Show commands:
SageMath
E = EllipticCurve("ep1")
E.isogeny_class()
Elliptic curves in class 14400ep
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14400.dx4 | 14400ep1 | \([0, 0, 0, -1740, -38000]\) | \(-24389/12\) | \(-286654464000\) | \([2]\) | \(12288\) | \(0.90376\) | \(\Gamma_0(N)\)-optimal |
14400.dx2 | 14400ep2 | \([0, 0, 0, -30540, -2054000]\) | \(131872229/18\) | \(429981696000\) | \([2]\) | \(24576\) | \(1.2503\) | |
14400.dx3 | 14400ep3 | \([0, 0, 0, -16140, 3792400]\) | \(-19465109/248832\) | \(-5944066965504000\) | \([2]\) | \(61440\) | \(1.7085\) | |
14400.dx1 | 14400ep4 | \([0, 0, 0, -476940, 126365200]\) | \(502270291349/1889568\) | \(45137758519296000\) | \([2]\) | \(122880\) | \(2.0550\) |
Rank
sage: E.rank()
The elliptic curves in class 14400ep have rank \(0\).
Complex multiplication
The elliptic curves in class 14400ep do not have complex multiplication.Modular form 14400.2.a.ep
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.