Show commands:
SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 235200.ef
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
235200.ef1 | 235200ef2 | \([0, -1, 0, -129033, 17624937]\) | \(31554496/525\) | \(3953006400000000\) | \([2]\) | \(1769472\) | \(1.7913\) | |
235200.ef2 | 235200ef1 | \([0, -1, 0, -408, 775062]\) | \(-64/2205\) | \(-259416045000000\) | \([2]\) | \(884736\) | \(1.4447\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 235200.ef have rank \(1\).
Complex multiplication
The elliptic curves in class 235200.ef do not have complex multiplication.Modular form 235200.2.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 LMFDB 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 LMFDB labels.