Show commands:
SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 466578.cj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
466578.cj1 | 466578cj3 | \([1, -1, 0, -41723509221, 3280354179500047]\) | \(632678989847546725777/80515134\) | \(1022257497735180307455246\) | \([2]\) | \(778567680\) | \(4.4670\) | \(\Gamma_0(N)\)-optimal* |
466578.cj2 | 466578cj4 | \([1, -1, 0, -2983537881, 35521459010995]\) | \(231331938231569617/90942310746882\) | \(1154645771593235997718605687858\) | \([2]\) | \(778567680\) | \(4.4670\) | |
466578.cj3 | 466578cj2 | \([1, -1, 0, -2607942591, 51246807493657]\) | \(154502321244119857/55101928644\) | \(699599651613296661840902436\) | \([2, 2]\) | \(389283840\) | \(4.1204\) | \(\Gamma_0(N)\)-optimal* |
466578.cj4 | 466578cj1 | \([1, -1, 0, -139744971, 1037250589045]\) | \(-23771111713777/22848457968\) | \(-290094623322307610514471792\) | \([2]\) | \(194641920\) | \(3.7738\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 466578.cj have rank \(0\).
Complex multiplication
The elliptic curves in class 466578.cj do not have complex multiplication.Modular form 466578.2.a.cj
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.