According to Hindry p.7 Conj 3.1 and Stein Szpiro's conjecture states that the Szpiro ratio is:

$$ \sigma_{E/K}=\frac{\log{|N_{K/Q}\Delta_{E/K}|}}{\log{|N_{K/Q} f_{E/K}}|}$$

Given $ \varepsilon >0$ there are only finitely many $ E/K$ with $ \sigma_{E/K}\geq 6+\varepsilon $. In particular, $ \sigma_{E/K}$ is bounded.

$\Delta_{E/K}$ is the minimal discriminant.

Szpiro's conjecture implies abc with a bit higher exponent over the integers.

Over number fields, the uniform abc conjecture depends on the discriminant of the number field. Maybe this is because every integer can be arbitrary large power in some number field. To make $d$ a $k$-th power, work with defining polynomial $x^k -d$. So start with EC over the rationals with discriminant $\Delta$ and compute $\sigma_{E/K}$ for $K$ with defining polynomial $x^k - \Delta$ for $k$ sufficiently large.

Unless the minimal discriminant takes care of this case, this will give unbounded Szpiro ratio over some number fields.

Why the Szpiro conjecture over number fields doesn't depend on the discriminant of the number field while the uniform abc conjecture depends?

Over $K$ with defining polynomial $x^{16} + 22384$ take $E/K : y^2 = x^3 + 7x -1$.

Not sure if the discriminant is minimal, but the norm is $2^{64} \cdot 1399^{16}$ while the norm of the conductor is $2^{10} \cdot 1399$ giving Szpiro ratio $11.305664847$

Some experiments in degree 14 and sage's global_minimal_model() suggest the global minimal model preserves the high powers in the discriminant:

```
a4,a6= -2 -1 global minimal model= Elliptic Curve defined by y^2 + (1/14*w^12-1/28*w^10+1/7*w^8+1/4*w^7-1/14*w^6+2/7*w^4-1/7*w^2-3/7)*y = x^3 + (243/56*w^13+17/16*w^12-297/28*w^11+51/2*w^10-561/14*w^9+111/2*w^8-1581/28*w^7+81/2*w^6+255/28*w^5-99*w^4+1530/7*w^3-374*w^2+3330/7*w-527)*x + (-531/112*w^13-9419/56*w^12+2945/7*w^11-40829/56*w^10+55889/56*w^9-31119/28*w^8+47939/56*w^7-1209/28*w^6-10470/7*w^5+53021/14*w^4-90715/14*w^3+125729/14*w^2-69170/7*w+107855/14) over Number Field in w with defining polynomial x^14 - 80 Delta= 2^32 * 5^14 f= 2^32 * 5 d_q= 2^4 * 5 ratio= 1.87946880927217
a4,a6= -2 1 global minimal model= Elliptic Curve defined by y^2 + (1/14*w^12-1/28*w^10+1/7*w^8+1/4*w^7-1/14*w^6+2/7*w^4-1/7*w^2-3/7)*y = x^3 + (243/56*w^13+17/16*w^12-297/28*w^11+51/2*w^10-561/14*w^9+111/2*w^8-1581/28*w^7+81/2*w^6+255/28*w^5-99*w^4+1530/7*w^3-374*w^2+3330/7*w-527)*x + (533/112*w^13+4709/28*w^12-5891/14*w^11+40819/56*w^10-55887/56*w^9+31125/28*w^8-47933/56*w^7+1185/28*w^6+10460/7*w^5-26504/7*w^4+90725/14*w^3-125767/14*w^2+69150/7*w-107857/14) over Number Field in w with defining polynomial x^14 - 80 Delta= 2^32 * 5^14 f= 2^18 * 5 d_q= 2^4 * 5 ratio= 3.17425556409935
```

The last two examples give the abc triples $c_4^3-c_6^2=w^{14}$

Curves of Szpiro ratios $ > 66$ are in this question and they appear to contradict boundedness of the ratio.

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