Example: A Curve of Rank $2$

We only consider primes $p$ of good ordinary reduction for a given curve $E$ in this section. If $E$ is an elliptic curve with analytic rank $0$, then the $p$-adic and classical BSD conjecture are the same, so there is nothing new to illustrate. We will thus consider only curves of rank $\geq 1$ in this section.

We consider the elliptic curve 446d1 of rank $2$ at the prime $p=5$.

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 $2$, that $p$ is a good ordinary prime, and that there are $10$ points on $E$ modulo $p$ (so $E$ is ananomolous at $p$, i.e., $p\mid \char93 E(\mathbb{F}_p)$).

sage: E.rank()
2
sage: E.is_ordinary(p)
True
sage: E.Np(p)
10

Next we compute the $p$-adic $L$-series of $E$ at $p$. We add $O(T^7)$ 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 $p$-adic modular form $E_2$ evaluated on our elliptic curve with differential $\omega$ to precision $O(p^{8})$. This is the key difficult input to the computation of the $p$-adic regulator $\Reg _p(E)$.

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 $p$-adic regulator, normalized to the choice of $1+p$ as a topological generator of $1+p \mathbb{Z}_p$.

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 $\mathcal{L}_p^*(E,0)$, which is the leading term of the $p$-adic $L$-function. It is not a unit, so we call the prime $p$ an irregular prime.

sage: Lpstar = LpT[2]; Lpstar
5 + 5^2 + O(5^3)

Finally, putting everything together we compute the conjectural $p$-adic order of $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$. In particular, we see that conjecturally $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})(5)$ is trivial.

sage: eps = (1-1/Lp.alpha(20))^2
sage: Lpstar / (eps*reg*(2*1)) * (1)^2
1 + O(5^2)