Module of Supersingular Points¶

The module of divisors on the modular curve $X_0(N)$ over $F_p$ supported at supersingular points.

AUTHORS:

William Stein
David Kohel
Iftikhar Burhanuddin

EXAMPLES:

sage: x = SupersingularModule(389)
sage: m = x.T(2).matrix()
sage: a = m.change_ring(GF(97))
sage: D = a.decomposition()
sage: D[:3]
[
(Vector space of degree 33 and dimension 1 over Finite Field of size 97
Basis matrix:
[ 0  0  0  1 96 96  1 96 96  0  2 96 96  0  1  0  1  2 95  0  1  1  0  1  0 95  0 96 95  1 96  0  2], True),
(Vector space of degree 33 and dimension 1 over Finite Field of size 97
Basis matrix:
[ 0  1 96 75 16 81 22 17 17  0  0 80 80  1 16 40 74  0  0 96 81 23 57 74  0  0  0 24  0 23 73  0  0], True),
(Vector space of degree 33 and dimension 1 over Finite Field of size 97
Basis matrix:
[ 0  1 96 90 90  7  7  6 91  0  0 91  6 13  7  0 91  0  0 84 90  6  0  6  0  0  0 90  0 91  7  0  0], True)
]
sage: len(D)
9

We compute a Hecke operator on a space of huge dimension!:

sage: X = SupersingularModule(next_prime(100000))
sage: t = X.T(2).matrix()            # long time (but still less than a minute!)
sage: t.nrows()                      # long time
8334

TESTS:

sage: X = SupersingularModule(389)
sage: T = X.T(2).matrix().change_ring(QQ)
sage: d = T.decomposition()
sage: len(d)
6
sage: [a[0].dimension() for a in d]
[1, 1, 2, 3, 6, 20]
sage: loads(dumps(X)) == X
True
sage: loads(dumps(d)) == d
True

sage.modular.ssmod.ssmod.Phi2_quad(J3, ssJ1, ssJ2)¶

This function returns a certain quadratic polynomial over a finite field in indeterminate J3.

The roots of the polynomial along with ssJ1 are the neighboring/2-isogenous supersingular j-invariants of ssJ2.

INPUT:

J3 – indeterminate of a univariate polynomial ring defined over a finite field with p^2 elements where p is a prime number
ssJ2, ssJ2 – supersingular j-invariants over the finite field

OUTPUT:

polynomial – defined over the finite field

EXAMPLES: The following code snippet produces a factor of the modular polynomial $\Phi_{2}(x,j_{in})$ , where $j_{in}$ is a supersingular j-invariant defined over the finite field with $37^2$ elements:

sage: F = GF(37^2, 'a')
sage: X = PolynomialRing(F, 'x').gen()
sage: j_in = supersingular_j(F)
sage: poly = sage.modular.ssmod.ssmod.Phi_polys(2,X,j_in)
sage: poly.roots()
[(8, 1), (27*a + 23, 1), (10*a + 20, 1)]
sage: sage.modular.ssmod.ssmod.Phi2_quad(X, F(8), j_in)
x^2 + 31*x + 31

Note

Given a root (j1,j2) to the polynomial $Phi_2(J1,J2)$ , the pairs (j2,j3) not equal to (j2,j1) which solve $Phi_2(j2,j3)$ are roots of the quadratic equation:

J3^2 + (-j2^2 + 1488*j2 + (j1 - 162000))*J3 + (-j1 + 1488)*j2^2 + (1488*j1 + 40773375)*j2 + j1^2 - 162000*j1 + 8748000000

This will be of use to extend the 2-isogeny graph, once the initial three roots are determined for $Phi_2(J1,J2)$ .

AUTHORS:

David Kohel – kohel@maths.usyd.edu.au
Iftikhar Burhanuddin – burhanud@usc.edu

sage.modular.ssmod.ssmod.Phi_polys(L, x, j)¶

This function returns a certain polynomial of degree $L+1$ in the indeterminate x over a finite field.

The roots of the modular polynomial $\Phi(L, x, j)$ are the $L$ -isogenous supersingular j-invariants of j.

INPUT:

L – integer, either 2,3,5,7 or 11
x – indeterminate of a univariate polynomial ring defined over a finite field with p^2 elements, where p is a prime number
j – supersingular j-invariant over the finite field

OUTPUT:

polynomial – defined over the finite field

EXAMPLES: The following code snippet produces the modular polynomial $\Phi_{L}(x,j_{in})$ , where $j_{in}$ is a supersingular j-invariant defined over the finite field with $7^2$ elements:

sage: F = GF(7^2, 'a')
sage: X = PolynomialRing(F, 'x').gen()
sage: j_in = supersingular_j(F)
sage: sage.modular.ssmod.ssmod.Phi_polys(2,X,j_in)
x^3 + 3*x^2 + 3*x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(3,X,j_in)
x^4 + 4*x^3 + 6*x^2 + 4*x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(5,X,j_in)
x^6 + 6*x^5 + x^4 + 6*x^3 + x^2 + 6*x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(7,X,j_in)
x^8 + x^7 + x + 1
sage: sage.modular.ssmod.ssmod.Phi_polys(11,X,j_in)
x^12 + 5*x^11 + 3*x^10 + 3*x^9 + 5*x^8 + x^7 + x^5 + 5*x^4 + 3*x^3 + 3*x^2 + 5*x + 1

AUTHORS:

William Stein – wstein@gmail.com
David Kohel – kohel@maths.usyd.edu.au
Iftikhar Burhanuddin – burhanud@usc.edu

class sage.modular.ssmod.ssmod.SupersingularModule(prime=2, level=1, base_ring=Integer Ring)¶

Bases: sage.modular.hecke.module.HeckeModule_free_module

The module of supersingular points in a given characteristic, with given level structure.

The characteristic must not divide the level.

NOTE: Currently, only level 1 is implemented.

EXAMPLES:

sage: S = SupersingularModule(17)
sage: S
Module of supersingular points on X_0(1)/F_17 over Integer Ring
sage: S = SupersingularModule(16)
...
ValueError: the argument prime must be a prime number
sage: S = SupersingularModule(prime=17, level=34)
...
ValueError: the argument level must be coprime to the argument prime
sage: S = SupersingularModule(prime=17, level=5)
...
NotImplementedError: supersingular modules of level > 1 not yet implemented

dimension()¶

Return the dimension of the space of modular forms of weight 2 and level equal to the level associated to self.

INPUT:

self – SupersingularModule object

OUTPUT:: integer – dimension, nonnegative

EXAMPLES:

sage: S = SupersingularModule(7)
sage: S.dimension()
1

sage: S = SupersingularModule(15073)
sage: S.dimension()
1256

sage: S = SupersingularModule(83401)
sage: S.dimension()
6950

NOTES:: The case of level > 1 has not yet been implemented.

AUTHORS:

David Kohel – kohel@maths.usyd.edu.au
Iftikhar Burhanuddin – burhanud@usc.edu

hecke_matrix(L)¶

This function returns the $L^{\text{th}}$ Hecke matrix.

INPUT:

self – SupersingularModule object
L – integer, positive

OUTPUT:: matrix – sparse integer matrix

EXAMPLES: This example computes the action of the Hecke operator $T_2$ on the module of supersingular points on $X_0(1)/F_{37}$ :

sage: S = SupersingularModule(37)
sage: M = S.hecke_matrix(2)
sage: M
[1 1 1]
[1 0 2]
[1 2 0]

This example computes the action of the Hecke operator $T_3$ on the module of supersingular points on $X_0(1)/F_{67}$ :

sage: S = SupersingularModule(67)
sage: M = S.hecke_matrix(3)
sage: M
[0 0 0 0 2 2]
[0 0 1 1 1 1]
[0 1 0 2 0 1]
[0 1 2 0 1 0]
[1 1 0 1 0 1]
[1 1 1 0 1 0]            

Note

The first list — list_j — returned by the supersingular_points function are the rows and column indexes of the above hecke matrices and its ordering should be kept in mind when interpreting these matrices.

AUTHORS:

David Kohel – kohel@maths.usyd.edu.au
Iftikhar Burhanuddin – burhanud@usc.edu

level()¶

This function returns the level associated to self.

INPUT:

self – SupersingularModule object

OUTPUT:: integer – the level, positive

EXAMPLES:

sage: S = SupersingularModule(15073)
sage: S.level()
1

AUTHORS:

David Kohel – kohel@maths.usyd.edu.au
Iftikhar Burhanuddin – burhanud@usc.edu

prime()¶

This function returns the characteristic of the finite field associated to self.

INPUT:

self – SupersingularModule object

OUTPUT:

integer – characteristic, positive

EXAMPLES:

sage: S = SupersingularModule(19)
sage: S.prime()
19

AUTHORS:

David Kohel – kohel@maths.usyd.edu.au
Iftikhar Burhanuddin – burhanud@usc.edu

rank()¶

Return the dimension of the space of modular forms of weight 2 and level equal to the level associated to self.

INPUT:

self – SupersingularModule object

OUTPUT:: integer – dimension, nonnegative

EXAMPLES:

sage: S = SupersingularModule(7)
sage: S.dimension()
1

sage: S = SupersingularModule(15073)
sage: S.dimension()
1256

sage: S = SupersingularModule(83401)
sage: S.dimension()
6950

NOTES:: The case of level > 1 has not yet been implemented.

AUTHORS:

David Kohel – kohel@maths.usyd.edu.au
Iftikhar Burhanuddin – burhanud@usc.edu

supersingular_points()¶

This function computes the supersingular j-invariants over the finite field associated to self.

INPUT:

self – SupersingularModule object

OUTPUT: list_j, dict_j – list_j is the list of supersingular: j-invariants, dict_j is a dictionary with these j-invariants as keys and their indexes as values. The latter is used to speed up j-invariant look-up. The indexes are based on the order of their discovery.

EXAMPLES:

The following examples calculate supersingular j-invariants over finite fields with characteristic 7, 11 and 37:

sage: S = SupersingularModule(7)
sage: S.supersingular_points()
([6], {6: 0})

sage: S = SupersingularModule(11)
sage: S.supersingular_points()
([1, 0], {0: 1, 1: 0})

sage: S = SupersingularModule(37)
sage: S.supersingular_points()
([8, 27*a + 23, 10*a + 20], {8: 0, 10*a + 20: 2, 27*a + 23: 1})

AUTHORS:

David Kohel – kohel@maths.usyd.edu.au
Iftikhar Burhanuddin – burhanud@usc.edu

upper_bound_on_elliptic_factors(p=None, ellmax=2)¶

Return an upper bound (provably correct) on the number of elliptic curves of conductor equal to the level of this supersingular module.

INPUT:

p - (default: 997) prime to work modulo

ALGORITHM: Currently we only use $T_2$ . Function will be extended to use more Hecke operators later.

The prime p is replaced by the smallest prime that doesn’t divide the level.

EXAMPLE:

sage: SupersingularModule(37).upper_bound_on_elliptic_factors()
2

(There are 4 elliptic curves of conductor 37, but only 2 isogeny classes.)

weight()¶

This function returns the weight associated to self.

INPUT:

self – SupersingularModule object

OUTPUT:: integer – weight, positive

EXAMPLES:

sage: S = SupersingularModule(19)
sage: S.weight()
2

AUTHORS:

David Kohel – kohel@maths.usyd.edu.au
Iftikhar Burhanuddin – burhanud@usc.edu

sage.modular.ssmod.ssmod.dimension_supersingular_module(prime, level=1)¶

This function returns the dimension of the Supersingular module, which is equal to the dimension of the space of modular forms of weight $2$ and conductor equal to prime times level.

INPUT:

prime – integer, prime
level – integer, positive

OUTPUT:: dimension – integer, nonnegative

EXAMPLES: The code below computes the dimensions of Supersingular modules with level=1 and prime = 7, 15073 and 83401:

sage: dimension_supersingular_module(7)
1

sage: dimension_supersingular_module(15073)
1256

sage: dimension_supersingular_module(83401)
6950

NOTES: The case of level > 1 has not been implemented yet.

AUTHORS:

David Kohel – kohel@maths.usyd.edu.au
Iftikhar Burhanuddin - burhanud@usc.edu

sage.modular.ssmod.ssmod.supersingular_D(prime)¶

This function returns a fundamental discriminant $D$ of an imaginary quadratic field, where the given prime does not split (see Silverman’s Advanced Topics in the Arithmetic of Elliptic Curves, page 184, exercise 2.30(d).)

INPUT:

prime – integer, prime

OUTPUT:: D – integer, negative

EXAMPLES:

These examples return supersingular discriminants for 7, 15073 and 83401:

sage: supersingular_D(7)
-4

sage: supersingular_D(15073)
-15

sage: supersingular_D(83401)
-7

AUTHORS:

David Kohel - kohel@maths.usyd.edu.au
Iftikhar Burhanuddin - burhanud@usc.edu

sage.modular.ssmod.ssmod.supersingular_j(FF)¶

This function returns a supersingular j-invariant over the finite field FF.

INPUT:

FF – finite field with p^2 elements, where p is a prime number

OUTPUT:: finite field element – a supersingular j-invariant defined over the finite field FF

EXAMPLES:

The following examples calculate supersingular j-invariants for a few finite fields:

sage: supersingular_j(GF(7^2, 'a'))
6

Observe that in this example the j-invariant is not defined over the prime field:

sage: supersingular_j(GF(15073^2,'a'))  # optional -- requires database
10630*a + 6033

sage: supersingular_j(GF(83401^2, 'a'))
67977

AUTHORS:

David Kohel – kohel@maths.usyd.edu.au
Iftikhar Burhanuddin – burhanud@usc.edu

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