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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 99450co
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 99450.ce4 | 99450co1 | \([1, -1, 1, 355495, -13922503]\) | \(436192097814719/259683840000\) | \(-2957961240000000000\) | \([2]\) | \(2211840\) | \(2.2336\) | \(\Gamma_0(N)\)-optimal |
| 99450.ce3 | 99450co2 | \([1, -1, 1, -1444505, -111122503]\) | \(29263955267177281/16463793153600\) | \(187532893890225000000\) | \([2, 2]\) | \(4423680\) | \(2.5801\) | |
| 99450.ce2 | 99450co3 | \([1, -1, 1, -14449505, 21035007497]\) | \(29291056630578924481/175463302795560\) | \(1998636683405675625000\) | \([2]\) | \(8847360\) | \(2.9267\) | |
| 99450.ce1 | 99450co4 | \([1, -1, 1, -17239505, -27499652503]\) | \(49745123032831462081/97939634471640\) | \(1115593648903524375000\) | \([2]\) | \(8847360\) | \(2.9267\) |
Rank
sage: E.rank()
The elliptic curves in class 99450co have rank \(1\).
Complex multiplication
The elliptic curves in class 99450co do not have complex multiplication.Modular form 99450.2.a.co
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.
