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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 404586x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 404586.x3 | 404586x1 | \([1, -1, 0, -2857737, -1324599795]\) | \(19804628171203875/5638671302656\) | \(734853313575946027008\) | \([2]\) | \(21233664\) | \(2.7105\) | \(\Gamma_0(N)\)-optimal |
| 404586.x4 | 404586x2 | \([1, -1, 0, 7525623, -8744548851]\) | \(361682234074684125/462672528510976\) | \(-60297261966077460000768\) | \([2]\) | \(42467328\) | \(3.0571\) | |
| 404586.x1 | 404586x3 | \([1, -1, 0, -212471817, -1192012309747]\) | \(11165451838341046875/572244736\) | \(54366730053257486592\) | \([2]\) | \(63700992\) | \(3.2598\) | |
| 404586.x2 | 404586x4 | \([1, -1, 0, -212106777, -1196312553955]\) | \(-11108001800138902875/79947274872976\) | \(-7595477316042381922573872\) | \([2]\) | \(127401984\) | \(3.6064\) |
Rank
sage: E.rank()
The elliptic curves in class 404586x have rank \(1\).
Complex multiplication
The elliptic curves in class 404586x do not have complex multiplication.Modular form 404586.2.a.x
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.
