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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 184041bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 184041.bn5 | 184041bm1 | \([1, -1, 0, -4420818, 5133179871]\) | \(-1532808577/938223\) | \(-5848571832825267643383\) | \([2]\) | \(10321920\) | \(2.8787\) | \(\Gamma_0(N)\)-optimal |
| 184041.bn4 | 184041bm2 | \([1, -1, 0, -78957423, 270021366720]\) | \(8732907467857/1656369\) | \(10325256445605102135849\) | \([2, 2]\) | \(20643840\) | \(3.2253\) | |
| 184041.bn1 | 184041bm3 | \([1, -1, 0, -1263261258, 17282072234961]\) | \(35765103905346817/1287\) | \(8022732280967445327\) | \([2]\) | \(41287680\) | \(3.5718\) | |
| 184041.bn3 | 184041bm4 | \([1, -1, 0, -87239268, 209920017555]\) | \(11779205551777/3763454409\) | \(23460129895130970397330689\) | \([2, 2]\) | \(41287680\) | \(3.5718\) | |
| 184041.bn6 | 184041bm5 | \([1, -1, 0, 246795147, 1430281349316]\) | \(266679605718863/296110251723\) | \(-1845853360702026571675376883\) | \([2]\) | \(82575360\) | \(3.9184\) | |
| 184041.bn2 | 184041bm6 | \([1, -1, 0, -553783203, -4857027042906]\) | \(3013001140430737/108679952667\) | \(677474875334541494255066307\) | \([2]\) | \(82575360\) | \(3.9184\) |
Rank
sage: E.rank()
The elliptic curves in class 184041bm have rank \(1\).
Complex multiplication
The elliptic curves in class 184041bm do not have complex multiplication.Modular form 184041.2.a.bm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
