Show commands:
SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 46800.fd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
46800.fd1 | 46800dk4 | \([0, 0, 0, -74662275, 248313253250]\) | \(986551739719628473/111045168\) | \(5180923358208000000\) | \([2]\) | \(3932160\) | \(3.0153\) | |
46800.fd2 | 46800dk3 | \([0, 0, 0, -8422275, -3190810750]\) | \(1416134368422073/725251155408\) | \(33837317906715648000000\) | \([2]\) | \(3932160\) | \(3.0153\) | |
46800.fd3 | 46800dk2 | \([0, 0, 0, -4678275, 3859141250]\) | \(242702053576633/2554695936\) | \(119191893590016000000\) | \([2, 2]\) | \(1966080\) | \(2.6687\) | |
46800.fd4 | 46800dk1 | \([0, 0, 0, -70275, 149701250]\) | \(-822656953/207028224\) | \(-9659108818944000000\) | \([2]\) | \(983040\) | \(2.3221\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 46800.fd have rank \(1\).
Complex multiplication
The elliptic curves in class 46800.fd do not have complex multiplication.Modular form 46800.2.a.fd
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.