Show commands:
SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 6864.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6864.h1 | 6864p4 | \([0, -1, 0, -7676908, 8189094508]\) | \(195453211868372997250000/14320648682977923\) | \(3666086062842348288\) | \([2]\) | \(186624\) | \(2.6128\) | |
| 6864.h2 | 6864p3 | \([0, -1, 0, -7676773, 8189396800]\) | \(3127086412733145284608000/16789083597\) | \(268625337552\) | \([2]\) | \(93312\) | \(2.2662\) | |
| 6864.h3 | 6864p2 | \([0, -1, 0, -193588, -15763556]\) | \(3134160907827154000/1390984039929627\) | \(356091914221984512\) | \([2]\) | \(62208\) | \(2.0635\) | |
| 6864.h4 | 6864p1 | \([0, -1, 0, -95173, 11162788]\) | \(5958673237147648000/102990700534293\) | \(1647851208548688\) | \([2]\) | \(31104\) | \(1.7169\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6864.h have rank \(0\).
Complex multiplication
The elliptic curves in class 6864.h do not have complex multiplication.Modular form 6864.2.a.h
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
