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SageMath
E = EllipticCurve("cx1")
E.isogeny_class()
Elliptic curves in class 61710cx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 61710.ct3 | 61710cx1 | \([1, 0, 0, -5311600, -7975015168]\) | \(-9354997870579612441/10093752054144000\) | \(-17881697482791398784000\) | \([2]\) | \(5529600\) | \(2.9647\) | \(\Gamma_0(N)\)-optimal |
| 61710.ct2 | 61710cx2 | \([1, 0, 0, -100427280, -387239277600]\) | \(63229930193881628103961/26218934428500000\) | \(46448441695087888500000\) | \([2]\) | \(11059200\) | \(3.3113\) | |
| 61710.ct4 | 61710cx3 | \([1, 0, 0, 44519225, 144030507017]\) | \(5508208700580085578359/8246033269590589440\) | \(-14608350945109174218915840\) | \([2]\) | \(16588800\) | \(3.5140\) | |
| 61710.ct1 | 61710cx4 | \([1, 0, 0, -292499655, 1447147708425]\) | \(1562225332123379392365961/393363080510106009600\) | \(696866692271563912472985600\) | \([2]\) | \(33177600\) | \(3.8606\) |
Rank
sage: E.rank()
The elliptic curves in class 61710cx have rank \(1\).
Complex multiplication
The elliptic curves in class 61710cx do not have complex multiplication.Modular form 61710.2.a.cx
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
