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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 259182cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 259182.cp4 | 259182cp1 | \([1, -1, 0, -399141, 12054037]\) | \(5445273626857/3103398144\) | \(4007939298030049536\) | \([2]\) | \(4423680\) | \(2.2594\) | \(\Gamma_0(N)\)-optimal |
| 259182.cp2 | 259182cp2 | \([1, -1, 0, -4668021, 3875390437]\) | \(8710408612492777/19986042384\) | \(25811333566012398096\) | \([2, 2]\) | \(8847360\) | \(2.6060\) | |
| 259182.cp1 | 259182cp3 | \([1, -1, 0, -74647161, 248256543145]\) | \(35618855581745079337/188166132\) | \(243010532328625908\) | \([2]\) | \(17694720\) | \(2.9525\) | |
| 259182.cp3 | 259182cp4 | \([1, -1, 0, -2990961, 6696540769]\) | \(-2291249615386537/13671036998388\) | \(-17655706386432006633972\) | \([2]\) | \(17694720\) | \(2.9525\) |
Rank
sage: E.rank()
The elliptic curves in class 259182cp have rank \(0\).
Complex multiplication
The elliptic curves in class 259182cp do not have complex multiplication.Modular form 259182.2.a.cp
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.
