We consider the elliptic curve 446d1 of rank at the prime .
sage: E = EllipticCurve('446d1'); p = 5; E Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 4*x + 4 over Rational Field
Next we verify that the rank is , that is a good ordinary prime, and that there are points on modulo (so is ananomolous at , i.e., ).
sage: E.rank() 2 sage: E.is_ordinary(p) True sage: E.Np(p) 10
Next we compute the -adic -series of at . We add so that the displayed series doesn't take several lines.
sage: Lp = E.padic_lseries(p) sage: LpT = Lp.series(4) sage: LpT = LpT.add_bigoh(7); LpT (5 + 5^2 + O(5^3))*T^2 + (2*5 + 3*5^2 + O(5^3))*T^3 + (4*5^2 + O(5^3))*T^4 + (4*5 + O(5^2))*T^5 + (1 + 2*5 + O(5^3))*T^6 + O(T^7)
We compute the -adic modular form evaluated on our elliptic curve with differential to precision . This is the key difficult input to the computation of the -adic regulator .
sage: E.padic_E2(p, prec=8) 3*5 + 4*5^2 + 5^3 + 5^4 + 5^5 + 2*5^6 + 4*5^7 + O(5^8)
We compute the normalized -adic regulator, normalized to the choice of as a topological generator of .
sage: Regp = E.padic_regulator(p, 10) sage: R = Regp.parent() sage: kg = log(R(1+p)) sage: reg = Regp * p^2 / log(R(1+p))^2 sage: reg*kg^2 2*5 + 2*5^2 + 5^4 + 4*5^5 + 2*5^7 + O(5^8)
We compute the Tamagawa numbers and torsion subgroup.
sage: E.tamagawa_numbers() [2, 1] sage: E.torsion_order() 1
We compute , which is the leading term of the -adic -function. It is not a unit, so we call the prime an irregular prime.
sage: Lpstar = LpT[2]; Lpstar 5 + 5^2 + O(5^3)
Finally, putting everything together we compute the conjectural -adic order of . In particular, we see that conjecturally is trivial.
sage: eps = (1-1/Lp.alpha(20))^2 sage: Lpstar / (eps*reg*(2*1)) * (1)^2 1 + O(5^2)
William 2007-05-25