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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 121968cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121968.bx3 | 121968cf1 | \([0, 0, 0, -639212871, 6220372323814]\) | \(87364831012240243408/1760913\) | \(582185660338098432\) | \([2]\) | \(22118400\) | \(3.3922\) | \(\Gamma_0(N)\)-optimal |
| 121968.bx2 | 121968cf2 | \([0, 0, 0, -639234651, 6219927232090]\) | \(21843440425782779332/3100814593569\) | \(4100713190811767696753664\) | \([2, 2]\) | \(44236800\) | \(3.7388\) | |
| 121968.bx4 | 121968cf3 | \([0, 0, 0, -581604771, 7386989931970]\) | \(-8226100326647904626/4152140742401883\) | \(-10982106668220235816520964096\) | \([2]\) | \(88473600\) | \(4.0854\) | |
| 121968.bx1 | 121968cf4 | \([0, 0, 0, -697213011, 5024378661874]\) | \(14171198121996897746/4077720290568771\) | \(10785270050428806222225610752\) | \([2]\) | \(88473600\) | \(4.0854\) |
Rank
sage: E.rank()
The elliptic curves in class 121968cf have rank \(0\).
Complex multiplication
The elliptic curves in class 121968cf do not have complex multiplication.Modular form 121968.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
