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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 205350.x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 205350.x1 | 205350ei8 | \([1, 1, 0, -182539750, -949334112500]\) | \(16778985534208729/81000\) | \(3247247486390625000\) | \([2]\) | \(29859840\) | \(3.1743\) | |
| 205350.x2 | 205350ei7 | \([1, 1, 0, -15521750, -3209998500]\) | \(10316097499609/5859375000\) | \(234899268402099609375000\) | \([2]\) | \(29859840\) | \(3.1743\) | |
| 205350.x3 | 205350ei6 | \([1, 1, 0, -11414750, -14820487500]\) | \(4102915888729/9000000\) | \(360805276265625000000\) | \([2, 2]\) | \(14929920\) | \(2.8278\) | |
| 205350.x4 | 205350ei4 | \([1, 1, 0, -9874625, 11939184375]\) | \(2656166199049/33750\) | \(1353019785996093750\) | \([2]\) | \(9953280\) | \(2.6250\) | |
| 205350.x5 | 205350ei5 | \([1, 1, 0, -2345125, -1191579125]\) | \(35578826569/5314410\) | \(213051907582088906250\) | \([2]\) | \(9953280\) | \(2.6250\) | |
| 205350.x6 | 205350ei2 | \([1, 1, 0, -633875, 175709625]\) | \(702595369/72900\) | \(2922522737751562500\) | \([2, 2]\) | \(4976640\) | \(2.2785\) | |
| 205350.x7 | 205350ei3 | \([1, 1, 0, -462750, -396703500]\) | \(-273359449/1536000\) | \(-61577433816000000000\) | \([2]\) | \(7464960\) | \(2.4812\) | |
| 205350.x8 | 205350ei1 | \([1, 1, 0, 50625, 13483125]\) | \(357911/2160\) | \(-86593266303750000\) | \([2]\) | \(2488320\) | \(1.9319\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 205350.x have rank \(1\).
Complex multiplication
The elliptic curves in class 205350.x do not have complex multiplication.Modular form 205350.2.a.x
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
