Defining parameters
| $A$: | $[3, 3]$ | $\alpha$: | \( \left[\frac{1}{3}, \frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right] \) |
| $B$: | $[5]$ | $\beta$: | \( \left[\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}\right] \) |
| $\gamma$: | $[-5, -1, 3, 3]$ | ||
Invariants
| Degree: | $4$ |
| Weight: | $1$ |
| Type: | Symplectic |
| Wild primes: |
\( 3 \), \( 5 \)
|
| Hodge vector: | $[2, 2]$ |
| Rotation number: | $1$ |
| Determinant character: | $\Q(-2)$ |
| Bezout matrix: | $\left(\begin{array}{rrrr} 1 & 2 & 1 & 0 \\ 2 & 2 & 0 & -1 \\ 1 & 0 & -2 & -2 \\ 0 & -1 & -2 & -1 \end{array}\right)$ |
| Bezout determinant: | $1$ |
| Bezout module: | $C_1$ |
| Levelt matrices: | ${h_{\infty}=\left(\begin{array}{rrrr} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & -2 \end{array}\right),\;}$ ${h_0=\left(\begin{array}{rrrr} -1 & 1 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ -1 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \end{array}\right),\;}$ $ {h_1=\left(\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right)}$ |
Zigzag plot
$p$-parts of defining parameters
| $p$ | $A_p$ | $B_p$ | $C_p$ |
|---|---|---|---|
| $2$ | $[]$ | $[]$ | $[1, 1, 1, 1]$ |
| $3$ | $[3, 3]$ | $[1, 1, 1, 1]$ | $[]$ |
| $5$ | $[1, 1, 1, 1]$ | $[5]$ | $[]$ |
| $7$ | $[]$ | $[]$ | $[1, 1, 1, 1]$ |
Monodromy groups modulo $\ell$
| $\ell$ | Index | Subimage | Image | $A^{\perp} _\ell$ | $B^{\perp} _\ell$ | $C^{\perp} _\ell$ | Imprimitivity |
|---|---|---|---|---|---|---|---|
| $2$ | $ $ | $\operatorname{Sp}(4,2)$ | $\operatorname{Sp}(4,2)$ | $[3, 3]$ | $[5]$ | $[\ ]$ | $1$ |
| $3$ | $ $ | $\operatorname{Sp}(4,3)$ | $\operatorname{Sp}(4,3)$ | $[1, 1, 1, 1]$ | $[5]$ | $[\ ]$ | $1$ |
| $5$ | $ $ | $\operatorname{Sp}(4,5)$ | $\operatorname{Sp}(4,5)$ | $[3, 3]$ | $[1, 1, 1, 1]$ | $[\ ]$ | $1$ |
| $7$ | $ $ | $\operatorname{Sp}(4,7)$ | $\operatorname{Sp}(4,7)$ | $[3, 3]$ | $[5]$ | $[\ ]$ | $1$ |
Good Euler factors
$p=7$
| $t$ | $\Gal(F_p)$ | $F_p(T)$ | Ordinary? |
|---|---|---|---|
| $2$ | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | |
| $3$ | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) | |
| $4$ | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) | |
| $5$ | $D_{4}$ | \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | |
| $6$ | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
$p=11$
| $t$ | $\Gal(F_p)$ | $F_p(T)$ | Ordinary? |
|---|---|---|---|
| $2$ | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) | |
| $3$ | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) | |
| $4$ | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | |
| $5$ | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | |
| $6$ | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | |
| $7$ | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) | |
| $8$ | $D_{4}$ | \( 1 + 3 T + T^{2} + 3 p T^{3} + p^{2} T^{4} \) | |
| $9$ | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | |
| $10$ | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
$p=13$
| $t$ | $\Gal(F_p)$ | $F_p(T)$ | Ordinary? |
|---|---|---|---|
| $2$ | $D_{4}$ | \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | |
| $3$ | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) | |
| $4$ | $D_{4}$ | \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | |
| $5$ | $D_{4}$ | \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | |
| $6$ | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | |
| $7$ | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | |
| $8$ | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) | |
| $9$ | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) | |
| $10$ | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | |
| $11$ | $D_{4}$ | \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | |
| $12$ | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
$p=17$
| $t$ | $\Gal(F_p)$ | $F_p(T)$ | Ordinary? |
|---|---|---|---|
| $2$ | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | |
| $3$ | $D_{4}$ | \( 1 + 3 T + 3 p T^{3} + p^{2} T^{4} \) | No |
| $4$ | $D_{4}$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) | |
| $5$ | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) | |
| $6$ | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | |
| $7$ | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | |
| $8$ | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) | |
| $9$ | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | |
| $10$ | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) | |
| $11$ | $D_{4}$ | \( 1 + 11 T + 63 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | |
| $12$ | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) | No |
| $13$ | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | |
| $14$ | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) | |
| $15$ | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | |
| $16$ | $D_{4}$ | \( 1 - 6 T + 21 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |