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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 163995y
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 163995.t6 | 163995y1 | \([1, 1, 0, -92527, 10794256]\) | \(147281603041/5265\) | \(3131744785065\) | \([2]\) | \(602112\) | \(1.4868\) | \(\Gamma_0(N)\)-optimal |
| 163995.t5 | 163995y2 | \([1, 1, 0, -96732, 9753939]\) | \(168288035761/27720225\) | \(16488636293367225\) | \([2, 2]\) | \(1204224\) | \(1.8333\) | |
| 163995.t7 | 163995y3 | \([1, 1, 0, 176593, 55180554]\) | \(1023887723039/2798036865\) | \(-1664337580319728665\) | \([2]\) | \(2408448\) | \(2.1799\) | |
| 163995.t4 | 163995y4 | \([1, 1, 0, -437337, -102168864]\) | \(15551989015681/1445900625\) | \(860055411598475625\) | \([2, 2]\) | \(2408448\) | \(2.1799\) | |
| 163995.t8 | 163995y5 | \([1, 1, 0, 508788, -482700339]\) | \(24487529386319/183539412225\) | \(-109173522714062499225\) | \([2]\) | \(4816896\) | \(2.5265\) | |
| 163995.t2 | 163995y6 | \([1, 1, 0, -6833142, -6877884681]\) | \(59319456301170001/594140625\) | \(353408699703515625\) | \([2, 2]\) | \(4816896\) | \(2.5265\) | |
| 163995.t3 | 163995y7 | \([1, 1, 0, -6669147, -7223487744]\) | \(-55150149867714721/5950927734375\) | \(-3539750597991943359375\) | \([2]\) | \(9633792\) | \(2.8731\) | |
| 163995.t1 | 163995y8 | \([1, 1, 0, -109330017, -440050177806]\) | \(242970740812818720001/24375\) | \(14498818449375\) | \([2]\) | \(9633792\) | \(2.8731\) |
Rank
sage: E.rank()
The elliptic curves in class 163995y have rank \(1\).
Complex multiplication
The elliptic curves in class 163995y do not have complex multiplication.Modular form 163995.2.a.y
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
