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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 222180.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 222180.r1 | 222180c3 | \([0, 1, 0, -159405, 22572528]\) | \(189123395584/16078125\) | \(38082232445250000\) | \([2]\) | \(2737152\) | \(1.9229\) | |
| 222180.r2 | 222180c1 | \([0, 1, 0, -32445, -2254500]\) | \(1594753024/4725\) | \(11191513208400\) | \([2]\) | \(912384\) | \(1.3736\) | \(\Gamma_0(N)\)-optimal |
| 222180.r3 | 222180c2 | \([0, 1, 0, -19220, -4095420]\) | \(-20720464/178605\) | \(-6768627188440320\) | \([2]\) | \(1824768\) | \(1.7202\) | |
| 222180.r4 | 222180c4 | \([0, 1, 0, 171220, 104303028]\) | \(14647977776/132355125\) | \(-5015886999828768000\) | \([2]\) | \(5474304\) | \(2.2695\) |
Rank
sage: E.rank()
The elliptic curves in class 222180.r have rank \(1\).
Complex multiplication
The elliptic curves in class 222180.r do not have complex multiplication.Modular form 222180.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
