Torsion points of an elliptic curve (example in Silverman)
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Let $E$ be the elliptic curve $$y^2=x(x-2)(x-10)$$ Silverman claims (Example. X. 1.5 p.315 Arithmetic of Elliptic Curves) that $E(mathbb{Q})_{tors}$ injects into the reduction $widetilde{E}(mathbb{F}_3)$. I understand by VII.3.1 earlier in his book that the $m$-torsion for all $m$ prime to $3$ injects into $widetilde{E}(mathbb{F}_3)$ . So my question is about the 3-torsion. Why does $E$ have no $mathbb{Q}$ points that are 3-torsion (i.e. $[3]P=0$)?
number-theory elliptic-curves arithmetic-geometry
asked Nov 16 at 22:44
usr0192
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Let $E$ be the elliptic curve $$y^2=x(x-2)(x-10)$$ Silverman claims (Example. X. 1.5 p.315 Arithmetic of Elliptic Curves) that $E(mathbb{Q})_{tors}$ injects into the reduction $widetilde{E}(mathbb{F}_3)$. I understand by VII.3.1 earlier in his book that the $m$-torsion for all $m$ prime to $3$ injects into $widetilde{E}(mathbb{F}_3)$ . So my question is about the 3-torsion. Why does $E$ have no $mathbb{Q}$ points that are 3-torsion (i.e. $[3]P=0$)?
number-theory elliptic-curves arithmetic-geometry
asked Nov 16 at 22:44
usr0192
1,152411
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $E$ be the elliptic curve $$y^2=x(x-2)(x-10)$$ Silverman claims (Example. X. 1.5 p.315 Arithmetic of Elliptic Curves) that $E(mathbb{Q})_{tors}$ injects into the reduction $widetilde{E}(mathbb{F}_3)$. I understand by VII.3.1 earlier in his book that the $m$-torsion for all $m$ prime to $3$ injects into $widetilde{E}(mathbb{F}_3)$ . So my question is about the 3-torsion. Why does $E$ have no $mathbb{Q}$ points that are 3-torsion (i.e. $[3]P=0$)?
number-theory elliptic-curves arithmetic-geometry
asked Nov 16 at 22:44
usr0192
1,152411
Let $E$ be the elliptic curve $$y^2=x(x-2)(x-10)$$ Silverman claims (Example. X. 1.5 p.315 Arithmetic of Elliptic Curves) that $E(mathbb{Q})_{tors}$ injects into the reduction $widetilde{E}(mathbb{F}_3)$. I understand by VII.3.1 earlier in his book that the $m$-torsion for all $m$ prime to $3$ injects into $widetilde{E}(mathbb{F}_3)$ . So my question is about the 3-torsion. Why does $E$ have no $mathbb{Q}$ points that are 3-torsion (i.e. $[3]P=0$)?
number-theory elliptic-curves arithmetic-geometry
number-theory elliptic-curves arithmetic-geometry
asked Nov 16 at 22:44
usr0192
1,152411
asked Nov 16 at 22:44
usr0192
1,152411
asked Nov 16 at 22:44
usr0192
1,152411
asked Nov 16 at 22:44
usr0192
1,152411
asked Nov 16 at 22:44
usr0192
1,152411
1,152411
add a comment |
add a comment |
1 Answer
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In the geometric picture the $3$-torsion points are precisely the inflection points of the curve. That is to say, writing an equation for the elliptic curve as $y^2=f(x)$, the $3$-torsion points are the points on the curve with with $x$-coordinate satisfying $f'(x)=0$ and $f''(x)=0$.
In this case there is no inflection point because $f'(x)=3x^2-24x+20$ and $f''(x)=6x-24$ have no common zero, so there is no $3$-torsion.
answered Nov 16 at 23:03
Servaes
20.6k33789
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
In the geometric picture the $3$-torsion points are precisely the inflection points of the curve. That is to say, writing an equation for the elliptic curve as $y^2=f(x)$, the $3$-torsion points are the points on the curve with with $x$-coordinate satisfying $f'(x)=0$ and $f''(x)=0$.
In this case there is no inflection point because $f'(x)=3x^2-24x+20$ and $f''(x)=6x-24$ have no common zero, so there is no $3$-torsion.
answered Nov 16 at 23:03
Servaes
20.6k33789
add a comment |
up vote
1
down vote
accepted
In the geometric picture the $3$-torsion points are precisely the inflection points of the curve. That is to say, writing an equation for the elliptic curve as $y^2=f(x)$, the $3$-torsion points are the points on the curve with with $x$-coordinate satisfying $f'(x)=0$ and $f''(x)=0$.
In this case there is no inflection point because $f'(x)=3x^2-24x+20$ and $f''(x)=6x-24$ have no common zero, so there is no $3$-torsion.
answered Nov 16 at 23:03
Servaes
20.6k33789
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
In the geometric picture the $3$-torsion points are precisely the inflection points of the curve. That is to say, writing an equation for the elliptic curve as $y^2=f(x)$, the $3$-torsion points are the points on the curve with with $x$-coordinate satisfying $f'(x)=0$ and $f''(x)=0$.
In this case there is no inflection point because $f'(x)=3x^2-24x+20$ and $f''(x)=6x-24$ have no common zero, so there is no $3$-torsion.
answered Nov 16 at 23:03
Servaes
20.6k33789
In the geometric picture the $3$-torsion points are precisely the inflection points of the curve. That is to say, writing an equation for the elliptic curve as $y^2=f(x)$, the $3$-torsion points are the points on the curve with with $x$-coordinate satisfying $f'(x)=0$ and $f''(x)=0$.
In this case there is no inflection point because $f'(x)=3x^2-24x+20$ and $f''(x)=6x-24$ have no common zero, so there is no $3$-torsion.
answered Nov 16 at 23:03
Servaes
20.6k33789
edited Nov 16 at 23:11
answered Nov 16 at 23:03
Servaes
20.6k33789
answered Nov 16 at 23:03
Servaes
20.6k33789
answered Nov 16 at 23:03
Servaes
20.6k33789
20.6k33789
add a comment |
add a comment |
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