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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 346560cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 346560.cf4 | 346560cf1 | \([0, -1, 0, 215675359, -2592351300159]\) | \(89962967236397039/287450726400000\) | \(-3545071004569556326809600000\) | \([2]\) | \(165888000\) | \(3.9695\) | \(\Gamma_0(N)\)-optimal |
| 346560.cf3 | 346560cf2 | \([0, -1, 0, -2031881761, -30371707791935]\) | \(75224183150104868881/11219310000000000\) | \(138365455083008163840000000000\) | \([2]\) | \(331776000\) | \(4.3160\) | |
| 346560.cf2 | 346560cf3 | \([0, -1, 0, -76277047841, -8108447418287679]\) | \(-3979640234041473454886161/1471455901872240\) | \(-18147164620384916011336335360\) | \([2]\) | \(829440000\) | \(4.7742\) | |
| 346560.cf1 | 346560cf4 | \([0, -1, 0, -1220432873761, -518941758497276735]\) | \(16300610738133468173382620881/2228489100\) | \(27483500185753249382400\) | \([2]\) | \(1658880000\) | \(5.1208\) |
Rank
sage: E.rank()
The elliptic curves in class 346560cf have rank \(1\).
Complex multiplication
The elliptic curves in class 346560cf do not have complex multiplication.Modular form 346560.2.a.cf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
