$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2\t^2 = 0\}?$

Assume that $ab \neq 0$. What is $$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2\}?$$$$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$$I need this result, but unfortunately I am not a number theorist. Could anyone provide me a reference/supply a computation? Thanks!

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edited Oct 16, 2015 at 21:04

Antonio

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$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^2^4 : 2y^2 = x^4 - 17\ax^4 + by^4 + z^2 + t^2\}?$

For each prime number $p \neq 2$,Assume that $17$, what$ab \neq 0$. What is the number$$N_p := \text{card}\{(x, y) \in (\textbf{F}_p)^2 : 2y^2 = x^4 - 17\}?$$$$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2\}?$$I need this result, but unfortunately I am not a number theorist. Could anyone provide me a reference/supply a computation? Thanks!

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asked Oct 16, 2015 at 20:55

Antonio

75
3

$N_p := \text{card}\{(x, y) \in (\textbf{F}_p)^2 : 2y^2 = x^4 - 17\}?$

For each prime number $p \neq 2$, $17$, what is the number$$N_p := \text{card}\{(x, y) \in (\textbf{F}_p)^2 : 2y^2 = x^4 - 17\}?$$I need this result, but unfortunately I am not a number theorist. Could anyone provide me a reference/supply a computation? Thanks!

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$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2\t^2 = 0\}?$

$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2\}?$

$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$

$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^2^4 : 2y^2 = x^4 - 17\ax^4 + by^4 + z^2 + t^2\}?$

$N_p := \text{card}\{(x, y) \in (\textbf{F}_p)^2 : 2y^2 = x^4 - 17\}?$

$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2\}?$

$N_p := \text{card}\{(x, y) \in (\textbf{F}_p)^2 : 2y^2 = x^4 - 17\}?$