Show commands:
SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 29400ef
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
29400.em2 | 29400ef1 | \([0, 1, 0, 5717, 1013438]\) | \(2048/45\) | \(-453978078750000\) | \([2]\) | \(129024\) | \(1.4927\) | \(\Gamma_0(N)\)-optimal |
29400.em1 | 29400ef2 | \([0, 1, 0, -122908, 15676688]\) | \(1272112/75\) | \(12106082100000000\) | \([2]\) | \(258048\) | \(1.8393\) |
Rank
sage: E.rank()
The elliptic curves in class 29400ef have rank \(0\).
Complex multiplication
The elliptic curves in class 29400ef do not have complex multiplication.Modular form 29400.2.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.