Defining parameters
| $A$: | $[6, 3, 2]$ | $\alpha$: | \( \left[\frac{1}{6}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{5}{6}\right] \) |
| $B$: | $[5, 1]$ | $\beta$: | \( \left[\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, 1\right] \) |
| $\gamma$: | $[-5, -1, 6]$ | ||
Invariants
| Degree: | $5$ |
| Weight: | $0$ |
| Type: | Orthogonal |
| Wild primes: |
\( 2 \), \( 3 \), \( 5 \)
|
| Hodge vector: | $[5]$ |
| Rotation number: | $1$ |
| Determinant character: | $\chi_{(1-t)} \otimes \Q(0)$ |
| Bezout matrix: | $\left(\begin{array}{rrrrr} -1 & -1 & -1 & -1 & -2 \\ -1 & -1 & -1 & -2 & -1 \\ -1 & -1 & -2 & -1 & -1 \\ -1 & -2 & -1 & -1 & -1 \\ -2 & -1 & -1 & -1 & -1 \end{array}\right)$ |
| Bezout determinant: | $-6$ |
| Bezout module: | $C_{2} \times C_{3}$ |
| Levelt matrices: | ${h_{\infty}=\left(\begin{array}{rrrrr} 0 & 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 1 & -1 \end{array}\right),\;}$ ${h_0=\left(\begin{array}{rrrrr} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \end{array}\right),\;}$ $ {h_1=\left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & -1 \end{array}\right)}$ |
Zigzag plot
$p$-parts of defining parameters
| $p$ | $A_p$ | $B_p$ | $C_p$ |
|---|---|---|---|
| $2$ | $[2, 2, 2]$ | $[1, 1, 1]$ | $[1, 1]$ |
| $3$ | $[3, 3]$ | $[1, 1, 1, 1]$ | $[1]$ |
| $5$ | $[1, 1, 1, 1]$ | $[5]$ | $[1]$ |
| $7$ | $[]$ | $[]$ | $[1, 1, 1, 1, 1]$ |
Monodromy groups modulo $\ell$
| $\ell$ | Index | Subimage | Image | $A^{\perp} _\ell$ | $B^{\perp} _\ell$ | $C^{\perp} _\ell$ | Imprimitivity |
|---|---|---|---|---|---|---|---|
| $2$ | $ $ | $\operatorname{Sp}(4,2)$ | $\operatorname{go}(5,2)$ | $[3, 3]$ | $[5]$ | $[1]$ | $1$ |
| $3$ | $80\cdot 3^{2}$ | ?? | $S_6$ | $[2, 2, 2, 1]$ | $[5]$ | $[1]$ | $1$ |
| $5$ | $ $ | $S_6$ | $S_6$ | $[6, 3, 2]$ | $[1, 1, 1, 1, 1]$ | $[\ ]$ | $1$ |
| $7$ | $ $ | $S_6$ | $S_6$ | $[6, 3, 2]$ | $[5, 1]$ | $[\ ]$ | $1$ |
Good Euler factors
$p=7$
| $t$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|---|---|
| $2$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T )^{2}( 1 + T^{2} ) \) |
| $3$ | $C_1$$\times$$C_4$ | \( ( 1 - T )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| $4$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| $5$ | $C_1$$\times$$C_4$ | \( ( 1 - T )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| $6$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T )^{2}( 1 + T + T^{2} ) \) |
$p=11$
| $t$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|---|---|
| $2$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T )^{2}( 1 + T^{2} ) \) |
| $3$ | $C_1$$\times$$C_4$ | \( ( 1 - T )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| $4$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| $5$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| $6$ | $C_1$$\times$$C_2$$\times$$C_2$ | \( ( 1 + T )( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| $7$ | $C_1$$\times$$C_4$ | \( ( 1 - T )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| $8$ | $C_1$$\times$$C_4$ | \( ( 1 - T )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| $9$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T )^{2}( 1 + T^{2} ) \) |
| $10$ | $C_1$$\times$$C_2$$\times$$C_2$ | \( ( 1 + T )( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
$p=13$
| $t$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|---|---|
| $2$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T )^{2}( 1 + T^{2} ) \) |
| $3$ | $C_1$$\times$$C_4$ | \( ( 1 - T )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| $4$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| $5$ | $C_1$$\times$$C_2$$\times$$C_2$ | \( ( 1 + T )( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| $6$ | $C_1$$\times$$C_1$ | \( ( 1 + T )^{2}( 1 - T )^{3} \) |
| $7$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T )^{2}( 1 + T^{2} ) \) |
| $8$ | $C_1$$\times$$C_4$ | \( ( 1 - T )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| $9$ | $C_1$$\times$$C_4$ | \( ( 1 - T )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| $10$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| $11$ | $C_1$$\times$$C_2$$\times$$C_2$ | \( ( 1 + T )( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| $12$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T )^{2}( 1 + T^{2} ) \) |
$p=17$
| $t$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|---|---|
| $2$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| $3$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| $4$ | $C_1$$\times$$C_4$ | \( ( 1 - T )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| $5$ | $C_1$$\times$$C_2$$\times$$C_2$ | \( ( 1 + T )( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| $6$ | $C_1$$\times$$C_2$ | \( ( 1 - T )^{3}( 1 + T + T^{2} ) \) |
| $7$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T )^{2}( 1 + T^{2} ) \) |
| $8$ | $C_1$$\times$$C_2$ | \( ( 1 - T )^{3}( 1 + T + T^{2} ) \) |
| $9$ | $C_1$$\times$$C_2$$\times$$C_2$ | \( ( 1 + T )( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| $10$ | $C_1$$\times$$C_2$$\times$$C_2$ | \( ( 1 + T )( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| $11$ | $C_1$$\times$$C_4$ | \( ( 1 - T )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| $12$ | $C_1$$\times$$C_4$ | \( ( 1 - T )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| $13$ | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + T^{2} )^{2} \) |
| $14$ | $C_1$$\times$$C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| $15$ | $C_1$$\times$$C_1$ | \( ( 1 + T )^{2}( 1 - T )^{3} \) |
| $16$ | $C_1$$\times$$C_2$$\times$$C_2$ | \( ( 1 + T )( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |