Show commands:
SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 22743n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22743.f6 | 22743n1 | \([1, -1, 1, 3181, -16014]\) | \(103823/63\) | \(-2160676176687\) | \([2]\) | \(27648\) | \(1.0560\) | \(\Gamma_0(N)\)-optimal |
22743.f5 | 22743n2 | \([1, -1, 1, -13064, -119982]\) | \(7189057/3969\) | \(136122599131281\) | \([2, 2]\) | \(55296\) | \(1.4026\) | |
22743.f3 | 22743n3 | \([1, -1, 1, -126779, 17301156]\) | \(6570725617/45927\) | \(1575132932804823\) | \([2]\) | \(110592\) | \(1.7492\) | |
22743.f2 | 22743n4 | \([1, -1, 1, -159269, -24390012]\) | \(13027640977/21609\) | \(741111928603641\) | \([2, 2]\) | \(110592\) | \(1.7492\) | |
22743.f4 | 22743n5 | \([1, -1, 1, -110534, -39634320]\) | \(-4354703137/17294403\) | \(-593136580192447347\) | \([2]\) | \(221184\) | \(2.0957\) | |
22743.f1 | 22743n6 | \([1, -1, 1, -2547284, -1564182084]\) | \(53297461115137/147\) | \(5041577745603\) | \([2]\) | \(221184\) | \(2.0957\) |
Rank
sage: E.rank()
The elliptic curves in class 22743n have rank \(1\).
Complex multiplication
The elliptic curves in class 22743n do not have complex multiplication.Modular form 22743.2.a.n
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.