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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 258570.ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 258570.ef1 | 258570ef7 | \([1, -1, 1, -10024911009158, 12217124280120623777]\) | \(31664865542564944883878115208137569/103216295812500\) | \(363191696923764800812500\) | \([2]\) | \(4459069440\) | \(5.6060\) | |
| 258570.ef2 | 258570ef6 | \([1, -1, 1, -626556946658, 190892678863373777]\) | \(7730680381889320597382223137569/441370202660156250000\) | \(1553068646901730407972656250000\) | \([2, 2]\) | \(2229534720\) | \(5.2594\) | |
| 258570.ef3 | 258570ef8 | \([1, -1, 1, -625420790078, 191619465502646081]\) | \(-7688701694683937879808871873249/58423707246780395507812500\) | \(-205578055369099004030227661132812500\) | \([2]\) | \(4459069440\) | \(5.6060\) | |
| 258570.ef4 | 258570ef4 | \([1, -1, 1, -123768737618, 16757519007800081]\) | \(59589391972023341137821784609/8834417507562311995200\) | \(31086051486804055651845461947200\) | \([2]\) | \(1486356480\) | \(5.0567\) | |
| 258570.ef5 | 258570ef3 | \([1, -1, 1, -39230827538, 2971343928843281]\) | \(1897660325010178513043539489/14258428094958372000000\) | \(50171754900801887029717092000000\) | \([2]\) | \(1114767360\) | \(4.9128\) | |
| 258570.ef6 | 258570ef2 | \([1, -1, 1, -8452601618, 210391515070481]\) | \(18980483520595353274840609/5549773448629762560000\) | \(19528230697329431006911388160000\) | \([2, 2]\) | \(743178240\) | \(4.7101\) | |
| 258570.ef7 | 258570ef1 | \([1, -1, 1, -3188238098, -66720369131503]\) | \(1018563973439611524445729/42904970360310988800\) | \(150971596771234213859740876800\) | \([2]\) | \(371589120\) | \(4.3635\) | \(\Gamma_0(N)\)-optimal |
| 258570.ef8 | 258570ef5 | \([1, -1, 1, 22633718062, 1398361438905617]\) | \(364421318680576777174674911/450962301637624725000000\) | \(-1586820785333592083953090725000000\) | \([2]\) | \(1486356480\) | \(5.0567\) |
Rank
sage: E.rank()
The elliptic curves in class 258570.ef have rank \(1\).
Complex multiplication
The elliptic curves in class 258570.ef do not have complex multiplication.Modular form 258570.2.a.ef
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
