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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 348726.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 348726.r1 | 348726r5 | \([1, 0, 1, -28567382, 58766768156]\) | \(54804145548726848737/637608031452\) | \(29996831572335049212\) | \([2]\) | \(28311552\) | \(2.8882\) | |
| 348726.r2 | 348726r4 | \([1, 0, 1, -6394762, -6224581684]\) | \(614716917569296417/19093020912\) | \(898247989756463472\) | \([2]\) | \(14155776\) | \(2.5416\) | |
| 348726.r3 | 348726r3 | \([1, 0, 1, -1831722, 868022860]\) | \(14447092394873377/1439452851984\) | \(67720327579549877904\) | \([2, 2]\) | \(14155776\) | \(2.5416\) | |
| 348726.r4 | 348726r2 | \([1, 0, 1, -416602, -88598260]\) | \(169967019783457/26337394944\) | \(1239065948385425664\) | \([2, 2]\) | \(7077888\) | \(2.1950\) | |
| 348726.r5 | 348726r1 | \([1, 0, 1, 45478, -7641844]\) | \(221115865823/664731648\) | \(-31272886008741888\) | \([2]\) | \(3538944\) | \(1.8485\) | \(\Gamma_0(N)\)-optimal |
| 348726.r6 | 348726r6 | \([1, 0, 1, 2262018, 4198689724]\) | \(27207619911317663/177609314617308\) | \(-8355786679977432708348\) | \([2]\) | \(28311552\) | \(2.8882\) |
Rank
sage: E.rank()
The elliptic curves in class 348726.r have rank \(2\).
Complex multiplication
The elliptic curves in class 348726.r do not have complex multiplication.Modular form 348726.2.a.r
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
