Do these two elliptic curves have good reduction in these fields?

Question

In Silverman's Arithmetic of Elliptic Curves, in Chapter VII, Exercise 7.5, one is asked to show that three elliptic curves have good reduction over given p-adic field extensions (where they have bad reduction over the base p-adic field). Specifically one is asked to show good reduction by "writing down"(!) a minmal Weirstrauss equation over the field.

However, both footling around doing Tate's Algorithm by hand, and kicking it into Magma seem to suggest that the first two don't actually have good reduction in the given fields, which seems curious.

The curves and extensions are as follows:

1. $E: y^2 = x^3 + x,\text{ defined over }\mathbb{Q}_2(\pi),\pi^8=2$
2. $E: y^2 +y = x^3,\text{ defined over }\mathbb{Q}_3(\pi),\pi^4=3$
3. $E: y^2 = x^3 + x^2 -3x-2 ,\text{ defined over }\mathbb{Q}_5(\pi),\pi^4=5$

Here is the Magma code I am executing for the first one:

K:=pAdicField(2,400);
R<x>:=PolynomialRing(K);
E:=EllipticCurve([K|0,0,0,1,0]);
new:=x^8-2;
F:=SplittingField(new);
EF:=BaseChange(E,F);
loc,min:=LocalInformation(EF);
loc;

And its output: <F.1 + O(F.1^6401), 12, 6, 4, I2*, true>.

So this would seem to suggest there is some measure of error in the text: but I thought it was more likely that I was missing something. I would be interested to hear what people think.

Answer 1

In the 2nd edition of The Arithmetic of Elliptic Curves, the problem is stated as follows:

Exercise 7.5 Show that the following elliptic curves have good reduction over a field of the indicated form by writing down a minimal equation for $E$ over that field.

(a) $E:y^2=x^3+x$, $\mathbb{Q}_2(\eta,i)$, $\eta^8=2$, $i^2=-1$.

(b) $E:y^2+y=x^3$, $\mathbb{Q}_3(\pi,\eta)$, $\pi^2=\sqrt{-3}$, $\eta^3=2$.

(c) $E:y^2=x^3+x^2-3x-2$, $\mathbb{Q}_5(\pi)$, $\pi^4=5$.

It is easy enough to do (b) by hand. We substitute $x\to \pi^2 X+\eta$ and $y\to \pi^3 Y+1$ and divide by $\pi^6$ to obtain the Weierstrass equation $$ Y^2 - \pi Y = X^3 - \eta\pi^2 X^2 + \eta^2X. $$ This Weierstrass equation has discriminant $\Delta=1$, so in fact represents a global everywhere-good-reduction model for $E$ over the field $\mathbb{Q}(\sqrt[4]{-3},\sqrt[3]{2})$.