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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 3150bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3150.bp7 | 3150bk1 | \([1, -1, 1, 47245, -2990253]\) | \(1023887723039/928972800\) | \(-10581580800000000\) | \([2]\) | \(24576\) | \(1.7622\) | \(\Gamma_0(N)\)-optimal |
| 3150.bp6 | 3150bk2 | \([1, -1, 1, -240755, -26606253]\) | \(135487869158881/51438240000\) | \(585913702500000000\) | \([2, 2]\) | \(49152\) | \(2.1087\) | |
| 3150.bp4 | 3150bk3 | \([1, -1, 1, -3390755, -2401706253]\) | \(378499465220294881/120530818800\) | \(1372921357893750000\) | \([2]\) | \(98304\) | \(2.4553\) | |
| 3150.bp5 | 3150bk4 | \([1, -1, 1, -1698755, 833613747]\) | \(47595748626367201/1215506250000\) | \(13845375878906250000\) | \([2, 2]\) | \(98304\) | \(2.4553\) | |
| 3150.bp2 | 3150bk5 | \([1, -1, 1, -27011255, 54040488747]\) | \(191342053882402567201/129708022500\) | \(1477455443789062500\) | \([2, 2]\) | \(196608\) | \(2.8019\) | |
| 3150.bp8 | 3150bk6 | \([1, -1, 1, 285745, 2663322747]\) | \(226523624554079/269165039062500\) | \(-3065958023071289062500\) | \([2]\) | \(196608\) | \(2.8019\) | |
| 3150.bp1 | 3150bk7 | \([1, -1, 1, -432180005, 3458268326247]\) | \(783736670177727068275201/360150\) | \(4102333593750\) | \([2]\) | \(393216\) | \(3.1485\) | |
| 3150.bp3 | 3150bk8 | \([1, -1, 1, -26842505, 54748901247]\) | \(-187778242790732059201/4984939585440150\) | \(-56781577465404208593750\) | \([2]\) | \(393216\) | \(3.1485\) |
Rank
sage: E.rank()
The elliptic curves in class 3150bk have rank \(0\).
Complex multiplication
The elliptic curves in class 3150bk do not have complex multiplication.Modular form 3150.2.a.bk
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
