Related to this question.

According to Hindry p.7 Conj 3.1 and Stein's notes Szpiro's conjecture over number fields 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 and $f_{E/K}$ is the conductor.

To my knowledge over the rationals the largest known Szpiro ratio is about $9.01$.

Consider the elliptic curve with cremona label 37b1 and ainvariants $[0, 1, 1, -23, -50]$. Over the rationals the discriminant is $37^3$. Over the number field with defining polynomial $x^{22} - 37^3$ the norm of the minimal discriminant is $37^{66}$ and the norm of the conductor is $37$ while the global minimal model is the same as over the rationals, giving Szpiro ratio $66$. If this pattern continue for number fields of the form $x^n - 37^3$, the Szpiro ratio probably will be unbounded.

Computation with sage, possibly assuming GRH since the number fields are not certified suggest the following large Szpiro ratios {33,39,42,48,51,66}:

```
a_i= [0, 1, 1, -23, -50] ratio= 33.00000 Delta= 37^33 f= 37 d_E/Q= 37^3 global minimal model= Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-23)*x + (-50) over Number Field in w with defining polynomial x^11 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 39.00000 Delta= 37^39 f= 37 d_E/Q= 37^3 global minimal model= Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-23)*x + (-50) over Number Field in w with defining polynomial x^13 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 42.00000 Delta= 37^42 f= 37 d_E/Q= 37^3 global minimal model= Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-23)*x + (-50) over Number Field in w with defining polynomial x^14 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 48.00000 Delta= 37^48 f= 37 d_E/Q= 37^3 global minimal model= Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-23)*x + (-50) over Number Field in w with defining polynomial x^16 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 51.00000 Delta= 37^51 f= 37 d_E/Q= 37^3 global minimal model= Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-23)*x + (-50) over Number Field in w with defining polynomial x^17 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 66.00000 Delta= 37^66 f= 37 d_E/Q= 37^3 global minimal model= Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-23)*x + (-50) over Number Field in w with defining polynomial x^22 - 50653
```

Q1 Are the computations correct?

Q2 Does increasing the degree of the number field continue to increase the ratio?

Q3 (

New) Is this a counterexample this formulation of Szpiro's conjecture? My concern isbounded, not infinitely many?

Working will large degree fields is not fast for me.

Small search found 46 ratios greater than $24$.

The largest is $134.6199$ from ainvariants $[1, 1, 1, -110, -880]$

```
a_i= (1, 1, 1, -110, -880) ratio= 134.6199 Delta= -1 * 3^304 * 5^19 f= 3 * 5 d_E/Q= -1 * 3^16 * 5 global minimal model= Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + (-110)*x + (-880) over Number Field in w with defining polynomial x^19 + 215233605
```

**Added** Addressing Jamie Weigandt answer.

With his code and possibly assuming GRH, I get the same ratio.

To assume GRH, replace `K.<a> = NumberField(x^11 - 50653)`

with `K.<a> = NumberField(x^11 - 50653,check=False)`

. This avoids
certification of the NF with pari's `bnfcertify`

.

Here is the output of

```
Nf.<w>=NumberField(x**D-d)
Nf.<w>=NumberField(x**D-d,check=False) #assume GRH
print 'a_i=',ai,' ratio=',szpiro_ratio(E2),Nf
a_i= [0, 1, 1, -23, -50] ratio= 30.0 Number Field in w with defining polynomial x^10 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 33.0 Number Field in w with defining polynomial x^11 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 39.0 Number Field in w with defining polynomial x^13 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 42.0 Number Field in w with defining polynomial x^14 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 48.0 Number Field in w with defining polynomial x^16 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 51.0 Number Field in w with defining polynomial x^17 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 57.0 Number Field in w with defining polynomial x^19 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 60.0 Number Field in w with defining polynomial x^20 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 66.0 Number Field in w with defining polynomial x^22 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 69.0 Number Field in w with defining polynomial x^23 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 75.0 Number Field in w with defining polynomial x^25 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 78.0 Number Field in w with defining polynomial x^26 - 50653
a_i= [0, 1, 1, -23, -50] ratio= 84.0 Number Field in w with defining polynomial x^28 - 50653
```

According to pari's documentation, the comment: `#Don't do this. It will take forever`

probably will mean GRH fails for this NF, since in this case
pari's bnfcertify doesn't return.

Regarding his comment about Q2 and 32a1: 32a1 has $a_6=0$, so the ratio degrades for trivial reasons. I didn't claim high ratio for it.

The claim that the ratio is bounded over number fields is from here and William Stein' notes

Similar definition of the Szpiro's conjecture is in Silverman's The Arithmetic of Elliptic Curves, p. 275.

With the above notation, there is constant $k$, depending on $\epsilon$ and $K$ s.t.:

$$ N_{K/Q} \Delta_{E/K} \le k(N_{K/Q} f_{E/K})^{6+ \epsilon}\qquad (2) $$

Silverman asks "It is very interesting to ask how the constants $k$ appearing in these conjectures depend on the field $K$."

In analogy with the uniform abc conjecture, probably Silverman should take $k$ the discriminant of $K$.