# Belyi maps downloaded from the LMFDB on 24 June 2026. # Search link: https://www.lmfdb.org/Belyi/?deg=5 # Query "{'deg': 5}" returned 23 maps, sorted by degree. # Each entry in the following data list has the form: # [Label, Degree, Group, abc, Ramification type, Genus, Orbit Size, Base field] # For more details, see the definitions at the bottom of the file. "5T1-5_5_1.1.1.1.1-a" 5 "5T1" [5, 5, 1] [[5], [5], [1, 1, 1, 1, 1]] 0 1 ["1.1.1.1", [-1, 1]] "5T1-5_5_5-a" 5 "5T1" [5, 5, 5] [[5], [5], [5]] 2 1 ["1.1.1.1", [-1, 1]] "5T1-5_5_5-b" 5 "5T1" [5, 5, 5] [[5], [5], [5]] 2 1 ["1.1.1.1", [-1, 1]] "5T1-5_5_5-c" 5 "5T1" [5, 5, 5] [[5], [5], [5]] 2 1 ["1.1.1.1", [-1, 1]] "5T2-5_2.2.1_2.2.1-a" 5 "5T2" [5, 2, 2] [[5], [2, 2, 1], [2, 2, 1]] 0 1 ["1.1.1.1", [-1, 1]] "5T3-4.1_4.1_2.2.1-a" 5 "5T3" [4, 4, 2] [[4, 1], [4, 1], [2, 2, 1]] 0 2 ["2.0.4.1", [1, 0, 1]] "5T3-5_4.1_4.1-a" 5 "5T3" [5, 4, 4] [[5], [4, 1], [4, 1]] 1 2 ["2.0.4.1", [1, 0, 1]] "5T4-5_2.2.1_3.1.1-a" 5 "5T4" [5, 2, 3] [[5], [2, 2, 1], [3, 1, 1]] 0 1 ["1.1.1.1", [-1, 1]] "5T4-5_3.1.1_3.1.1-a" 5 "5T4" [5, 3, 3] [[5], [3, 1, 1], [3, 1, 1]] 0 1 ["1.1.1.1", [-1, 1]] "5T4-5_5_2.2.1-a" 5 "5T4" [5, 5, 2] [[5], [5], [2, 2, 1]] 1 1 ["1.1.1.1", [-1, 1]] "5T4-5_5_3.1.1-a" 5 "5T4" [5, 5, 3] [[5], [5], [3, 1, 1]] 1 1 ["1.1.1.1", [-1, 1]] "5T4-5_5_3.1.1-b" 5 "5T4" [5, 5, 3] [[5], [5], [3, 1, 1]] 1 1 ["1.1.1.1", [-1, 1]] "5T4-5_5_5-a" 5 "5T4" [5, 5, 5] [[5], [5], [5]] 2 1 ["1.1.1.1", [-1, 1]] "5T5-3.2_3.2_2.2.1-a" 5 "5T5" [6, 6, 2] [[3, 2], [3, 2], [2, 2, 1]] 0 1 ["1.1.1.1", [-1, 1]] "5T5-3.2_3.2_3.1.1-a" 5 "5T5" [6, 6, 3] [[3, 2], [3, 2], [3, 1, 1]] 0 1 ["1.1.1.1", [-1, 1]] "5T5-4.1_2.2.1_3.2-a" 5 "5T5" [4, 2, 6] [[4, 1], [2, 2, 1], [3, 2]] 0 1 ["1.1.1.1", [-1, 1]] "5T5-4.1_3.2_3.1.1-a" 5 "5T5" [4, 6, 3] [[4, 1], [3, 2], [3, 1, 1]] 0 2 ["2.2.24.1", [-6, 0, 1]] "5T5-4.1_4.1_3.1.1-a" 5 "5T5" [4, 4, 3] [[4, 1], [4, 1], [3, 1, 1]] 0 1 ["1.1.1.1", [-1, 1]] "5T5-5_2.1.1.1_3.2-a" 5 "5T5" [5, 2, 6] [[5], [2, 1, 1, 1], [3, 2]] 0 1 ["1.1.1.1", [-1, 1]] "5T5-5_4.1_2.1.1.1-a" 5 "5T5" [5, 4, 2] [[5], [4, 1], [2, 1, 1, 1]] 0 1 ["1.1.1.1", [-1, 1]] "5T5-5_3.2_3.2-a" 5 "5T5" [5, 6, 6] [[5], [3, 2], [3, 2]] 1 1 ["1.1.1.1", [-1, 1]] "5T5-5_3.2_4.1-a" 5 "5T5" [5, 6, 4] [[5], [3, 2], [4, 1]] 1 2 ["2.2.24.1", [-6, 0, 1]] "5T5-5_4.1_4.1-a" 5 "5T5" [5, 4, 4] [[5], [4, 1], [4, 1]] 1 1 ["1.1.1.1", [-1, 1]] # Label -- # The **label** of a Belyi map $\phi$ has the form $d$T$G$-$\lambda_0\_\lambda_1\_\lambda_\infty$-m encoding the following data. # - $d$T$G$ is the transitive group label of the monodromy group of $\phi$. Here $d$ is the degree of $\phi$. # - $\lambda_i$ are partitions of $d$ specifying the ramification type of $\phi$, each written using `.` as a separator between the parts of the partition. # - m is the Galois orbit label, a non-negative integer written in base 26 using the 26 symbols a, b, ..., z. #Degree (deg) -- # The **degree** of a Belyi map $\phi:X\to\mathbb{P}^1$ is the degree of $\phi$ as a finite map of curves. # Group -- # The **monodromy group** of a Belyi map $\phi \colon X \to \mathbb{P}^1$ of degree $d$ is the transitive subgroup of $S_d$ generated by its associated transitive permutation triple $\sigma$. This group is the geometric monodromy group of the branched cover $\phi$, or equivalently it is the geometric Galois group of the corresponding extension of function fields $K(X) \supseteq K(\phi) \simeq K(\mathbb{P}^1)$. # abc -- # The **$(a,b,c)$ triple** for a Belyi map is obtained by taking the orders of the elements in the corresponding transitive permutation triple. #Ramification type (lambdas) -- # The **ramification type** of a Belyi map $\phi \colon X \to \mathbb{P}^1$ is the triple of partitions $\lambda_0,\lambda_1,\lambda_\infty$ giving the ramification orders of the preimages of $0,1,\infty$, respectively. #Genus (g) -- # The **genus** of a Belyi map $\phi:X\to\mathbb{P}^1$ is the genus of the source $X$. #Orbit Size (orbit_size) -- # The **orbit size** is the number of isomorphism classes of Belyi maps contained in a given Galois orbit. #Base field (field) -- # The **base field** of a Belyi map $\phi:X\to\mathbb{P}^1$ is the field $X$ and $\phi$ are defined over. # (In the database, we take this base field to be the field of moduli whenever possible.)