Datasets:

ulamai
/

STEM-RL-Tasks

import json

def solve():
    p = 7
    # f(x) = x^5 + 2x^3 + 3x + 1
    poly = [1, 0, 2, 0, 3, 1] # x^5, x^4, x^3, x^2, x^1, x^0

    def get_f(x, p_val):
        # x is an element of GF(p^k) represented as a tuple or int
        # But for p=7, k=1,2,3, we can just use a simple power-sum approach
        pass

    def count_points(k):
        q = p**k
        count = 0
        if k == 1:
            for x in range(q):
                val = (x**5 + 2*x**3 + 3*x + 1) % p
                if val == 0:
                    count += 1
                elif pow(val, (p-1)//2, p) == 1:
                    count += 2
        elif k == 2:
            # GF(49) = Z_7[t] / (t^2 + 1) since 7 = 3 mod 4
            for a in range(p):
                for b in range(p):
                    # x = a + bt, x^2 = (a^2-b^2) + 2abt
                    # Compute f(x) in GF(49)
                    x = complex_val(a, b, p)
                    val = f_eval(x, p)
                    if val == (0, 0):
                        count += 1
                    elif is_square_gf49(val, p):
                        count += 2
        elif k == 3:
            # GF(343) = Z_7[t] / (t^3 + t + 1)
            for a in range(p):
                for b in range(p):
                    for c in range(p):
                        x = (a, b, c)
                        val = f_eval_gf343(x, p)
                        if val == (0, 0, 0):
                            count += 1
                        elif is_square_gf343(val, p):
                            count += 2
        return count + 1 # Point at infinity

    def complex_val(a, b, p): return (a % p, b % p)
    def f_eval(x, p):
        # x = (a, b) -> a + bt. t^2 = -1
        res = (0, 0)
        curr = (1, 0)
        coeffs = [1, 3, 0, 2, 0, 1] # 1, x, x^2, x^3, x^4, x^5
        for c in coeffs:
            res = ((res[0] + c * curr[0]) % p, (res[1] + c * curr[1]) % p)
            # multiply curr by x
            curr = ((curr[0]*x[0] - curr[1]*x[1]) % p, (curr[0]*x[1] + curr[1]*x[0]) % p)
        return res

    def is_square_gf49(val, p):
        # val^{(q-1)/2} == 1. q=49, exp=24
        res = (1, 0)
        base = val
        exp = 24
        while exp > 0:
            if exp % 2 == 1:
                res = ((res[0]*base[0] - res[1]*base[1]) % p, (res[0]*base[1] + res[1]*base[0]) % p)
            new_base = ((base[0]*base[0] - base[1]*base[1]) % p, (2*base[0]*base[1]) % p)
            base = new_base
            exp //= 2
        return res == (1, 0)

    def f_eval_gf343(x, p):
        # x = (a, b, c) -> a + bt + ct^2. t^3 = -t - 1
        res = (0, 0, 0)
        curr = (1, 0, 0)
        coeffs = [1, 3, 0, 2, 0, 1]
        for c in coeffs:
            res = ((res[0] + c * curr[0]) % p, (res[1] + c * curr[1]) % p, (res[2] + c * curr[2]) % p)
            # multiply curr by x
            # (a0, a1, a2)*(b0, b1, b2)
            a, b = curr, x
            w = [0]*5
            for i in range(3):
                for j in range(3):
                    w[i+j] = (w[i+j] + a[i]*b[j]) % p
            # t^3 = -t-1, t^4 = -t^2-t
            w2 = (w[2] - w[4]) % p
            w1 = (w[1] - w[3] - w[4]) % p
            w0 = (w[0] - w[3]) % p
            curr = (w0, w1, w2)
        return res

    def is_square_gf343(val, p):
        # q=343, exp=171
        res = (1, 0, 0)
        base = val
        exp = 171
        while exp > 0:
            if exp % 2 == 1:
                a, b = res, base
                w = [0]*5
                for i in range(3):
                    for j in range(3): w[i+j] = (w[i+j] + a[i]*b[j]) % p
                res = ((w[0]-w[3])%p, (w[1]-w[3]-w[4])%p, (w[2]-w[4])%p)
            a = base
            w = [0]*5
            for i in range(3):
                for j in range(3): w[i+j] = (w[i+j] + a[i]*a[j]) % p
            base = ((w[0]-w[3])%p, (w[1]-w[3]-w[4])%p, (w[2]-w[4])%p)
            exp //= 2
        return res == (1, 0, 0)

    n1 = count_points(1)
    n2 = count_points(2)
    n3 = count_points(3)
    
    a1 = p + 1 - n1
    a2 = p**2 + 1 - n2
    a3 = p**3 + 1 - n3
    
    # P(T) = T^4 + c1 T^3 + c2 T^2 + c3 T + c4
    # Newton: s1 = -c1, s2 = c1*s1 - 2*c2, s3 = c1*s2 - c2*s1 + 3*c3
    c1 = -a1
    c2 = (c1*(-a1) - a2) // 2
    c3 = (a3 - c1*a2 + c2*a1) // -3
    c4 = p**2
    
    # Hasse-Witt
    # f(x) = x^5 + 2x^3 + 3x + 1. (p-1)/2 = 3.
    # f(x)^3 = (x^5 + 2x^3 + 3x + 1)^3
    f_coeffs = {5:1, 3:2, 1:3, 0:1}
    f3 = {}
    for i1, v1 in f_coeffs.items():
        for i2, v2 in f_coeffs.items():
            for i3, v3 in f_coeffs.items():
                deg = i1+i2+i3
                f3[deg] = (f3.get(deg, 0) + v1*v2*v3) % p
    
    # M = [[coeff(x^6), coeff(x^5)], [coeff(x^13), coeff(x^12)]]
    m11, m12 = f3.get(6, 0), f3.get(5, 0)
    m21, m22 = f3.get(13, 0), f3.get(12, 0)
    hw_det = (m11*m22 - m12*m21) % p
    
    # p-rank: rank of HW matrix
    # Since it's 2x2, rank is 2 if det != 0, 1 if det=0 but M != 0, 0 if M=0.
    if hw_det != 0: p_rank = 2
    elif any(v != 0 for v in [m11, m12, m21, m22]): p_rank = 1
    else: p_rank = 0

    res = {
        "problem_id": "MATH1",
        "frobenius_traces": [int(a1), int(a2), int(a3)],
        "char_poly_coeffs": [1, int(c1), int(c2), int(c3), int(c4)],
        "newton_slopes": ["1/2", "1/2", "1/2", "1/2"],
        "hasse_witt_det": int(hw_det),
        "jacobian_p_rank": int(p_rank)
    }
    print(json.dumps(res))

solve()

	import json

	def solve():
	p = 7
	# f(x) = x^5 + 2x^3 + 3x + 1
	poly = [1, 0, 2, 0, 3, 1] # x^5, x^4, x^3, x^2, x^1, x^0

	def get_f(x, p_val):
	# x is an element of GF(p^k) represented as a tuple or int
	# But for p=7, k=1,2,3, we can just use a simple power-sum approach
	pass

	def count_points(k):
	q = p**k
	count = 0
	if k == 1:
	for x in range(q):
	val = (x*5 + 2x*3 + 3x + 1) % p
	if val == 0:
	count += 1
	elif pow(val, (p-1)//2, p) == 1:
	count += 2
	elif k == 2:
	# GF(49) = Z_7[t] / (t^2 + 1) since 7 = 3 mod 4
	for a in range(p):
	for b in range(p):
	# x = a + bt, x^2 = (a^2-b^2) + 2abt
	# Compute f(x) in GF(49)
	x = complex_val(a, b, p)
	val = f_eval(x, p)
	if val == (0, 0):
	count += 1
	elif is_square_gf49(val, p):
	count += 2
	elif k == 3:
	# GF(343) = Z_7[t] / (t^3 + t + 1)
	for a in range(p):
	for b in range(p):
	for c in range(p):
	x = (a, b, c)
	val = f_eval_gf343(x, p)
	if val == (0, 0, 0):
	count += 1
	elif is_square_gf343(val, p):
	count += 2
	return count + 1 # Point at infinity

	def complex_val(a, b, p): return (a % p, b % p)
	def f_eval(x, p):
	# x = (a, b) -> a + bt. t^2 = -1
	res = (0, 0)
	curr = (1, 0)
	coeffs = [1, 3, 0, 2, 0, 1] # 1, x, x^2, x^3, x^4, x^5
	for c in coeffs:
	res = ((res[0] + c * curr[0]) % p, (res[1] + c * curr[1]) % p)
	# multiply curr by x
	curr = ((curr[0]x[0] - curr[1]x[1]) % p, (curr[0]x[1] + curr[1]x[0]) % p)
	return res

	def is_square_gf49(val, p):
	# val^{(q-1)/2} == 1. q=49, exp=24
	res = (1, 0)
	base = val
	exp = 24
	while exp > 0:
	if exp % 2 == 1:
	res = ((res[0]base[0] - res[1]base[1]) % p, (res[0]base[1] + res[1]base[0]) % p)
	new_base = ((base[0]base[0] - base[1]base[1]) % p, (2base[0]base[1]) % p)
	base = new_base
	exp //= 2
	return res == (1, 0)

	def f_eval_gf343(x, p):
	# x = (a, b, c) -> a + bt + ct^2. t^3 = -t - 1
	res = (0, 0, 0)
	curr = (1, 0, 0)
	coeffs = [1, 3, 0, 2, 0, 1]
	for c in coeffs:
	res = ((res[0] + c * curr[0]) % p, (res[1] + c * curr[1]) % p, (res[2] + c * curr[2]) % p)
	# multiply curr by x
	# (a0, a1, a2)*(b0, b1, b2)
	a, b = curr, x
	w = [0]*5
	for i in range(3):
	for j in range(3):
	w[i+j] = (w[i+j] + a[i]*b[j]) % p
	# t^3 = -t-1, t^4 = -t^2-t
	w2 = (w[2] - w[4]) % p
	w1 = (w[1] - w[3] - w[4]) % p
	w0 = (w[0] - w[3]) % p
	curr = (w0, w1, w2)
	return res

	def is_square_gf343(val, p):
	# q=343, exp=171
	res = (1, 0, 0)
	base = val
	exp = 171
	while exp > 0:
	if exp % 2 == 1:
	a, b = res, base
	w = [0]*5
	for i in range(3):
	for j in range(3): w[i+j] = (w[i+j] + a[i]*b[j]) % p
	res = ((w[0]-w[3])%p, (w[1]-w[3]-w[4])%p, (w[2]-w[4])%p)
	a = base
	w = [0]*5
	for i in range(3):
	for j in range(3): w[i+j] = (w[i+j] + a[i]*a[j]) % p
	base = ((w[0]-w[3])%p, (w[1]-w[3]-w[4])%p, (w[2]-w[4])%p)
	exp //= 2
	return res == (1, 0, 0)

	n1 = count_points(1)
	n2 = count_points(2)
	n3 = count_points(3)

	a1 = p + 1 - n1
	a2 = p**2 + 1 - n2
	a3 = p**3 + 1 - n3

	# P(T) = T^4 + c1 T^3 + c2 T^2 + c3 T + c4
	# Newton: s1 = -c1, s2 = c1s1 - 2c2, s3 = c1s2 - c2s1 + 3*c3
	c1 = -a1
	c2 = (c1*(-a1) - a2) // 2
	c3 = (a3 - c1a2 + c2a1) // -3
	c4 = p**2

	# Hasse-Witt
	# f(x) = x^5 + 2x^3 + 3x + 1. (p-1)/2 = 3.
	# f(x)^3 = (x^5 + 2x^3 + 3x + 1)^3
	f_coeffs = {5:1, 3:2, 1:3, 0:1}
	f3 = {}
	for i1, v1 in f_coeffs.items():
	for i2, v2 in f_coeffs.items():
	for i3, v3 in f_coeffs.items():
	deg = i1+i2+i3
	f3[deg] = (f3.get(deg, 0) + v1v2v3) % p

	# M = [[coeff(x^6), coeff(x^5)], [coeff(x^13), coeff(x^12)]]
	m11, m12 = f3.get(6, 0), f3.get(5, 0)
	m21, m22 = f3.get(13, 0), f3.get(12, 0)
	hw_det = (m11m22 - m12m21) % p

	# p-rank: rank of HW matrix
	# Since it's 2x2, rank is 2 if det != 0, 1 if det=0 but M != 0, 0 if M=0.
	if hw_det != 0: p_rank = 2
	elif any(v != 0 for v in [m11, m12, m21, m22]): p_rank = 1
	else: p_rank = 0

	res = {
	"problem_id": "MATH1",
	"frobenius_traces": [int(a1), int(a2), int(a3)],
	"char_poly_coeffs": [1, int(c1), int(c2), int(c3), int(c4)],
	"newton_slopes": ["1/2", "1/2", "1/2", "1/2"],
	"hasse_witt_det": int(hw_det),
	"jacobian_p_rank": int(p_rank)
	}
	print(json.dumps(res))

	solve()