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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 422370ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 422370.ck3 | 422370ck1 | \([1, -1, 0, -43929, 3448413]\) | \(273359449/9360\) | \(321014746250640\) | \([2]\) | \(1843200\) | \(1.5554\) | \(\Gamma_0(N)\)-optimal |
| 422370.ck2 | 422370ck2 | \([1, -1, 0, -108909, -9066735]\) | \(4165509529/1368900\) | \(46948406639156100\) | \([2, 2]\) | \(3686400\) | \(1.9020\) | |
| 422370.ck4 | 422370ck3 | \([1, -1, 0, 313461, -62538777]\) | \(99317171591/106616250\) | \(-3656558594011196250\) | \([2]\) | \(7372800\) | \(2.2486\) | |
| 422370.ck1 | 422370ck4 | \([1, -1, 0, -1570959, -757343925]\) | \(12501706118329/2570490\) | \(88158674689082010\) | \([2]\) | \(7372800\) | \(2.2486\) |
Rank
sage: E.rank()
The elliptic curves in class 422370ck have rank \(1\).
Complex multiplication
The elliptic curves in class 422370ck do not have complex multiplication.Modular form 422370.2.a.ck
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.
