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A272207 Number of solutions to the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod p) as p runs through the primes. 5
2, 5, 6, 5, 11, 11, 23, 23, 17, 23, 35, 35, 35, 53, 53, 59, 47, 59, 65, 83, 71, 71, 77, 95, 95, 95, 89, 113, 107, 119, 125, 131, 119, 143, 155, 131, 179, 173, 149, 179, 191, 191, 203, 167, 179, 191, 227, 233, 233, 215, 239, 263, 227, 251, 263, 281, 251, 251, 251, 275 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the fourth row, starting with conductor 20, as a strong Weil curve for the weight 2 newform (eta(2*z)*eta(10*z))^2, symbolically 2^2 10^2, with Im(z) > 0, and the Dedekind eta function. See A030205 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z)*(eta(z)*eta(5*z))^2. For the q-expansion of (eta(2*z)*eta(10*z))^2 one has interspersed 0's: 0, 1, 0, -2, 0, -1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 2, 0, -6, ... This modular cusp form of weight 2 appears as the 39th entry in Martin's Table I.
For the p-defect prime(n) - a(n) see A273163(n), n >= 1.
The discriminant of this elliptic curve is -400 = -2^4*5^2 (bad primes 2 and 5, also the prime divisors of the conductor).
The congruence y^2 == x^3 + x^2 - x has the same number of solutions modulo prime(n). See a comment on A030205. The discriminant equals +5.
LINKS
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
FORMULA
a(n) gives the number of solutions of the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod prime(n)), n >= 1.
a(n) gives also the number of solutions of the congruence y^2 == x^3 + x^2 - x (mod prime(n)), n >= 1.
EXAMPLE
The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.
The solutions (x, y) of y^2 == x^3 + x^2 + 4*x + 4 (mod prime(n)) begin:
n, prime(n), a(n)\ solutions (x, y)
1, 2, 2: (0, 0), (1, 0)
2, 3, 5: (0, 1), (0, 2), (1, 1),
(1, 2) (2, 0)
3, 5, 6: (0, 2), (0, 3), (1, 0),
(2, 2), (2, 3), (4, 0)
4, 7, 5: (0, 2), (0, 5), (4, 3),
(4, 4), (6, 0)
5, 11, 11: (0, 2), (0, 9), (4, 1),
(4, 10), (5, 3), (5, 8),
(6, 4), (6, 7), (9, 5),
(9, 6), (10, 0)
...
The solutions (x, y) of y^2 == x^3 + x^2 - x (mod prime(n)) begin:
n, prime(n), a(n)\ solutions (x, y)
1, 2, 2: (0, 0), (1, 1)
2, 3, 5: (0, 0), (1, 1), (1, 2),
(2, 1) (2, 2)
3, 5, 6: (0, 0), (1, 1), (1, 4),
(2, 0), (4, 1), (4, 4)
4, 7, 5: (0, 0), (1, 1), (1, 6),
(6, 1), (6, 6)
5, 11, 11: (0, 0), (1, 1), (1, 10),
(3, 0), (6, 2), (6, 9),
(7, 0), (9, 3), (9, 8),
(10, 1), (10, 10)
...
CROSSREFS
Sequence in context: A239049 A161017 A198231 * A155947 A008294 A019694
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, May 20 2016
STATUS
approved

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