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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 85698j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85698.e4 | 85698j1 | \([1, -1, 0, 1488, 3050]\) | \(3375/2\) | \(-215836326162\) | \([]\) | \(71280\) | \(0.86416\) | \(\Gamma_0(N)\)-optimal |
| 85698.e3 | 85698j2 | \([1, -1, 0, -22317, 1350413]\) | \(-140625/8\) | \(-69930969676488\) | \([]\) | \(213840\) | \(1.4135\) | |
| 85698.e1 | 85698j3 | \([1, -1, 0, -569832, -165422656]\) | \(-189613868625/128\) | \(-13813524874368\) | \([]\) | \(498960\) | \(1.8371\) | |
| 85698.e2 | 85698j4 | \([1, -1, 0, -450807, -236547235]\) | \(-1159088625/2097152\) | \(-18331984114873270272\) | \([]\) | \(1496880\) | \(2.3864\) |
Rank
sage: E.rank()
The elliptic curves in class 85698j have rank \(1\).
Complex multiplication
The elliptic curves in class 85698j do not have complex multiplication.Modular form 85698.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 7 & 21 \\ 3 & 1 & 21 & 7 \\ 7 & 21 & 1 & 3 \\ 21 & 7 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
