I am a little confused on the $p-$ adic regulator on Elliptic Curves and what happens when you switch to different Weierstrass models. Restrict to ell. curves over $\mathbb Q$ for simplicity.

From my understanding, to define the p-adic height (and more precisely the $p-$adic sigma function), you need an integral Weierstrass model minimal at $p$. As long as that condition is cleared, the $p-$adic regulator should be independent of the model.

However, when I try this in practice, I get different results. For example, (and this is the case with most examples), take the rank 1 curves on Sage:

E=EllipticCurve('143a1')
E.is_p_minimal(7)
E1=E.short_weierstrass_model()
E1.is_p_minimal(7)

It's easy to check, they're both minimal at $p=7$ and that $p=7$ is a good ordinary non-anomalous prime. (The model of $E$ is automatically The minimal Weierstrass model).

However, their regulators are completely different.

E.padic_regulator(7)
2*7^2 + 5*7^3 + 7^4 + 5*7^5 + 3*7^6 + 5*7^7 + 7^8 + 5*7^9 + O(7^10)
E1.padic_regulator(7)
3*7 + 3*7^2 + 4*7^3 + 3*7^4 + 5*7^6 + 7^8 + 3*7^9 + 5*7^11 + 7^12 +O(7^13)

Not only are they different, but their $p-$valuation as you can see is also different, which, without going through details, would imply in the first case that the Iwasawa $\lambda$ invariant would be greater than 1, whereas in the second case it would imply that $\lambda_p(E)= 1$, which is absurd since $\lambda_p(E)$ does not depend on the model. The correct regulator is the first one, of The minimal curve, since in this example one can check $\lambda_p(E) >1$.

At first I was convinced it was an implementation error by Sage, so I posted pretty much the same question on Ask Sage with the assumption that it was a bug, but for various reasons I could go through, now I have started to doubt that it was due to implementation errors, which is why I also decided to post the question here.

So in short, am I misunderstanding something? Shouldn't both the regulators be the same, since both models are minimal at $p$?

I am a little confused on the $p$-adic regulator on elliptic curves and what happens when you switch to different Weierstrass models. Restrict to ell. curves over $\mathbb Q$ for simplicity.

From my understanding, to define the p-adic height (and more precisely the $p$-adic sigma function), you need an integral Weierstrass model minimal at $p$. As long as that condition is cleared, the $p$-adic regulator should be independent of the model.

However, when I try this in practice, I get different results. For example, (and this is the case with most examples), take the rank 1 curves on Sage:

E=EllipticCurve('143a1')
E.is_p_minimal(7)
E1=E.short_weierstrass_model()
E1.is_p_minimal(7)

It's easy to check, they're both minimal at $p=7$ and that $p=7$ is a good ordinary non-anomalous prime. (The model of $E$ is automatically The minimal Weierstrass model).

However, their regulators are completely different.

E.padic_regulator(7)
2*7^2 + 5*7^3 + 7^4 + 5*7^5 + 3*7^6 + 5*7^7 + 7^8 + 5*7^9 + O(7^10)
E1.padic_regulator(7)
3*7 + 3*7^2 + 4*7^3 + 3*7^4 + 5*7^6 + 7^8 + 3*7^9 + 5*7^11 + 7^12 +O(7^13)

Not only are they different, but their $p$-valuation as you can see is also different, which, without going through details, would imply in the first case that the Iwasawa $\lambda$ invariant would be greater than 1, whereas in the second case it would imply that $\lambda_p(E)= 1$, which is absurd since $\lambda_p(E)$ does not depend on the model. The correct regulator is the first one, of The minimal curve, since in this example one can check $\lambda_p(E) >1$.

At first I was convinced it was an implementation error by Sage, so I posted pretty much the same question on Ask Sage with the assumption that it was a bug, but for various reasons I could go through, now I have started to doubt that it was due to implementation errors, which is why I also decided to post the question here.

So in short, am I misunderstanding something? Shouldn't both the regulators be the same, since both models are minimal at $p$?