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Finding out $p$-torsion elements of an elliptic curve $E$ over $\mathbb{Q}_p$

asked 2016-05-26 05:30:16 -0600

Suman gravatar image

updated 2017-01-08 05:04:30 -0600

FrédéricC gravatar image

Let $E$ be an elliptic curve over $\mathbb{Q}$. Then how to compute the $p$-torsion elements of $E$ over the $p$-adic field $\mathbb{Q}_p$ using SAGE ? At least can we say whether $E(\mathbb{Q}_p)[p]=0$ or not ?

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answered 2016-05-26 15:00:23 -0600

nbruin gravatar image

updated 2016-05-30 00:24:06 -0600

You can probably adapt this code:

sage: E=EllipticCurve([1,2,3,4,5])
sage: K=Qp(7)
sage: KX=K['X']
sage: f,h=E.hyperelliptic_polynomials()
sage: n=3
sage: xvals=E.division_polynomial(n).roots(K,multiplicities=false)
sage: [(x1,y1) for x1 in xvals for y1 in KX([-f(x1),h(x1),1]).roots(multiplicities=false)]
[(3*7 + 5*7^2 + 5*7^3 + 5*7^4 + 4*7^5 + 4*7^6 + 2*7^7 + 5*7^8 + 2*7^9 + 6*7^10 + 2*7^12 + 5*7^13 + 3*7^14 + 7^15 + 6*7^17 + 2*7^18 + 4*7^19 + O(7^20),
  6 + 7 + 2*7^3 + 2*7^4 + 4*7^5 + 5*7^7 + 3*7^8 + 2*7^9 + 3*7^10 + 4*7^11 + 5*7^12 + 3*7^13 + 2*7^14 + 3*7^15 + 6*7^16 + 4*7^17 + 6*7^18 + 4*7^19 + O(7^20)),
 (3*7 + 5*7^2 + 5*7^3 + 5*7^4 + 4*7^5 + 4*7^6 + 2*7^7 + 5*7^8 + 2*7^9 + 6*7^10 + 2*7^12 + 5*7^13 + 3*7^14 + 7^15 + 6*7^17 + 2*7^18 + 4*7^19 + O(7^20),
  5 + 7 + 7^2 + 6*7^3 + 5*7^4 + 4*7^5 + 7^6 + 6*7^7 + 4*7^8 + 7^9 + 4*7^10 + 7^11 + 6*7^12 + 4*7^13 + 2*7^15 + 3*7^17 + 4*7^18 + 4*7^19 + O(7^20))]
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Asked: 2016-05-26 05:30:16 -0600

Seen: 58 times

Last updated: May 30 '16