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SageMath
E = EllipticCurve("ct1")
E.isogeny_class()
Elliptic curves in class 7350.ct
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7350.ct1 | 7350cj8 | \([1, 0, 0, -6533563, 6427416617]\) | \(16778985534208729/81000\) | \(148899515625000\) | \([2]\) | \(165888\) | \(2.3418\) | |
| 7350.ct2 | 7350cj7 | \([1, 0, 0, -555563, 21646617]\) | \(10316097499609/5859375000\) | \(10771087646484375000\) | \([2]\) | \(165888\) | \(2.3418\) | |
| 7350.ct3 | 7350cj6 | \([1, 0, 0, -408563, 100291617]\) | \(4102915888729/9000000\) | \(16544390625000000\) | \([2, 2]\) | \(82944\) | \(1.9953\) | |
| 7350.ct4 | 7350cj4 | \([1, 0, 0, -353438, -80904258]\) | \(2656166199049/33750\) | \(62041464843750\) | \([2]\) | \(55296\) | \(1.7925\) | |
| 7350.ct5 | 7350cj5 | \([1, 0, 0, -83938, 8055242]\) | \(35578826569/5314410\) | \(9769297220156250\) | \([2]\) | \(55296\) | \(1.7925\) | |
| 7350.ct6 | 7350cj2 | \([1, 0, 0, -22688, -1193508]\) | \(702595369/72900\) | \(134009564062500\) | \([2, 2]\) | \(27648\) | \(1.4460\) | |
| 7350.ct7 | 7350cj3 | \([1, 0, 0, -16563, 2683617]\) | \(-273359449/1536000\) | \(-2823576000000000\) | \([2]\) | \(41472\) | \(1.6487\) | |
| 7350.ct8 | 7350cj1 | \([1, 0, 0, 1812, -91008]\) | \(357911/2160\) | \(-3970653750000\) | \([2]\) | \(13824\) | \(1.0994\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 7350.ct have rank \(0\).
Complex multiplication
The elliptic curves in class 7350.ct do not have complex multiplication.Modular form 7350.2.a.ct
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.
