Show commands:
SageMath
E = EllipticCurve("em1")
E.isogeny_class()
Elliptic curves in class 44352em
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
44352.ee4 | 44352em1 | \([0, 0, 0, 996, -478928]\) | \(9148592/8301447\) | \(-99151951675392\) | \([2]\) | \(131072\) | \(1.3647\) | \(\Gamma_0(N)\)-optimal |
44352.ee3 | 44352em2 | \([0, 0, 0, -86124, -9504560]\) | \(1478729816932/38900169\) | \(1858484515700736\) | \([2, 2]\) | \(262144\) | \(1.7112\) | |
44352.ee2 | 44352em3 | \([0, 0, 0, -197004, 19989520]\) | \(8849350367426/3314597517\) | \(316714724870455296\) | \([2]\) | \(524288\) | \(2.0578\) | |
44352.ee1 | 44352em4 | \([0, 0, 0, -1369164, -616639088]\) | \(2970658109581346/2139291\) | \(204412438315008\) | \([2]\) | \(524288\) | \(2.0578\) |
Rank
sage: E.rank()
The elliptic curves in class 44352em have rank \(0\).
Complex multiplication
The elliptic curves in class 44352em do not have complex multiplication.Modular form 44352.2.a.em
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.