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SageMath
E = EllipticCurve("mt1")
E.isogeny_class()
Elliptic curves in class 286650.mt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286650.mt1 | 286650mt8 | \([1, -1, 1, -443099766605, -113527211776349353]\) | \(7179471593960193209684686321/49441793310\) | \(66256731679413289218750\) | \([2]\) | \(1019215872\) | \(4.9148\) | |
| 286650.mt2 | 286650mt6 | \([1, -1, 1, -27693752855, -1773855145296853]\) | \(1752803993935029634719121/4599740941532100\) | \(6164092783751500218501562500\) | \([2, 2]\) | \(509607936\) | \(4.5682\) | |
| 286650.mt3 | 286650mt7 | \([1, -1, 1, -27353411105, -1819578698042353]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-120476589429246821421142324218750\) | \([2]\) | \(1019215872\) | \(4.9148\) | |
| 286650.mt4 | 286650mt5 | \([1, -1, 1, -5472865355, -155579505021853]\) | \(13527956825588849127121/25701087819771000\) | \(34441915746595402785796875000\) | \([2]\) | \(339738624\) | \(4.3655\) | |
| 286650.mt5 | 286650mt3 | \([1, -1, 1, -1752148355, -26999381475853]\) | \(443915739051786565201/21894701746029840\) | \(29340994362639164485166250000\) | \([2]\) | \(254803968\) | \(4.2216\) | |
| 286650.mt6 | 286650mt2 | \([1, -1, 1, -456490355, -663812271853]\) | \(7850236389974007121/4400862921000000\) | \(5897577215420303765625000000\) | \([2, 2]\) | \(169869312\) | \(4.0189\) | |
| 286650.mt7 | 286650mt1 | \([1, -1, 1, -283618355, 1828656224147]\) | \(1882742462388824401/11650189824000\) | \(15612368595596136000000000\) | \([2]\) | \(84934656\) | \(3.6723\) | \(\Gamma_0(N)\)-optimal |
| 286650.mt8 | 286650mt4 | \([1, -1, 1, 1793932645, -5268177729853]\) | \(476437916651992691759/284661685546875000\) | \(-381473883854331756591796875000\) | \([2]\) | \(339738624\) | \(4.3655\) |
Rank
sage: E.rank()
The elliptic curves in class 286650.mt have rank \(1\).
Complex multiplication
The elliptic curves in class 286650.mt do not have complex multiplication.Modular form 286650.2.a.mt
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.
