A251913 -id:A251913 - cp's OEIS frontend

A378125 Triangle T(n, k) read by rows. Let m be a nonzero rational number then T(n, m mod (n+1)) is the n-th coefficient in the Hasse-Weil L-series (q^(n+1) in the q-expansion) associated to the elliptic equation -4x^3 + ((m+1)^2 + 8)x^2 - 2(m+3)x + 1 - y^2 = 0.

1, -1, -2, 0, -3, -1, 1, 2, 1, 2, -1, -2, -1, -3, 1, 0, 6, 1, 0, 3, 2, 1, -1, -3, 1, -2, -2, -2, -1, 0, -1, 0, -1, 0, -1, 0, 0, 6, -2, 0, 6, -2, 0, 6, -2, 1, 4, 1, 6, -1, 2, 2, 2, 3, -2, -1, -5, 4, 3, 1, -2, -4, -5, -3, -1, 1, 0, -6, -1, 0, -3, -2, 0, -6, -1, 0, -3, -2, 1, -2, -7, 0, 2, -2, -1, 0, -5, -2, -5, 3, 4, -1, 2, 3, -2, 2, 4, 2, -2, 1, 6, -1, 4, 2, 4

Offset: 0

Thomas Scheuerle, Nov 17 2024

Unfortunately, if m is a fraction m = b/c, this triangle can only be used for those coefficients where c and (n+1) are coprime. This is not only because the modulo operation is undefined otherwise, but also because rows of the triangle where (n+1) divides c contain these coefficients with the wrong sign.
The parametrization model for elliptic equations defined by -4*x^3 + ((m+1)^2 + 8)*x^2 - 2*(m+3)*x + 1 - y^2 is also used in A377441. From its relation to Somos-4 sequences, it is known that there is at least one generator point of infinite order if m is an integer > 0 or < -1. If we assume the Birch and Swinnerton-Dyer conjecture to be true, then we expect the associated L-function L(E, s) to be zero at s = 1 for such m.
The relation of m to the J-invariant is given by J(m) = (m^12 + 12*m^11 + 114*m^10 + 628*m^9 + 2823*m^8 + 8184*m^7 + 19036*m^6 + 24552*m^5 + 25407*m^4 + 16956*m^3 + 9234*m^2 + 2916*m + 729)/(m^5 + 4*m^4 + 23*m^3 + 9*m^2) for rational m.
The row sums of the triangle show some connection to the Dedekind psi function (A001615), but will deviate for at least many nonsquarefree n+1.
A short table which shows the Cremona label which corresponds to the L-series obtained for some rational m:
.
  m  | label
 -------------
  -5   655a1
  -4   166a1
  -3   153a1
  -2   58a1
  -1   11a3
 -1/2  26b1
 -1/3  141a1
   1   37a1
   2   158b1
   3   423g1
   4   458a1
   5   1745b1
.

			The triangle T(n, k) begins:
  q^(n+1) 0, 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15     sum  A001615
  --------------------------------------------------------------------
  [q^1]   1                                                   1     1
  [q^2]  -1,-2                                               -3     3
  [q^3]   0,-3,-1                                            -4     4
  [q^4]   1, 2, 1, 2                                          6     6
  [q^5]  -1,-2,-1,-3, 1                                       6     6
  [q^6]   0, 6, 1, 0, 3, 2                                   12    12
  [q^7]   1,-1,-3, 1,-2,-2,-2                                -8     8
  [q^8]  -1, 0,-1, 0,-1, 0,-1, 0                             -4    12 <- not equal
  [q^9]   0, 6,-2, 0, 6,-2, 0, 6,-2                          12    12
  [q^10]  1, 4, 1, 6,-1, 2, 2, 2, 3,-2                       18    18
  [q^11] -1,-5, 4, 3, 1,-2,-4,-5,-3,-1, 1                    12    12
  [q^12]  0,-6,-1, 0,-3,-2, 0,-6,-1, 0,-3,-2                -24    24
  [q^13]  1,-2,-7, 0, 2,-2,-1, 0,-5,-2,-5, 3, 4             -14    14
  [q^14] -1, 2, 3,-2, 2, 4, 2,-2, 1, 6,-1, 4, 2, 4           24    24
  [q^15]  0, 6, 1, 0,-3, 1, 0, 3, 3, 0, 3, 2, 0, 9,-1        24    24
  [q^16]  1,-4, 1,-4, 1,-4, 1,-4, 1,-4, 1,-4, 1,-4, 1,-4     24    24

Cf. A001615, A251913, A377441.

Cf. A006571 (main diagonal), A007653 (column 1).

T(n, k) = ellak(ellinit(ellfromeqn(-4*x^3 + ((k+n+2)^2 + 8)*x^2 - 2*(k+n+4)*x + 1 - y^2)),n+1);

T(n, n) = A006571(n), case m =-1. Also the expansion of (eta(q) * eta(q^11))^2 in powers of q.
T(n, 1) = A007653(n), case m = 1.
T(2*n, n) = A251913(2*n+1), case m = -1/2. See first comment.
Let p be an odd prime with good reduction, then T(p-1, k) is odd iff -4*x^3 + ((k+1)^2 + 8)*x^2 - 2*(k+3)*x + 1 == 0 (mod p) has no solution.

cp's OEIS Frontend

A378125 Triangle T(n, k) read by rows. Let m be a nonzero rational number then T(n, m mod (n+1)) is the n-th coefficient in the Hasse-Weil L-series (q^(n+1) in the q-expansion) associated to the elliptic equation -4x^3 + ((m+1)^2 + 8)x^2 - 2(m+3)x + 1 - y^2 = 0.

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cp's OEIS Frontend

A378125 Triangle T(n, k) read by rows. Let m be a nonzero rational number then T(n, m mod (n+1)) is the n-th coefficient in the Hasse-Weil L-series (q^(n+1) in the q-expansion) associated to the elliptic equation -4*x^3 + ((m+1)^2 + 8)*x^2 - 2*(m+3)*x + 1 - y^2 = 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A378125 Triangle T(n, k) read by rows. Let m be a nonzero rational number then T(n, m mod (n+1)) is the n-th coefficient in the Hasse-Weil L-series (q^(n+1) in the q-expansion) associated to the elliptic equation -4x^3 + ((m+1)^2 + 8)x^2 - 2(m+3)x + 1 - y^2 = 0.