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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 287490bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 287490.bc7 | 287490bc1 | \([1, 0, 1, 287461, 44725286]\) | \(1023887723039/928972800\) | \(-2383490046202675200\) | \([2]\) | \(6488064\) | \(2.2136\) | \(\Gamma_0(N)\)-optimal |
| 287490.bc6 | 287490bc2 | \([1, 0, 1, -1464859, 400095782]\) | \(135487869158881/51438240000\) | \(131976450800480160000\) | \([2, 2]\) | \(12976128\) | \(2.5602\) | |
| 287490.bc4 | 287490bc3 | \([1, 0, 1, -20630859, 36056522182]\) | \(378499465220294881/120530818800\) | \(309249104893553689200\) | \([2]\) | \(25952256\) | \(2.9067\) | |
| 287490.bc5 | 287490bc4 | \([1, 0, 1, -10335979, -12505609594]\) | \(47595748626367201/1215506250000\) | \(3118656485929556250000\) | \([2, 2]\) | \(25952256\) | \(2.9067\) | |
| 287490.bc8 | 287490bc5 | \([1, 0, 1, 1738601, -39972864178]\) | \(226523624554079/269165039062500\) | \(-690603849102172851562500\) | \([2]\) | \(51904512\) | \(3.2533\) | |
| 287490.bc2 | 287490bc6 | \([1, 0, 1, -164348479, -810968014594]\) | \(191342053882402567201/129708022500\) | \(332795298787416202500\) | \([2, 2]\) | \(51904512\) | \(3.2533\) | |
| 287490.bc3 | 287490bc7 | \([1, 0, 1, -163321729, -821600626894]\) | \(-187778242790732059201/4984939585440150\) | \(-12789991141633304743921350\) | \([2]\) | \(103809024\) | \(3.5999\) | |
| 287490.bc1 | 287490bc8 | \([1, 0, 1, -2629575229, -51901313272294]\) | \(783736670177727068275201/360150\) | \(924046366201350\) | \([2]\) | \(103809024\) | \(3.5999\) |
Rank
sage: E.rank()
The elliptic curves in class 287490bc have rank \(2\).
Complex multiplication
The elliptic curves in class 287490bc do not have complex multiplication.Modular form 287490.2.a.bc
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
