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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 388416x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388416.x4 | 388416x1 | \([0, -1, 0, -816880849, -9069630529871]\) | \(-152435594466395827792/1646846627220711\) | \(-651278321204546600821702656\) | \([2]\) | \(159252480\) | \(3.9610\) | \(\Gamma_0(N)\)-optimal |
| 388416.x3 | 388416x2 | \([0, -1, 0, -13103796769, -577351778745791]\) | \(157304700372188331121828/18069292138401\) | \(28583446824341450730307584\) | \([2, 2]\) | \(318504960\) | \(4.3076\) | |
| 388416.x2 | 388416x3 | \([0, -1, 0, -13137505729, -574231994272415]\) | \(79260902459030376659234/842751810121431609\) | \(2666263998192805981849979584512\) | \([2]\) | \(637009920\) | \(4.6542\) | |
| 388416.x1 | 388416x4 | \([0, -1, 0, -209660742529, -36950725639692767]\) | \(322159999717985454060440834/4250799\) | \(13448505480659730432\) | \([2]\) | \(637009920\) | \(4.6542\) |
Rank
sage: E.rank()
The elliptic curves in class 388416x have rank \(0\).
Complex multiplication
The elliptic curves in class 388416x do not have complex multiplication.Modular form 388416.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
