Show commands:
SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 405600ef
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
405600.ef2 | 405600ef1 | \([0, 1, 0, 282, 16368]\) | \(64/3\) | \(-115843416000\) | \([2]\) | \(442368\) | \(0.80272\) | \(\Gamma_0(N)\)-optimal* |
405600.ef1 | 405600ef2 | \([0, 1, 0, -8168, 269868]\) | \(195112/9\) | \(2780241984000\) | \([2]\) | \(884736\) | \(1.1493\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 405600ef have rank \(1\).
Complex multiplication
The elliptic curves in class 405600ef do not have complex multiplication.Modular form 405600.2.a.ef
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.