| Locus | Group ID | Signature |
Generators | Comments and Favorite Equations | Additional files |
| 1 | (120,34)=\(S_5\) | (2,4,5) |
| \( (-x_0,-x_1,-x_2,-x_2+x_3) \) |
| \( (x_0+x_1,-x_0-x_2,-x_0-x_3,-x_3) \) |
|
| Bring's curve. General trigonal |
| \( x_0^2 + x_0 x_1 + x_1^2 - x_1x_2+ x_2^2 - x_2 x_3 + x_3^2 \) |
| \( x_0^2 x_1 + x_0 x_1^2 + x_1^2 x_2 - x_1x_2^2+ x_2^2 x_3 - x_2x_3^2 \) |
| Genus-4-120-34.htm |
| 2 | (72,42) | (2,3,12) |
\(\left[ \begin{array}{rrrr}
-1 & 0 & 0 & 0 \\
0 & 0 & 0 & -i \\
0 & 0 & -1 & 0 \\
0 & i & 0 & 0
\end{array} \right] , \qquad
\left[ \begin{array}{rrrr}
-\zeta_6 & 0 & 0 & 0 \\
0 & -\frac{1}{2}\zeta_{12} & \frac{1}{2}\zeta_{3}& \frac{1}{2}\zeta_{12} \\
0 & \zeta_{12} & 0 & \zeta_{12} \\
0 & -\frac{1}{2}\zeta_{12}& -\frac{1}{2}\zeta_{3} & \frac{1}{2}\zeta_{12}
\end{array} \right]\)
|
| Cyclic trigonal \(y^3 = x(x^4-1)\) |
| Canonical ideal: \(x_1x_3-x_2^2 \) |
| \(x_0^3 - x_1^2 x_2 +x_2 x_3^2 \) |
| Genus-4-72-42.htm |
| 3 | (72,40) | (2,4,6) |
| \((-x_0,x_2,x_1,-x_3)\) |
| \( (-x_2,\zeta_6^2x_0,\zeta_6x_3,x_1)\) |
|
| Cyclic trigonal \( y^3 = (x^3-1)^2(x^3+1)\) |
| Canonical ideal: \(x_0 x_3-x_1 x_2,\) |
| \( x_1^3-x_0^3-x_3^3-x_2^3\) |
| Genus-4-72-40.htm |
| 4 | (40,8) | (2,4,10) |
| \( (1, 14)(2, 12)(3, 9)(4, 7)(5, 17)(8, 20)(11, 18)(13, 16)(15, 19)\) |
| \( (1, 5, 3, 2)(4, 11, 7, 15)(6, 8, 10, 13)(9, 18, 14, 19)(12, 16, 17, 20)\) |
|
Hyperelliptic \(y^2 = x^{10}-1\)
| Genus-4-40-8.txt |
| 5 | (36,12) | (2,6,6) |
| \((-x_0,\zeta_3 x_3,-x_2,-\zeta_6 x_1) \) |
| \((\zeta_3 x_0,-\zeta_3 x_3, \zeta_6 x_2, -x_1) \) |
|
| Cyclic trigonal \( y^3 = (x^3-1)(x^3+1)\) |
| Canonical ideal: \( x_1 x_3-x_2^2,\) |
| \( x_0^3-x_3^3+x_1^3\) |
| Genus-4-36-12.htm |
| 6 | (32,19) | (2,4,16) |
| \( (1, 4)(2, 14)(3, 12)(5, 11)(6, 8)(7, 9)(13, 15)\) |
| \( (1, 13, 5, 8)(2, 4, 9, 11)(3, 7, 12, 14)(6, 10, 15, 16)\) |
| Hyperelliptic \( y^2 = x^9-x\) | Genus-4-32-19.txt |
| 7 | (24,3) | (3,4,6) |
| \( (2, 3, 5)(6, 8, 7) \) |
| \((1, 8, 4, 3)(2, 7, 6, 5) \) |
| Hyperelliptic \( y^2 = x(x^4-1)(x^4+2i\sqrt{3}+1) \) | Genus-4-24-3.txt |
| 8 | \( (18,2) = Z_{18} \) | (2,9,18) |
| \( (1, 2)(3, 7)(4, 8)(5, 9)(6, 10)(11, 15)(12, 16)(13, 17)(14, 18) \) |
| \( (1, 3, 11, 4, 12, 14, 6, 13, 5)(2, 7, 15, 8, 16, 18, 10, 17, 9)\) |
| Hyperelliptic \(y^2=x^9-1\) | Genus-4-18-2.txt |
| 9 | \( (15,1) = Z_{15}\) | (3,5,15) |
| \( (\zeta_{3}^{2} x_0,\zeta_{3} x_1, \zeta_{3}x_2, \zeta_{3} x_3)\) |
| \( (\zeta_{5} x_0,\zeta_{5}^3 x_1,\zeta_{5}^2 x_2,\zeta_{5} x_3)\) |
|
| Cyclic trigonal \(y^3 = x^5-1 \) |
| Canonical ideal: \( x_1 x_3 -x_2^2\) |
| \( x_0^3 - x_1^2 x_2 +x_3^3 \) |
| Genus-4-15-1.htm |
| Locus | Group ID | Signature |
Generators | Comments and Favorite Equations | Additional files |
| 10 | (36,10) | (2,2,2,3) |
| \( (-x_0,-\zeta_3^2 x_2,-\zeta_3 x_1,-x_3) \) |
| \((x_3,-x_1,-x_2,x_0) \) |
| \( (-\zeta_3 x_3, x_2,x_1,-\zeta_3^2 x_0)\) |
|
| Cyclic trigonal \( y^3 = (x^3+t)^2(tx^3-1) \) |
| Canonical ideal: \(x_0x_3-x_1x_2 \) |
| \( x_1^3+t x_0^3+x_2^3- t x_3^3 \) |
| Genus-4-36-10.htm |
| 11 | (24,12)=\(S_4\) | (2,2,2,4) |
| \( (-x_0,-x_2,-x_1,-x_4)\) |
| \( (x_2,-x_1,x_0,-x_4)\) |
| \((-x_0,x_2,x_1,-x_4) \) |
|
| General trigonal |
| \( x_0^2+x_1^2+x_2^2+tx_3^2\) |
| \( x_0x_1x_2-x_3^3\) |
|
|
| 12 | (20,4) | (2,2,2,5) |
| \((1, 2)(3, 4)(5, 7)(6, 8)(9, 10)\) |
| \( (1, 3)(2, 4)(5, 6)(7, 8)\) |
| \((1, 10)(2, 9)(3, 8)(4, 6)(5, 7) \) |
| Hyperelliptic \( y^2 = (x^5-t^5)(x^5-t^{-5})\) | Genus-4-20-4.txt |
| 13 | (18,3) | (2,2,3,3) |
| \( (-x_0, x_3,-x_2,x_1) \) |
| \( (-x_0,\zeta_3x_3,-x_2,\zeta_3^2 x_1,)\) |
| \( (x_0,\zeta_3^2x_1,\zeta_3,x_2,x_3) \) |
|
| Cyclic trigonal \( y^3 = (x^3+t)(tx^3-1)\) |
| Canonical ideal: \( x_1 x_3 - x_2^2 \) |
| \( x_0^3-x_3^3+t x_2^3+x_1^3 \) |
| Genus-4-18-3.htm |
| 14 | (16,7) | (2,2,2,8) |
| \( (1, 4)(2, 6)(3, 7)(5, 8) \) |
| \( (1, 8)(2, 7)(3, 6)(4, 5) \) |
| \( (1, 7)(3, 4)(5, 8) \) |
| Hyperelliptic \( y^2 = x(x^4-t^4)(x^4-t^{-4}) \) | Genus-4-16-7.txt |
