A029937 -id:A029937 - cp's OEIS frontend

A029938 a(n) = (p-5)(p-7)/24, where p=prime(n).

0, 0, 1, 2, 5, 7, 12, 22, 26, 40, 51, 57, 70, 92, 117, 126, 155, 176, 187, 222, 247, 287, 345, 376, 392, 425, 442, 477, 610, 651, 715, 737, 852, 876, 950, 1027, 1080, 1162, 1247, 1276, 1426, 1457, 1520, 1552, 1751

Offset: 3

N. J. A. Sloane

Also the dimension of the space of cusp forms of weight two on Gamma1(p), where p=5, 7, 11, 13, ... ranges over all primes exceeding 3. - Steven Finch, Apr 03 2009

F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg, 2nd ed. 1994, p. 161.

T. D. Noe, Table of n, a(n) for n = 3..1000

Cf. A029937. - Steven Finch, Apr 03 2009

Cf. A000040.

[(p-5)*(p-7)/24: p in PrimesInInterval(4, 300)]; // Vincenzo Librandi, Feb 28 2014

A029938:=n->(ithprime(n)-5)*(ithprime(n)-7)/24; seq(A029938(n), n=3..60); # Wesley Ivan Hurt, Feb 25 2014

((#-5)(#-7))/24&/@Prime[Range[3,60]] (* Harvey P. Dale, Feb 01 2012 *)
Table[(Prime[n] - 5) (Prime[n] - 7)/24, {n, 3, 30}] (* Wesley Ivan Hurt, Feb 25 2014 *)

a(n) = (A000040(n) - 5)*(A000040(n) - 7)/24. - Wesley Ivan Hurt, Feb 25 2014

A146879 Minimal degree of X_1(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 5, 3, 4, 4, 7, 4, 5, 6, 6, 6, 11, 6, 12, 8, 10, 10, 12, 8, 18, 12, 14, 12

Offset: 1

Andrew V. Sutherland, Nov 03 2008

a(n) is the least d>0 for which there exists a plane curve f(x,y)=0 of degree d in x or y which is birationally equivalent to the modular curve X_1(n). There exist infinitely many non-isomorphic elliptic curves defined over number fields of degree a(n) which contain a point of order n. a(n)=1 if and only if X_1(n) has genus 0 and these values of n represent the possible finite orders of a point on an elliptic curve over Q.
By Mazur's theorem, these are 1,2,3,4,5,6,7,8,9,10 and 12. a(n)=2 if and only if X_1(n) is elliptic or hyperelliptic, which occurs only for n=11,13,14,15,16 and 18 [Mestre 1981]. The lower bound a(17)>3 follows from [Parent 1999] and the upper bound a(17)<=4 appears (for example) in [Reichert 1986]. a(20)=3 since it cannot be 1 or 2 and an explicit example of degree 3 is known (see below).
From [Jeon-Kim-Schweizer 2006] it follows that this is the only case when a(n)=3. The results a(21)=4 and a(22)=4 then follow from explicit examples [Sutherland 2008]. a(24) is either 4 or 5 and a(n) is not 4 for any n other than 17, 21, 22, or 24 by the results of [Jeon-Kim-Park 2006]. a(23) must be 5, 6, or 7. See [Sutherland 2008] for these and other upper bounds for n <= 50.
For n = 23 to 40, a(n) has been computed by M. Derickx and M. van Hoeij. For n = 41 to 100, upper bounds for a(n) have been computed by M. van Hoeij (see link). - Mark van Hoeij, Apr 17 2012

			a(20)<=3 because y^3+(x^2+3)y^2+(x^3+4)y+2=0 is an explicit plane model for X_1(20) and a(20)=3 because it is not 1 or 2 (these are all known).

Daeyeol Jeon, Chang Heon Kim and Andreas Schweizer, On the torsion of elliptic curves over cubic number fields, Acta Arithmetica 113 (2004), pp. 291-301.
Mark van Hoeij, Upper bounds
J.-F. Mestre, Corps euclidiens, unités exceptionnelles et courbes elliptiques, J. Number Theory, vol. 13, 1981, pp. 123-137
Markus Reichert, Explicit Determination of Nontrivial Torsion Structures of Elliptic Curves Over Quadratic Number Fields, Math. Comp. 46 (1986), pp. 637-658.
Andrew V. Sutherland, Constructing elliptic curves with prescribed torsion over finite fields, preprint, arXiv:0811.0296 [math.NT], 2008-2012.
A. V. Sutherland, Notes on torsion subgroups of elliptic curves over number fields, 2012. - From _N. J. A. Sloane_, Feb 02 2013
A. V. Sutherland, Torsion subgroups of elliptic curves over number fields, 2012. - From _N. J. A. Sloane_, Feb 03 2013

Cf. A029937.

A159046 Dimension of the space of newforms of weight 2 on the subgroup Gamma_1(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 2, 5, 2, 7, 3, 5, 4, 12, 5, 12, 6, 13, 8, 22, 7, 26, 13, 19, 11, 25, 13, 40, 14, 29, 19, 51, 13, 57, 25, 39, 21, 70, 23, 69, 24, 55, 37, 92, 22, 79, 42, 71, 34, 117, 34, 126, 39, 87, 61, 117, 31, 155, 68, 109, 45, 176, 55, 187, 56, 119, 87

Offset: 1

Steven Finch, Apr 03 2009

nonn

			a(p) = A029937(p) = (p-5)*(p-7)/24 for any prime p>3.
G.f. = x^11 + 2*x^13 + x^14 + x^15 + 2*x^16 + 5*x^17 + 2*x^18 + 7*x^19 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
G. Martin, Dimensions of the spaces of cusp forms and newforms on Gamma_0(N) and Gamma_1(N), J. Numb. Theory 112 (2005) 298-331.

Cf. A007427, A029937, A029938, A127788.

a[ n_] := If[ n < 1, 0, Sum[ DivisorSum[ n/j, MoebiusMu[#] MoebiusMu[n/j/#] &] If[ j < 5, 0, 1 + DivisorSum[ j, #^2 MoebiusMu[ j/#] / 24 - EulerPhi [#] EulerPhi[j/#] / 4 &]], {j, Divisors@n}]]; (* Michael Somos, May 10 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, j, sumdiv(n/j, k, moebius(k) * moebius(n/j/k)) * if( j<5, 0, 1 + sumdiv(j, k, k^2 * moebius(j/k) / 24 - eulerphi(k) * eulerphi(j/k) / 4))))}; /* Michael Somos, May 10 2015 */

a(n) = A029937(n) - sum a(m)*d(n/m), where the summation is over all divisors 1 < m < n of n and d is the divisor function.

Dirichlet convolution of A007247 and A029937. - Michael Somos, May 10 2015

A159050 Dimension of the space of weight 2 modular forms for Gamma_1(n).

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 5, 5, 7, 7, 10, 9, 13, 12, 16, 15, 20, 17, 24, 22, 28, 25, 33, 28, 39, 33, 42, 39, 49, 40, 55, 48, 60, 52, 72, 56, 75, 63, 80, 72, 90, 72, 98, 85, 104, 88, 115, 92, 128, 103, 128, 114, 143, 111, 160, 132, 156, 133, 174, 136, 185, 150, 192, 164, 216

Offset: 1

Steven Finch, Apr 03 2009

nonn

a(n) = A029937(n) + A029936(n) - 1.

Klaus Brockhaus, Table of n, a(n) for n = 1..1000
S. R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
W. A. Stein, Modular Forms: A Computational Approach.

Cf. A029936, A029937, A111248.

[ Dimension(ModularForms(Gamma1(n), 2)) : n in [1..100] ]; // Klaus Brockhaus, Mar 11 2011

def A159050(n): return dimension_modular_forms(Gamma1(n),k=2) # D. S. McNeil, Mar 11 2011

A365138 Genus of the quotient of the modular curve X_1(n) by the Fricke involution.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 3, 1, 2, 3, 5, 2, 6, 4, 6, 5, 10, 4, 12, 8, 10, 10, 11, 8, 20, 13, 15, 12, 24, 12, 28, 17, 20, 22, 33, 18, 34, 23, 31, 27, 45, 25, 39, 29, 42, 39, 56, 28, 62, 44, 47, 46, 59, 39, 77, 51, 65, 48, 85, 48, 93, 66, 71, 67, 89, 60, 109

Offset: 1

David Jao, Aug 23 2023

nonn

C. H. Kim and J. K. Koo, Estimation of Genus for Certain Arithmetic Groups, Communications in Algebra, 32:7 (2004), 2479-2495.

Cf. A000003, A001617, A029937, A276183.

A000003[n_] :=
 Length[Select[
   Flatten[#, 1] &@
    Table[{i, j, (j^2 + 4 n)/(4 i)}, {i, Sqrt[4 n/3]}, {j, 1 - i, i}],
    Mod[#3, 1] == 0 && #3 >= # &&
       GCD[##] == 1 && ! (# == #3 && #2 < 0) & @@ # &]];
A001617[n_] :=
  If[n < 1, 0,
   1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d,
      Divisors@n}] -
    Count[(#^2 - # + 1)/n & /@ Range[n], _?IntegerQ]/3 -
    Count[(#^2 + 1)/n & /@ Range[n], _?IntegerQ]/4];
A029937[n_] =
  If[n < 5, 0,
   1 + Sum[d^2*MoebiusMu[n/d]/24 - EulerPhi[d]*EulerPhi[n/d]/4, {d,
      Divisors[n]}]];
A276183[n_] :=
 If[0 <= n <= 4,
  0, (A001617[n] + 1)/2 -
   If[Mod[n, 8] == 3, 4, If[Mod[n, 8] == 7, 6, 3]]*A000003[n]/12];
A365138[n_] := (A029937[n] - A001617[n])/2 + A276183[n]

a(n) = (A029937(n) - A001617(n))/2 + A276183(n).

cp's OEIS Frontend

A029938 a(n) = (p-5)(p-7)/24, where p=prime(n).

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A146879 Minimal degree of X_1(n).

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A159046 Dimension of the space of newforms of weight 2 on the subgroup Gamma_1(n).

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PARI

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A159050 Dimension of the space of weight 2 modular forms for Gamma_1(n).

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Magma

Sage

A365138 Genus of the quotient of the modular curve X_1(n) by the Fricke involution.

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