Computation programs and log files for the paper "Automorphism groups of superspecial curves of genus 4 over F11"

To prove Propositions 4.3.1 -- 4.3.4, we did the following computation: For each $F_{11}$-isomorphism class (resp. $\overline{F_{11}}$-isomorphism class) of nonhyperelliptic superspecial curves of genus 4 over $F_{11}$, compute its automorphism group over $F_{11}$ (resp. $\overline{F_{11}}$). Each automorphism is computed as an element of the general linear group of degree 4 over $F_{11}$ (resp. $\overline{F_{11}}$). For our algorithm, see Section 3.1 in our paper. In the following, we show our computation programs implemented over Magma V2-22.3, and log files. (EV) OS: Windows 10, 64bit, CPU: intel Core i5, 2.60GHZ, 8GB memory.

  • Codes and log files（uploaded at September 11th, 2017）

    1. "ZeroDimensionalIdealVariety.txt" (txt): Code for solving zero-dimensional multivariate systems (Algorithm 2.3.1)

    2. Automorphism groups over the prime field $K = F_{11}$ (Propositions 4.3.1 -- 4.3.3)
      (N1): Q = 2xw + 2yz

      • Code file "AutPrime-N1.txt" (txt): Code for computing matrices representing automorphisms over $F_{11}$ of each $F_{11}$-isomorphism class

      • Log file "result_AutPrime-N1.txt" (txt): Log file obtained by executing "AutPrime-N1.txt"

      • List "List_of_Matrices_AutPrime-N1.txt" (txt): List of matrices obtained by executing "AutPrime-N1.txt", see also the log file

      • Code file "Group_Structure_AutPrime-N1.txt" (txt): Code for determining the finite group structure of each automorphism group over the prime field $F_{11}$

      • Log file "result_Group_Structure_AutPrime-N1.txt" (txt): Log file obtained by executing "Group_Structure_AutPrime-N1.txt"

    3. (N2): Q = 2xw + y^2 - (epsilon)z^2

      • Code file "AutPrime-N2.txt" (txt): Code for computing matrices representing automorphisms over $F_{11}$ of each $F_{11}$-isomorphism class

      • Log file "result_AutPrime-N2.txt" (txt): Log file obtained by executing "AutPrime-N2.txt"

      • List "List_of_Matrices_AutPrime-N2.txt" (txt): List of matrices obtained by executing "AutPrime-N2.txt", see also the log file

      • Code file "Group_Structure_AutPrime-N2.txt" (txt): Code for determining the finite group structure of each automorphism group over the prime field $F_{11}$

      • Log file "result_Group_Structure_AutPrime-N2.txt" (txt): Log file obtained by executing "Group_Structure_AutPrime-N2.txt"

    4. (Dege): Q = z^2 + 2yw

      • Code file "AutPrime-Dege.txt" (txt): Code for computing matrices representing automorphisms over $F_{11}$ of each $F_{11}$-isomorphism class

      • Log file "result_AutPrime-Dege.txt" (txt): Log file obtained by executing "AutPrime-Dege.txt"

      • List "List_of_Matrices_AutPrime-Dege.txt" (txt): List of matrices obtained by executing "AutPrime-Dege.txt", see also the log file

      • Code file "Group_Structure_AutPrime-Dege.txt" (txt): Code for determining the finite group structure of each automorphism group over the prime field $F_{11}$

      • Log file "result_Group_Structure_AutPrime-Dege.txt" (txt): Log file obtained by executing "Group_Structure_AutPrime-Dege.txt"

    5. Automorphism groups over the algebraic closure (Proposition 4.3.4)
      N1 type:

      • Code file "AutAC-N1.txt" (txt): Code for computing matrices representing automorphisms over $\overline{F_{11}}$ of each $\overline{F_{11}}$-isomorphism class

      • Log file "result_AutAC-N1.txt" (txt): Log file obtained by executing "AutAC-N1.txt"

      • List "List_of_Matrices_AutAC-N1.txt" (txt): List of matrices obtained by executing "AutAC-N1.txt", see also the log file

      • Code file "Group_Structure_AutAC-N1.txt" (txt): Code for determining the finite group structure of each automorphism group over the algebraic closure $\overline{F_{11}}$

      • Log file "result_Group_Structure_AutAC-N1.txt" (txt): Log file obtained by executing "Group_Structure_AutAC-N1.txt"

    6. Dege type: Q = z^2 + 2yw

      • Code file "AutAC-Dege.txt" (txt): Code for computing matrices representing automorphisms over $\overline{F_{11}}$ of each $\overline{F_{11}}$-isomorphism class

      • Log file "result_AutAC-Dege.txt" (txt): Log file obtained by executing "AutAC-Dege.txt"

      • List "List_of_Matrices_AutAC-Dege.txt" (txt): List of matrices obtained by executing "AutAC-Dege.txt", see also the log file

      • Code file "Group_Structure_AutAC-Dege.txt" (txt): Code for determining the finite group structure of each automorphism group over the algebraic closure $\overline{F_{11}}$

      • Log file "result_Group_Structure_AutAC-Dege.txt" (txt): Log file obtained by executing "Group_Structure_AutAC-Dege.txt"

Other computations

The following are our computation programs (over Magma V2-22.3) in Section 5 of our paper.

  • Computations in Section 5
    Computing the Galois cohomology:

    • Code file "GaloisCohomology_AC.txt" (txt): Code for enumerating $F_{11}$-isomorphism classes

    • Log file "result_GaloisCohomology_AC.txt" (txt): Log file obtained by executing "GaloisCohomology_AC.txt"