Fordham
    University
Equations of Riemann surfaces with automorphisms

Genus 6, \( \delta = 0\)

CheckGenus6.txt
Locus Group ID Signature Generators Comments and Favorite Equations Additional files
1 (150,5) (2,3,10) \( \left[\begin{array}{rrr} 0 & -\zeta_5^2 & 0 \\ -\zeta_5^3 & 0 & 0 \\ 0 & 0 & -1 \end{array} \right], \quad \left[\begin{array}{rrr} 0 & 0 & \zeta_5^2\\ 1 & 0 & 0 \\ 0 & \zeta_5^3 & 0 \end{array} \right] \)
Plane quintic
\( y_0^5+y_1^5+y_2^5\)
Genus-6-150-5.htm
2 (120,34) (2,4,6) \( \left[\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & -1 & 1 & 0 & -1 & 1 \\ -1 & 0 & 0 & 0 & 0 & 1 \\ -1 & 0 & 1 & -1 & 0 & 1 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -1 & 0 \end{array} \right], \quad \left[\begin{array}{rrrrrr} 0 & 0 & 1 & 0 & -1 & 0 \\ -1 & 0 & 1 & -1 & 0 & 1\\ -1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & -1 & 1 & 0 & -1 & 0 \end{array} \right] \)
Inoue-Kato
\(-x_0 x_2 + x_1 x_2 - x_0 x_3 + x_1 x_4, \)
\( -x_0 x_1 + x_1 x_2 - x_0 x_3 + x_2 x_5,\)
\(-x_0 x_1 - x_0 x_2 - 2 x_0 x_3 - x_3 x_4 - x_3 x_5, \)
\( -x_0 x_1 - x_0 x_2 - x_0 x_3 - x_1 x_4 - x_3 x_4 - x_4 x_5,\)
\( -x_0 x_1 - x_0 x_2 - x_0 x_3 - x_2 x_5 - x_3 x_5 - x_4 x_5,\)
\( 4 (\sum_{i=0}^{5} x_i^2 ) + 2 x_0 x_1 + 2 x_0 x_2 + 2 x_1 x_2 + 4 x_1 x_3 + 4 x_2 x_3 \)
\( \mbox{ } + 4 x_0 x_4 + 4 x_2 x_4 + 2 x_3 x_4 + 4 x_0 x_5 + 4 x_1 x_5 + 2 x_3 x_5 + 2 x_4 x_5\)
Genus-6-120-34.htm
3 (72,15) (2,4,9)
\( \left[\begin{array}{rrrrrr} 0 & 0 & 0 & -\frac{1}{2}\zeta_9 & \frac{1}{2}\zeta_{36}^{13} & \frac{1}{2}\zeta_9\\ 0 & 0 & 0 & -\zeta_{36}^{13} & 0 & -\zeta_{36}^{13}\\ 0 & 0 & 0 & \frac{1}{2}\zeta_9 & \frac{1}{2}\zeta_{36}^{13} & -\frac{1}{2}\zeta_9\\ \frac{1}{2}\zeta_{36}^{14} & \frac{1}{2}\zeta_{36}^5 & -\frac{1}{2}\zeta_{36}^{14} & 0 & 0 & 0\\ -\zeta_{36}^5 & 0 & -\zeta_{36}^5 & 0 & 0 & 0\\ -\frac{1}{2}\zeta_{36}^{14} & \frac{1}{2}\zeta_{36}^5 & \frac{1}{2}\zeta_{36}^{14} & 0 & 0 & 0 \end{array} \right],\)
\( (\zeta_{12}x_3,\zeta_3 x_4, -\zeta_{12} x_5, \zeta_{12}^5x_0,\zeta_3^2 x_1,-\zeta_{12}^5x_2) \)
Cyclic trigonal \( y^3 = (x^4-2 \sqrt{3} i x^2+1)(x^4+2 \sqrt{3}ix^2 + 1)^2\)
Canonical ideal: \( x_0 x_2 - x_1^2\),
\( x_0 x_4 - x_1 x_3,\)
\( x_0 x_5-x_1 x_4 \),
\( x_1 x_4 - x_2 x_3\),
\( x_1x_5-x_2x_4 \),
\( x_3x_5-x_4^2 \),
\( x_0^3+(4 \zeta_6-2) x_0^2 x_2+x_0 x_2^2+x_3^3+(-4 \zeta_6+2) x_3^2 x_5+x_3 x_5^2\),
\( x_0^2 x_1+(4 \zeta_6-2) x_0 x_1 x_2+x_1 x_2^2+x_3^2 x_4+(-4 \zeta_6+2) x_3 x_4 x_5+x_4 x_5^2\),
\( x_0^2 x_2+(4 \zeta_6-2) x_0 x_2^2+x_2^3+x_3^2 x_5+(-4 \zeta_6+2) x_3 x_5^2+x_5^3\)
Genus-6-72-15.htm
4 (56,7) (2,4,14)
\( (1, 17)(2, 25)(3, 22)(4, 9)(5, 20)(6, 18)(7, 14)(8, 24)(10, 12)(11, 27)(13, 28)(15, 26)(19, 23)\)
\( (1, 12, 3, 18)(2, 14, 5, 9)(4, 6, 7, 10)(8, 20, 13, 25)(11, 17, 15, 22)(16, 27, 21, 26)(19, 24, 23, 28) \)
Hyperelliptic
\( y^2 = x^{14}-1\)
Genus-6-56-7.txt
5 (48,6) (2,4,24)
\( (1, 11)(2, 22)(3, 10)(4, 5)(6, 15)(7, 14)(8, 13)(9, 18)(12, 21)(16, 24)(17, 23)\)
\( (1, 7, 4, 2)(3, 6, 9, 13)(5, 24, 11, 17)(8, 12, 15, 21)(10, 16, 18, 23)(14, 19, 22, 20) \)
Hyperelliptic
\( y^2 = x^{13}-x\)
Genus-6-48-6.txt
6 (48,29) (2,6,8)
\( (1, 11)(2, 22)(3, 10)(4, 5)(6, 15)(7, 14)(8, 13)(9, 18)(12, 21)(16, 24)(17, 23)\)
\( (1, 7, 4, 2)(3, 6, 9, 13)(5, 24, 11, 17)(8, 12, 15, 21)(10, 16, 18, 23)(14, 19, 22, 20)\)
Hyperelliptic
\( y^2 =x(x^4-1) (x^8 + 14x^4+1)\)
Genus-6-48-29.txt
7 (48,15) (2,6,8)
\( (\zeta_8^3 x_5, -i x_4, \zeta_8 x_3, -\zeta_8^3 x_2, i x_1, -\zeta_8 x_0) \)
\((-\zeta_6 x_2,\zeta_6 x_1, -\zeta_6 x_0, -\zeta_3 x_5,\zeta_3 x_4,-\zeta_3 x_3) \)
Cyclic trigonal \( y^3 = (x^4-1)^2(x^4+1) \)
Canonical ideal: \( x_0 x_2 - x_1^2 \),
\( x_0 x_4 - x_1 x_3,\)
\( x_0 x_5-x_1 x_4 \),
\( x_1 x_4 - x_2 x_3\),
\( x_1x_5-x_2x_4 \),
\( x_3x_5-x_4^2 \),
\( x_0 x_1^2 - x_2^3 - x_3 x_4^2 - x_5^3\),
\( x_0^2 x_1 - x_1 x_2^2 - x_3^2 x_4 - x_4 x_5^2 \),
\( x_0^3 - x_1^2 x_2 - x_3^3 - x_4^2 x_5 \)
Genus-6-48-15.htm
8 (39,1) (3,3,13) \( \left[\begin{array}{rrr} 0 & 0 & \zeta_{13}^{12} \\ \zeta_{13}^{4} & 0 & 0 \\ 0 & \zeta_{13}^{10}& 0 \end{array} \right], \quad \left[\begin{array}{rrr} 0 & \zeta_{13}^{7} & 0 \\ 0 & 0 & \zeta_{13}^{11} \\ \zeta_{13}^{8} & 0 & 0 \end{array} \right] \)
Plane quintic
\( y_0^4 y_1+y_1^4 y_2+y_2^4 y_0\)
Genus-6-39-1.htm
9 (30,1) (2,10,15) \( \left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & \zeta_5^4 \\ 0 & \zeta_5 & 0 \end{array} \right], \quad \left[\begin{array}{rrr} \zeta_5^3& 0 & 0 \\ 0 & -\zeta_5 & 0 \\ 0 & -\zeta_5^2 & \zeta_5 \end{array} \right] \)
Plane quintic
\( y_0^5+y_1^4 y_2+2 \zeta_5 y_1^3 y_2^2+2 \zeta_5^2 y_1^2 y_2^3+\zeta_5^3 y_1 y_2^4\)
Genus-6-30-1.htm
10 (26,2) (2,13,26)
\( (1, 2)(3, 5)(4, 6)(7, 9)(8, 10)(11, 13)(12, 14)(15, 17)(16, 18)(19, 21)(20, 22)(23, 25)(24, 26) \)
\( (1, 16, 19, 3, 12, 23, 7, 8, 24, 11, 4, 20, 15)(2, 18, 21, 5, 14, 25, 9, 10, 26, 13, 6, 22, 17)\)
Hyperelliptic
\( y^2 = x^{13}-1\)
Genus-6-26-2.txt
11 (21,2) (3,7,21)
\( (x_0,\zeta_7 x_1, x_2, \zeta_7 x_3,\zeta_7^2 x_4 \zeta_7^3 x_5)\)
\( (\zeta_3^2 x_0, \zeta_3^2 x_1, \zeta_3 x_2, \zeta_3 x_3, \zeta_3 x_4, \zeta_3 x_5) \)
Cyclic trigonal \( y^3 = (x^4-1)^2(x^4+1) \)
Canonical ideal: \( x_0 x_3-x_1 x_2 \),
\( x_0 x_4-x_1 x_3,\)
\( x_0 x_5-x_1 x_4 \),
\( x_2 x_4-x_3^2 \),
\( x_2 x_5-x_3 x_4 \),
\( x_3 x_5-x_4^2 \),
\( x_0 x_1^2 - x_2^3 - x_3 x_4^2 - x_5^3\),
\( x_0^2 x_1 - x_1 x_2^2 - x_3^2 x_4 - x_4 x_5^2 \),
\( x_0^3 - x_1^2 x_2 - x_3^3 - x_4^2 x_5 \)
Genus-6-21-2.htm

Genus 6, \( \delta = 1\)

Locus Group ID Signature Generators Comments and Favorite Equations Additional files
12 (60,5) (2,2,2,3) \( \left[\begin{array}{rrrrrr} -\varphi & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ -\varphi & 0 & \varphi & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & -1 & 0 \end{array} \right], \quad \left[\begin{array}{rrrrrr} 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -\varphi & -\varphi & -1 \end{array} \right], \quad \left[\begin{array}{rrrrrr} -\varphi & \varphi-1 & \varphi-1 & 0 & 0 & 0 \\ -\varphi & \varphi-1 & \varphi & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & -\varphi \\ 0 & 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right] \) Here \( \varphi = \frac{1+\sqrt{5}}{2}\)
\( x_0^2+x_0 x_1+\varphi x_0 x_2+x_1^2+(\varphi-1) x_1 x_2+x_2^2+c_1 (x_3^2-\varphi x_3 x_4-\varphi x_3 x_5+x_4^2+x_4 x_5+x_5^2),\)
\( 24 x_0^2+24 x_0 x_3+(12 \varphi+12) x_0 x_5+(36 \varphi+12) x_1 x_4+(36 \varphi+12) x_1 x_5+(-8 \varphi-16) x_2^2+(-36 \varphi-12) x_2 x_4+(48 \varphi+36) x_2 x_5+(-288 \varphi-180) x_3^2+(396 \varphi+252) x_3 x_4+(540 \varphi+324) x_3 x_5+(-126 \varphi-72) x_4^2+(-252 \varphi-144) x_4 x_5+(-234 \varphi-144) x_5^2,\)
\( 24 x_0 x_1+(-24 \varphi-36) x_0 x_3+(36 \varphi+18) x_0 x_4+(30 \varphi+6) x_0 x_5+(-54 \varphi-36) x_1 x_3+(24 \varphi+18) x_1 x_4+(42 \varphi+18) x_1 x_5+8 x_2^2+(-18 \varphi-18) x_2 x_3+(48 \varphi+18) x_2 x_4+(6 \varphi+6) x_2 x_5+(117 \varphi+72) x_3^2+(-108 \varphi-72) x_3 x_4+(-108 \varphi-54) x_3 x_5+(-18 \varphi-9) x_4^2+(-36 \varphi-18) x_4 x_5+(27 \varphi+18) x_5^2,\)
\(24 x_0 x_2+(6 \varphi-6) x_0 x_3+(-18 \varphi-18) x_0 x_4+12 \varphi x_0 x_5+(-6 \varphi-12) x_1 x_4+(-6 \varphi-12) x_1 x_5+(8 \varphi+8) x_2^2+18 \varphi x_2 x_3+(-12 \varphi-6) x_2 x_4+(-18 \varphi-6) x_2 x_5+(90 \varphi+54) x_3^2+(-18 \varphi-18) x_3 x_4+(-270 \varphi-162) x_3 x_5+(-45 \varphi-27) x_4^2+(72 \varphi+54) x_4 x_5+(153 \varphi+99) x_5^2,\)
\( (-24 \varphi+12) x_0 x_3+(36 \varphi+12) x_0 x_5+24 x_1^2+(36 \varphi+36) x_1 x_3+(-12 \varphi-12) x_1 x_4+(-12 \varphi-12) x_1 x_5+(8 \varphi-24) x_2^2+(-24 \varphi+12) x_2 x_4+(12 \varphi+24) x_2 x_5+(-216 \varphi-126) x_3^2+(360 \varphi+216) x_3 x_4+(216 \varphi+108) x_3 x_5+(-108 \varphi-72) x_4^2+(-216 \varphi-144) x_4 x_5+18 \varphi x_5^2,\)
\( (6 \varphi+12) x_0 x_3+(-24 \varphi-18) x_0 x_5+24 x_1 x_2+(18 \varphi+18) x_1 x_3+(-6 \varphi+6) x_1 x_4+(-24 \varphi-12) x_1 x_5+(8 \varphi-16) x_2^2+(-12 \varphi-6) x_2 x_4+(-18 \varphi-6) x_2 x_5+(-45 \varphi-27) x_3^2+(-18 \varphi-18) x_3 x_4+(108 \varphi+54) x_3 x_5+(90 \varphi+54) x_4^2+18 \varphi x_4 x_5+(-90 \varphi-63) x_5^2\)
Genus-6-60-5.htm
13 (28,3) (2,2,2,7)
\( (1, 2)(3, 4)(5, 7)(6, 8)(9, 11)(10, 12)(13, 14)\)
\((1, 6)(2, 8)(5, 10)(7, 12)(9, 13)(11, 14) \)
\((1, 4)(2, 3)(5, 8)(6, 7)(9, 12)(10, 11)(13, 14)\)
Hyperelliptic
\( y^2 = (x^7-t^7)(x^7-t^{-7}) \)
Genus-6-28-3.txt
14 (24,12) (2,2,3,4) \( \left[\begin{array}{rrrrrr} -1 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 & -1 & -1 \end{array} \right], \quad \left[\begin{array}{rrrrrr} -1 & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \end{array} \right], \quad \left[\begin{array}{rrrrrr} 1 & 1 & 0 & 0 & 0 & 0 \\ -1 & -1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & -1 & -1 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 1 & 0 & 1 \end{array} \right] \)
\( x_0^2 - x_0 x_1 + x_0 x_2 + x_1^2 + x_2^2+c_2 (x_3^2 - x_3 x_5 + x_4^2 - x_4 x_5 + x_5^2), \)
\( x_0 x_3 + x_0 x_4 - \frac{1}{2} x_0 x_5 - \frac{1}{2} x_1 x_3 - \frac{3}{2} x_1 x_4 + x_1 x_5 + \frac{3}{2} x_2 x_3 + \frac{1}{2} x_2 x_4 - x_2 x_5+\frac{\sqrt{3}}{2} (x_3^2 - 2 x_3 x_4 + x_4^2 - x_5^2),\)
\( \frac{1}{4} x_0 x_3 + \frac{1}{4} x_0 x_4 - \frac{1}{2} x_0 x_5 + \frac{1}{4} x_1 x_3 - \frac{3}{4} x_1 x_4 + \frac{1}{4} x_1 x_5 + \frac{3}{4} x_2 x_3 - \frac{1}{4} x_2 x_4 - \frac{1}{4} x_2 x_5+\frac{\sqrt{3}}{2} (\frac{1}{2} x_3^2 + x_3 x_4 - x_3 x_5 + \frac{1}{2} x_4^2 - x_4 x_5),\)
\( x_0^2 + 2 x_0 x_2+ \frac{2}{\sqrt{3}} (x_0 x_3 - x_1 x_5) -2 x_4 x_5 + x_5^2,\)
\( x_0 x_1 + x_0 x_2 + \frac{2}{\sqrt{3}} (\frac{1}{2} x_0 x_3 - \frac{1}{2} x_0 x_4 - \frac{1}{2} x_1 x_5 - \frac{1}{2} x_2 x_5) + x_3 x_5 - x_4 x_5,\)
\( x_1^2 - x_2^2 + \frac{2}{\sqrt{3}} (x_0 x_3 - x_0 x_4 + x_1 x_4 - x_1 x_5 + x_2 x_3 - x_2 x_5) + x_3^2 - x_4^2\)
Genus-6-24-12.htm
15 (24,8) (2,2,3,4) \(\left[ \begin{array}{rrrrrr} 0& 0& 0& 0& 0& \zeta_{12}^5\\ 0& 0& 0& 0& \zeta_6& 0\\ 0& 0& 0& \zeta_{12}^{-1}& 0& 0\\ 0& 0& \zeta_{12}& 0& 0& 0\\ 0& \zeta_6^{-1}& 0& 0& 0& 0\\ \zeta_{12}^{-5}& 0& 0& 0& 0& 0 \end{array} \right], \qquad \left[ \begin{array}{rrrrrr} 0& 0& 1& 0& 0& 0\\ 0& -1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 1& 0& 0] \end{array} \right], \qquad \left[ \begin{array}{rrrrrr} \zeta_{3}& 0& 0& 0& 0& 0\\ 0& \zeta_{3}& 0& 0& 0& 0\\ 0& 0& \zeta_{3}& 0& 0& 0\\ 0& 0& 0& \zeta_{3}^2& 0& 0\\ 0& 0& 0& 0& \zeta_{3}^2& 0\\ 0& 0& 0& 0& 0& \zeta_{3}^2 \end{array} \right]\)
Cyclic trigonal \( y^3 = (x^4-tx^2+1)^2(x_4+tx^2+1) \)
\( 2\times 2\) minors of \( \left[ \begin{array}{rrrr} x_0 & x_1 & x_3 & x_4 \\ x_1 & x_2 & x_4 & x_5 \end{array} \right]\), and
\( x_0^3+t x_0^2 x_2+x_0 x_2^2+x_3^3-t x_3^2 x_5+x_3 x_5^2, \)
\( x_0^2 x_1+t x_0 x_1 x_2+x_1 x_2^2+x_3^2 x_4-t x_3 x_4 x_5+x_4 x_5^2, \)
\( x_0^2 x_2+t x_0 x_2^2+x_2^3+x_3^2 x_5-t x_3 x_5^2+x_5^3 \)
Genus-6-24-8.htm
16 (24,6) (2,2,2,12)
\( (1, 5)(2, 3)(4, 10)(6, 12)(7, 11)(8, 9) \)
\( (1, 12)(2, 11)(3, 8)(4, 7)(5, 10) \)
\( (1, 3)(2, 5)(4, 7)(6, 9)(8, 12)(10, 11) \)
Hyperelliptic
\( y^2 = x(x^6-t^6)(x^6-t^{-6})\)
Genus-6-24-6a.txt
17 (24,6) (2,2,3,4) \( \left[ \begin{array}{rrrrrr} 0 & 0 & 0 & 0 & 0 & -\zeta_{12}^-1 \\ 0 & 0 & 0 & 0 & \zeta_6 & 0 \\ 0 & 0 & 0 & \zeta_{12}^-1 & 0 & 0 \\ 0 & 0 & \zeta_{12} & 0 & 0 & 0 \\ 0 & -\zeta_{3} & 0 & 0 & 0 & 0 \\ -\zeta_{12} & 0 & 0 & 0 & 0 & 0 \end{array} \right], \qquad \left[ \begin{array}{rrrrrr} 0 & 0 & 0 & 0 & 0 & -\zeta_{3} \\ 0 & 0 & 0 & 0 & \zeta_{3} & 0 \\ 0 & 0 & 0 & -\zeta_{3} & 0 & 0 \\ 0 & 0 & \zeta_6 & 0 & 0 & 0 \\ 0 & -\zeta_6 & 0 & 0 & 0 & 0 \\ \zeta_6 & 0 & 0 & 0 & 0 & 0 \end{array} \right], \qquad \left[ \begin{array}{rrrrrr} \zeta_{3}^2 & 0 & 0 & 0 & 0 & 0 \\ 0 & \zeta_{3}^2 & 0 & 0 & 0 & 0 \\ 0 & 0 & \zeta_{3}^2 & 0 & 0 & 0 \\ 0 & 0 & 0 & \zeta_{3} & 0 & 0 \\ 0 & 0 & 0 & 0 & \zeta_{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & \zeta_{3} \end{array} \right] \)
Cyclic trigonal \( y^3 = (x^4-t)^2(t x_4+1) \)
\( 2\times 2\) minors of \( \left[ \begin{array}{rrrr} x_0 & x_1 & x_3 & x_4 \\ x_1 & x_2 & x_4 & x_5 \end{array} \right]\), and
\( x_0 x_2^2+t x_0^3+t x_5^2 x_3+x_3^3,\)
\( x_1 x_2^2+t x_0^2 x_1+t x_5^2 x_4+x_3^2 x_4, \)
\( x_2^3+t x_0^2 x_2+t x_5^3+x_3^2 x_5 \)
Genus-6-24-6b.htm