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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 291312.bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
291312.bp1 | 291312bp6 | \([0, 0, 0, -570913424931, -166036463004685534]\) | \(285531136548675601769470657/17941034271597192\) | \(1293089187801614706177862828032\) | \([2]\) | \(1698693120\) | \(5.2382\) | |
291312.bp2 | 291312bp4 | \([0, 0, 0, -35749975971, -2583952540697950]\) | \(70108386184777836280897/552468975892674624\) | \(39818867101414332709082979631104\) | \([2, 2]\) | \(849346560\) | \(4.8916\) | |
291312.bp3 | 291312bp5 | \([0, 0, 0, -12177008931, -5940634611545566]\) | \(-2770540998624539614657/209924951154647363208\) | \(-15130213814796461103815666178490368\) | \([2]\) | \(1698693120\) | \(5.2382\) | |
291312.bp4 | 291312bp2 | \([0, 0, 0, -3775570851, 22441092658850]\) | \(82582985847542515777/44772582831427584\) | \(3226956812680852421648805003264\) | \([2, 2]\) | \(424673280\) | \(4.5450\) | |
291312.bp5 | 291312bp1 | \([0, 0, 0, -2923275171, 60758772299426]\) | \(38331145780597164097/55468445663232\) | \(3997854653498821000531279872\) | \([2]\) | \(212336640\) | \(4.1985\) | \(\Gamma_0(N)\)-optimal |
291312.bp6 | 291312bp3 | \([0, 0, 0, 14562103389, 176503229018786]\) | \(4738217997934888496063/2928751705237796928\) | \(-211088006770826419766401742143488\) | \([2]\) | \(849346560\) | \(4.8916\) |
Rank
sage: E.rank()
The elliptic curves in class 291312.bp have rank \(0\).
Complex multiplication
The elliptic curves in class 291312.bp do not have complex multiplication.Modular form 291312.2.a.bp
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.