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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 117975.n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 117975.n1 | 117975p8 | \([1, 1, 1, -393250063, 3001425766406]\) | \(242970740812818720001/24375\) | \(674715615234375\) | \([2]\) | \(11796480\) | \(3.1931\) | |
| 117975.n2 | 117975p6 | \([1, 1, 1, -24578188, 46889360156]\) | \(59319456301170001/594140625\) | \(16446193121337890625\) | \([2, 2]\) | \(5898240\) | \(2.8465\) | |
| 117975.n3 | 117975p7 | \([1, 1, 1, -23988313, 49247680406]\) | \(-55150149867714721/5950927734375\) | \(-164725492000579833984375\) | \([2]\) | \(11796480\) | \(3.1931\) | |
| 117975.n4 | 117975p4 | \([1, 1, 1, -1573063, 695069156]\) | \(15551989015681/1445900625\) | \(40023455580087890625\) | \([2, 2]\) | \(2949120\) | \(2.4999\) | |
| 117975.n5 | 117975p2 | \([1, 1, 1, -347938, -66958594]\) | \(168288035761/27720225\) | \(767313586269140625\) | \([2, 2]\) | \(1474560\) | \(2.1534\) | |
| 117975.n6 | 117975p1 | \([1, 1, 1, -332813, -74037094]\) | \(147281603041/5265\) | \(145738572890625\) | \([2]\) | \(737280\) | \(1.8068\) | \(\Gamma_0(N)\)-optimal |
| 117975.n7 | 117975p3 | \([1, 1, 1, 635187, -375659844]\) | \(1023887723039/2798036865\) | \(-77451452915566640625\) | \([2]\) | \(2949120\) | \(2.4999\) | |
| 117975.n8 | 117975p5 | \([1, 1, 1, 1830062, 3295056656]\) | \(24487529386319/183539412225\) | \(-5080488510323956640625\) | \([2]\) | \(5898240\) | \(2.8465\) |
Rank
sage: E.rank()
The elliptic curves in class 117975.n have rank \(1\).
Complex multiplication
The elliptic curves in class 117975.n do not have complex multiplication.Modular form 117975.2.a.n
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
