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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 221760r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.nw4 | 221760r1 | \([0, 0, 0, 110868, -2900144]\) | \(788632918919/475398000\) | \(-90849972584448000\) | \([2]\) | \(2359296\) | \(1.9433\) | \(\Gamma_0(N)\)-optimal |
| 221760.nw3 | 221760r2 | \([0, 0, 0, -453612, -23447216]\) | \(54014438633401/30015562500\) | \(5736063320064000000\) | \([2, 2]\) | \(4718592\) | \(2.2898\) | |
| 221760.nw2 | 221760r3 | \([0, 0, 0, -4445292, 3586628176]\) | \(50834334659676121/338378906250\) | \(64665216000000000000\) | \([2]\) | \(9437184\) | \(2.6364\) | |
| 221760.nw1 | 221760r4 | \([0, 0, 0, -5493612, -4948535216]\) | \(95946737295893401/168104301750\) | \(32125232342827008000\) | \([2]\) | \(9437184\) | \(2.6364\) |
Rank
sage: E.rank()
The elliptic curves in class 221760r have rank \(0\).
Complex multiplication
The elliptic curves in class 221760r do not have complex multiplication.Modular form 221760.2.a.r
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.
