While experimenting with the data base (related to #6796 ), I found a few inconsistent entries.
3T1-3_3_3-a, 9T11-6.2.1_3.3.3_2.2.2.2.1-a: although the coefficients of the curve/map lie in $\mathbb{Q}$, the base field is not $\mathbb{Q}$. In the embeddings, there is only one entry (which is ok because the equations do not depend on the embedding).
9T9-4.4.1_4.4.1_2.2.2.2.1-a, 8T13-6.2_3.3.1.1_2.2.2.2-a: although the base field has degree $2$, there is only one entry in the embedding section. Contrary the two examples above, the coefficients of the curve/map do not lie in $\mathbb{Q}$.
I tried to compute monodromy triples using certified homotopy continuation software for all examples that are maps from $\mathbb{P}^1$ to $\mathbb{P}^1$ in the database (I use the map directly instead of the planar model, so hopefully the problems raised in issue #6796 do not happen here). My computations match (up to simultaneous conjugacy) the database on all but the following examples.
7T3-3.3.1_3.3.1_3.3.1-a, 6T10-4.2_4.2_2.2.1.1-a, 7T4-6.1_3.3.1_2.2.2.1-a: all those examples are over a base field of degree $2$, and what I get matches the database if I swap the entries in the embeddings. For 7T4-6.1_3.3.1_2.2.2.1-a (I only tried with this example), the monodromy permutations I get seems consistant with what MonodromyRepresentation in magma returns (although I do not know what are the precise conventions of magma for monodromy permutations and fundamental group generators).
9T33-5.3.1_5.1.1.1.1_4.3.2-a: in the embeddings section, the value $0.6015913043995355+2.186160957891682\sqrt{-1}$ appears twice (and one embedding is missing). If I replace the value on line $4$ by $2.54704593520662$, what I get matches the database.
9T34-6.1.1.1_5.2.2_4.3.1.1-a: same problem, the value $-4.68092434938512-1.104809145531491\sqrt{-1}$ appears twice, value on line $1$ should be $3.13824200205890 - 2.91398703000310\sqrt{-1}$.
For examples that are not maps from $\mathbb{P}^1$ to $\mathbb{P}^1$, I tried to compute the triple for the planar model and check if it matches the database up to $S_3$-action. When doing so, I also found a few mismatches:
6T16-6_5.1_3.2.1-a
7T7-6.1_6.1_3.2.2-b
8T45-6.2_6.2_4.2.1.1-a
7T6-7_5.1.1_5.1.1-c
I did not investigate those examples too much for now; investigation should be easier once there is no hidden $S_3$ action. Even though there may be some mismatch at the level of triples, the monodromy groups are correct.
