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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 221778v
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221778.h3 | 221778v1 | \([1, -1, 0, -6417, 208965]\) | \(-140625/8\) | \(-1662590713032\) | \([]\) | \(308448\) | \(1.1019\) | \(\Gamma_0(N)\)-optimal |
| 221778.h4 | 221778v2 | \([1, -1, 0, 34653, 375983]\) | \(3375/2\) | \(-2727064417050738\) | \([]\) | \(925344\) | \(1.6512\) | |
| 221778.h2 | 221778v3 | \([1, -1, 0, -129627, -36458331]\) | \(-1159088625/2097152\) | \(-435838179877060608\) | \([]\) | \(2159136\) | \(2.0748\) | |
| 221778.h1 | 221778v4 | \([1, -1, 0, -13272027, -18607019995]\) | \(-189613868625/128\) | \(-174532122691247232\) | \([]\) | \(6477408\) | \(2.6241\) |
Rank
sage: E.rank()
The elliptic curves in class 221778v have rank \(1\).
Complex multiplication
The elliptic curves in class 221778v do not have complex multiplication.Modular form 221778.2.a.v
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.
