Curves with Exceptional L-polynomial Distributions

accessibility Curves with Exceptional L-polynomial Distributions

Genus 1 generic curve
y² = x³ + 314159x² + 271828 M[a₁] = (1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, ...)
a₁:

s₂:

s₃:

Genus 1 exceptional case
  y² + y = x³    M[a₁] = (1, 0, 1, 0, 3, 0, 10, 0, 35, 0, 126, ...)
         a₁:

s₂:

s₃:

Genus 2 generic curve
#1)   y² = x⁵ + x + 1
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 3, 0, 14, 0, 84, 0, 594, ...)    M[a₂] = (1, 1, 2, 4, 10, 27, 82, 268, 940, ...)

Genus 2 exceptional cases
#2)   y² = x⁶ + x² + 1
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 10, 0, 70, 0, 588, 0, 5544, ...)    M[a₂] = (1, 2, 5, 14, 44, 152, 569, 2270, 9524, ...)

#3)   y² = x⁶ + 3x⁴ - 2
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 11, 0, 90, 0, 889, 0, 9723, ...)    M[a₂] = (1, 2, 5, 14, 46, 172, 714, 3180, 14858, ...)

#4)   y² = x⁵ + x⁴ - 3x³ - 4x² - x
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 12, 0, 110, 0, 1204, 0, 14364, ...)    M[a₂] = (1, 1, 4, 13, 52, 222, 1014, 4839, 23860, ...)

#5)   y² = x⁶ + x⁴ + x² + 1
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 4, 0, 32, 0, 320, 0, 3584, 0, 43008, ...)    M[a₂] = (1, 3, 10, 37, 150, 654, 3012, 14445, 71398, ...)

#6)   y² = x⁵ + 2x⁴ - x³ - 3x² - x
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 12, 0, 100, 0, 980, 0, 10584, ...)    M[a₂] = (1, 1, 4, 11, 44, 172, 754, 3397, 16020, 77516, ...)

#7)   y² = x⁵ + x⁴ + x² - x
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 12, 0, 100, 0, 1008, 0, 11424, ...)    M[a₂] = (1, 1, 4, 11, 46, 182, 824, 3817, 18582, 92678, ...)

#8)   y² = x⁵ + 3x⁴ + x² - 1
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 12, 0, 110, 0, 1260, 0, 16002, ...)    M[a₂] = (1, 2, 5, 14, 49, 202, 944, 4720, 24553, 130658, ...)

#9)  y² = x⁶ + x⁵ + x - 1
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 5, 0, 35, 0, 294, 0, 2772, ...)    M[a₂] = (1, 1, 3, 7, 23, 76, 287, 1135, 4769, 20788, ...)

#10)   y² = x⁶ + x³ + 4
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 6, 0, 55, 0, 602, 0, 7182, ...)    M[a₂] = (1, 1, 3, 8, 29, 116, 517, 2437, 11965, 60326, ...)

#11)   y² = x⁵ + x³ + x
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1 0 2 0 16 0 160 0 1792 0 21504 ...)    M[a₂] = (1, 2, 6, 20, 78, 332, 1516, 7240, 35734, ...)

#11b)   y² = x⁵ + x⁴ + 2x³ - 2x² -2x + 2
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 16, 0, 160, 0, 1792, 0, 21504, ...)    M[a₂] = (1, 1, 6, 17, 78, 322, 1516, 7205, 35734, ...)

#12)   y² = x⁶ + 4
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 18, 0, 220, 0, 3010, 0, 43092, ...)    M[a₂] = (1, 2, 6, 21, 90, 437, 2285, 12427, 69074, ...)

#12b)   y² = x⁶ + 3x⁵ - 25x³ +30x² - 9x + 1
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 18, 0, 220, 0, 3010, 0, 43092, ...)    M[a₂] = (1, 1, 5, 16, 85, 416, 2264, 12342, 68989, ...)

#13)   y² = x⁶ + 1
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 4, 0, 48, 0, 640, 0, 8960, 0, 129024, ...)    M[a₂] = (1, 1, 11, 40, 235, 1196, 6650, 36632, 205859, ...)

#14)   y² = x⁶ + x³ - 2
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 6, 0, 50, 0, 490, 0, 5292, ...)    M[a₂] = (1, 1, 3, 7, 25, 91, 387, 1716, 8045, 38821, ...)

#15)   y² = x⁶ + 2
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 18, 0, 200, 0, 2450, 0, 31752, ...)    M[a₂] = (1, 2, 6, 20, 82, 372, 1825, 9326, 49026, ...)

#15b)   y² = x⁶ - 15x⁴ - 20x³ + 6x + 1
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 18, 0, 200, 0, 2450, 0, 31752, ...)    M[a₂] = (1, 1, 5, 16, 77, 356, 1804, 9262, 48941, ...)

#16)   y² = x⁵ + x³ + 2x
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 6, 0, 50, 0, 504, 0, 5712, ...)    M[a₂] = (1, 1, 3, 7, 26, 96, 422, 1926, 9326, 46402, ...)

#17)   y² = x⁶ + x⁵ + 10x³ + 5x² + x - 2
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 18, 0, 200, 0, 2520, 0, 34272, ...)    M[a₂] = (1, 2, 6, 20, 83, 382, 1920, 10096, 54787, ...)

#17b)   y² = x⁶ + x⁵ - 5x⁴ - 5x² - x + 1
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 18, 0, 200, 0, 2520, 0, 34272, ...)    M[a₂] = (1, 1, 5, 16, 79, 366, 1904, 10032, 54723, ...)

#18)   y² = x⁵ + x³ - x
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 8, 0, 80, 0, 896, 0, 10752, ...)    M[a₂] = (1, 1, 4, 10, 42, 166, 768, 3620, 17902, 90310, ...)

#19)   y² = x⁵ + 1
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 9, 0, 100, 0, 1225, 0, 15876, ...)    M[a₂] = (1, 1, 3, 10, 41, 186, 912, 4656, 24425, ...)

#20)   y² = x⁶ + 2x³ + 2
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 9, 0, 110, 0, 1505, 0, 21546, ...)    M[a₂] = (1, 1, 4, 10, 48, 216, 1153, 6203, 34576, ...)

#20b)   y² = x⁶ + 6x⁵ - 30x⁴ - 40x³ + 60x² +24x - 8
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 9, 0, 110, 0, 1505, 0, 21546, ...)    M[a₂] = (1, 1, 4, 11, 48, 221, 1153, 6224, 34576,...)

#21)   y² = x⁵ + x
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 24, 0, 320, 0, 4480, 0, 64512, ...)    M[a₂] = (1, 2, 7, 26, 123, 622, 3346, 18412, 103011, ...)

#21b)   y² = x⁵ - x
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 2, 0, 24, 0, 320, 0, 4480, 0, 64512, ...)    M[a₂] = (1, 1, 7, 22, 123, 606, 3346, 18348, 103011, ...)

#22)   y² = x⁵ + 3x
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 9, 0, 100, 0, 1260, 0, 17136, ...)    M[a₂] = (1, 1, 4, 10, 45, 191, 973, 5048, 27443, ...)

#23)   y² = x⁵ + 2x
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 12, 0, 160, 0, 2240, 0, 32256, ...)    M[a₂] = (1, 1, 4, 13, 63, 311, 1678, 9206, 51523, ...)

#23b)   y² = x⁵ + 9x;
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 12, 0, 160, 0, 2240, 0, 32256, ...)    M[a₂] = (1, 1, 5, 13, 67, 311, 1694, 9206, 51587, ...)

#24)   y² = x⁶ + 3x⁵ + 10x³ - 15x² + 15x - 6
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 9, 0, 100, 0, 1225, 0, 15876, ...)    M[a₂] = (1, 1, 4, 10, 44, 186, 923, 4663, 24552, ...)

#25)   y² = x⁶ + 6x⁵ - 20x⁴ + 20x³ - 20x² - 8x + 8
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] = (1, 0, 1, 0, 6, 0, 60, 0, 770, 0, 10836, ...)    M[a₂] = (1, 1, 3, 7, 29, 121, 612, 3200, 17565, ...)

#25b)   y² = x⁶ + 7x⁵ + 10x⁴ + 10x³ + 15x² + 17x + 4
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

         M[a₁] =(1, 0, 1, 0, 6, 0, 60, 0, 770, 0, 10836, ...)    M[a₂] = (1, 1, 3, 8, 30, 126, 617, 3221, 17586, ...)

#26)   y² = x⁶ - 5x⁴ + 10x³ - 5x² + 2x - 1
         a₁:

a₂:

s₂:

s₃:

s₄:

s₅:

M[a₁] = (1, 0, 1, 0, 6, 0, 50, 0, 525, 0, 6426, ...) M[a₂] = (1, 1, 3, 7, 26, 96, 432, 2045, 10432, ...)