Show commands:
SageMath
E = EllipticCurve("qh1")
E.isogeny_class()
Elliptic curves in class 486720qh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
486720.qh3 | 486720qh1 | \([0, 0, 0, -2695212, 1721850416]\) | \(-63378025803/812500\) | \(-27757935968256000000\) | \([2]\) | \(18579456\) | \(2.5401\) | \(\Gamma_0(N)\)-optimal* |
486720.qh2 | 486720qh2 | \([0, 0, 0, -43255212, 109497882416]\) | \(261984288445803/42250\) | \(1443412670349312000\) | \([2]\) | \(37158912\) | \(2.8867\) | \(\Gamma_0(N)\)-optimal* |
486720.qh4 | 486720qh3 | \([0, 0, 0, 9472788, 8759280816]\) | \(3774555693/3515200\) | \(-87547020012162750873600\) | \([2]\) | \(55738368\) | \(3.0894\) | |
486720.qh1 | 486720qh4 | \([0, 0, 0, -48933612, 78917048496]\) | \(520300455507/193072360\) | \(4808520074168039091732480\) | \([2]\) | \(111476736\) | \(3.4360\) |
Rank
sage: E.rank()
The elliptic curves in class 486720qh have rank \(0\).
Complex multiplication
The elliptic curves in class 486720qh do not have complex multiplication.Modular form 486720.2.a.qh
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.