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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 16170.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16170.b1 | 16170e7 | \([1, 1, 0, -12564948, -16089573912]\) | \(1864737106103260904761/129177711985836360\) | \(15197628637421661917640\) | \([2]\) | \(1327104\) | \(3.0042\) | |
16170.b2 | 16170e4 | \([1, 1, 0, -12348123, -16706417517]\) | \(1769857772964702379561/691787250\) | \(81388078175250\) | \([2]\) | \(442368\) | \(2.4549\) | |
16170.b3 | 16170e6 | \([1, 1, 0, -2480748, 1200795408]\) | \(14351050585434661561/3001282273281600\) | \(353097858169306958400\) | \([2, 2]\) | \(663552\) | \(2.6577\) | |
16170.b4 | 16170e3 | \([1, 1, 0, -2339628, 1376376912]\) | \(12038605770121350841/757333463040\) | \(89099524593192960\) | \([2]\) | \(331776\) | \(2.3111\) | |
16170.b5 | 16170e2 | \([1, 1, 0, -771873, -261196767]\) | \(432288716775559561/270140062500\) | \(31781708213062500\) | \([2, 2]\) | \(221184\) | \(2.1084\) | |
16170.b6 | 16170e5 | \([1, 1, 0, -626343, -362514753]\) | \(-230979395175477481/348191894531250\) | \(-40964428199707031250\) | \([2]\) | \(442368\) | \(2.4549\) | |
16170.b7 | 16170e1 | \([1, 1, 0, -57453, -2433843]\) | \(178272935636041/81841914000\) | \(9628619340186000\) | \([2]\) | \(110592\) | \(1.7618\) | \(\Gamma_0(N)\)-optimal |
16170.b8 | 16170e8 | \([1, 1, 0, 5345532, 7256770872]\) | \(143584693754978072519/276341298967965000\) | \(-32511277482282114285000\) | \([2]\) | \(1327104\) | \(3.0042\) |
Rank
sage: E.rank()
The elliptic curves in class 16170.b have rank \(0\).
Complex multiplication
The elliptic curves in class 16170.b do not have complex multiplication.Modular form 16170.2.a.b
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.