Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 20097a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20097.e2 | 20097a1 | \([1, -1, 1, -2249, 153480]\) | \(-46574399618739/347190316781\) | \(-9374138553087\) | \([2]\) | \(35712\) | \(1.1721\) | \(\Gamma_0(N)\)-optimal |
20097.e1 | 20097a2 | \([1, -1, 1, -58844, 5496048]\) | \(834563889111074499/2244268390133\) | \(60595246533591\) | \([2]\) | \(71424\) | \(1.5187\) |
Rank
sage: E.rank()
The elliptic curves in class 20097a have rank \(1\).
Complex multiplication
The elliptic curves in class 20097a do not have complex multiplication.Modular form 20097.2.a.a
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.