Show commands:
SageMath
E = EllipticCurve("fc1")
E.isogeny_class()
Elliptic curves in class 257600fc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 257600.fc2 | 257600fc1 | \([0, -1, 0, -22433, -323263]\) | \(304821217/164864\) | \(675282944000000\) | \([2]\) | \(983040\) | \(1.5365\) | \(\Gamma_0(N)\)-optimal |
| 257600.fc1 | 257600fc2 | \([0, -1, 0, -278433, -56387263]\) | \(582810602977/829472\) | \(3397517312000000\) | \([2]\) | \(1966080\) | \(1.8831\) |
Rank
sage: E.rank()
The elliptic curves in class 257600fc have rank \(1\).
Complex multiplication
The elliptic curves in class 257600fc do not have complex multiplication.Modular form 257600.2.a.fc
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
