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SageMath
E = EllipticCurve("ez1")
E.isogeny_class()
Elliptic curves in class 258570ez
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 258570.ez6 | 258570ez1 | \([1, -1, 1, -121712, -20050221]\) | \(-56667352321/16711680\) | \(-58804119735828480\) | \([2]\) | \(2359296\) | \(1.9322\) | \(\Gamma_0(N)\)-optimal |
| 258570.ez5 | 258570ez2 | \([1, -1, 1, -2068592, -1144568109]\) | \(278202094583041/16646400\) | \(58574416143110400\) | \([2, 2]\) | \(4718592\) | \(2.2788\) | |
| 258570.ez4 | 258570ez3 | \([1, -1, 1, -2190272, -1002251181]\) | \(330240275458561/67652010000\) | \(238050088106609610000\) | \([2, 2]\) | \(9437184\) | \(2.6254\) | |
| 258570.ez2 | 258570ez4 | \([1, -1, 1, -33096992, -73279392429]\) | \(1139466686381936641/4080\) | \(14356474544880\) | \([2]\) | \(9437184\) | \(2.6254\) | |
| 258570.ez3 | 258570ez5 | \([1, -1, 1, -10981652, 13120221651]\) | \(41623544884956481/2962701562500\) | \(10424987638751826562500\) | \([2, 2]\) | \(18874368\) | \(2.9719\) | |
| 258570.ez7 | 258570ez6 | \([1, -1, 1, 4654228, -6017900781]\) | \(3168685387909439/6278181696900\) | \(-22091312676391268040900\) | \([2]\) | \(18874368\) | \(2.9719\) | |
| 258570.ez1 | 258570ez7 | \([1, -1, 1, -172587902, 872736186651]\) | \(161572377633716256481/914742821250\) | \(3218745595192975721250\) | \([2]\) | \(37748736\) | \(3.3185\) | |
| 258570.ez8 | 258570ez8 | \([1, -1, 1, 9962518, 57237021339]\) | \(31077313442863199/420227050781250\) | \(-1478671313139953613281250\) | \([2]\) | \(37748736\) | \(3.3185\) |
Rank
sage: E.rank()
The elliptic curves in class 258570ez have rank \(0\).
Complex multiplication
The elliptic curves in class 258570ez do not have complex multiplication.Modular form 258570.2.a.ez
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
