Show commands:
SageMath
E = EllipticCurve("ge1")
E.isogeny_class()
Elliptic curves in class 221760ge
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.is2 | 221760ge1 | \([0, 0, 0, -26592, -1669064]\) | \(75216478666752/326095\) | \(9015874560\) | \([2]\) | \(331776\) | \(1.1170\) | \(\Gamma_0(N)\)-optimal |
| 221760.is3 | 221760ge2 | \([0, 0, 0, -26172, -1724336]\) | \(-4481782160112/310023175\) | \(-137144331878400\) | \([2]\) | \(663552\) | \(1.4636\) | |
| 221760.is1 | 221760ge3 | \([0, 0, 0, -37152, -222696]\) | \(281370820608/161767375\) | \(3260484855936000\) | \([2]\) | \(995328\) | \(1.6663\) | |
| 221760.is4 | 221760ge4 | \([0, 0, 0, 148068, -1778544]\) | \(1113258734352/648484375\) | \(-209127308544000000\) | \([2]\) | \(1990656\) | \(2.0129\) |
Rank
sage: E.rank()
The elliptic curves in class 221760ge have rank \(0\).
Complex multiplication
The elliptic curves in class 221760ge do not have complex multiplication.Modular form 221760.2.a.ge
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.
