.cm have rank \(2\).
Complex multiplication
The elliptic curves in class 338130.cm do not have complex multiplication.Modular form 338130.2.a.cm
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.