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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 41070bl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
41070.bh8 | 41070bl1 | \([1, 0, 0, 2025, 107865]\) | \(357911/2160\) | \(-5541969043440\) | \([2]\) | \(103680\) | \(1.1272\) | \(\Gamma_0(N)\)-optimal |
41070.bh6 | 41070bl2 | \([1, 0, 0, -25355, 1405677]\) | \(702595369/72900\) | \(187041455216100\) | \([2, 2]\) | \(207360\) | \(1.4737\) | |
41070.bh7 | 41070bl3 | \([1, 0, 0, -18510, -3173628]\) | \(-273359449/1536000\) | \(-3940955764224000\) | \([2]\) | \(311040\) | \(1.6765\) | |
41070.bh5 | 41070bl4 | \([1, 0, 0, -93805, -9532633]\) | \(35578826569/5314410\) | \(13635322085253690\) | \([2]\) | \(414720\) | \(1.8203\) | |
41070.bh4 | 41070bl5 | \([1, 0, 0, -394985, 95513475]\) | \(2656166199049/33750\) | \(86593266303750\) | \([2]\) | \(414720\) | \(1.8203\) | |
41070.bh3 | 41070bl6 | \([1, 0, 0, -456590, -118563900]\) | \(4102915888729/9000000\) | \(23091537681000000\) | \([2, 2]\) | \(622080\) | \(2.0231\) | |
41070.bh1 | 41070bl7 | \([1, 0, 0, -7301590, -7594672900]\) | \(16778985534208729/81000\) | \(207823839129000\) | \([2]\) | \(1244160\) | \(2.3696\) | |
41070.bh2 | 41070bl8 | \([1, 0, 0, -620870, -25679988]\) | \(10316097499609/5859375000\) | \(15033553177734375000\) | \([2]\) | \(1244160\) | \(2.3696\) |
Rank
sage: E.rank()
The elliptic curves in class 41070bl have rank \(0\).
Complex multiplication
The elliptic curves in class 41070bl do not have complex multiplication.Modular form 41070.2.a.bl
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.