Coverage for local/lib/python2.7/site-packages/sage/quadratic_forms/quadratic_form__genus.py : 58%

""" 

Local and Global Genus Symbols 

""" 

############################################################# 

## ## 

## Wrappers for the Genus/Genus Symbol Code in ../genera/ ## 

## ## 

############################################################# 

from sage.quadratic_forms.genera.genus import Genus, LocalGenusSymbol, \ 

is_GlobalGenus, is_2_adic_genus, canonical_2_adic_compartments, \ 

canonical_2_adic_trains, canonical_2_adic_reduction, \ 

basis_complement, p_adic_symbol, is_even_matrix, \ 

split_odd, trace_diag_mod_8, two_adic_symbol 

#is_trivial_symbol 

#GenusSymbol_p_adic_ring, GenusSymbol_global_ring 

## Removed signature_pair_of_matrix due to a circular import issue. 

## NOTE: Removed the signature routine here... and rewrote it for now. 

from sage.rings.integer_ring import IntegerRing 

from sage.arith.all import is_prime, prime_divisors 

def global_genus_symbol(self): 

""" 

Returns the genus of a two times a quadratic form over ZZ. These 

are defined by a collection of local genus symbols (a la Chapter 

15 of Conway-Sloane), and a signature. 

EXAMPLES:: 

sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3,4]) 

sage: Q.global_genus_symbol() 

Genus of 

[2 0 0 0] 

[0 4 0 0] 

[0 0 6 0] 

[0 0 0 8] 

Genus symbol at 2: [2^-2 4^1 8^1]_6 

Genus symbol at 3: 1^3 3^-1 

:: 

sage: Q = QuadraticForm(ZZ, 4, range(10)) 

sage: Q.global_genus_symbol() 

Genus of 

[ 0 1 2 3] 

[ 1 8 5 6] 

[ 2 5 14 8] 

[ 3 6 8 18] 

Genus symbol at 2: 1^-4  

Genus symbol at 563: 1^3 563^-1 

""" 

## Check that the form is defined over ZZ 

if not self.base_ring() == IntegerRing(): 

raise TypeError("Oops! The quadratic form is not defined over the integers.") 

## Return the result 

try: 

return Genus(self.Hessian_matrix()) 

except Exception: 

raise TypeError("Oops! There is a problem computing the genus symbols for this form.") 

def local_genus_symbol(self, p): 

""" 

Returns the Conway-Sloane genus symbol of 2 times a quadratic form 

defined over ZZ at a prime number p. This is defined (in the 

Genus_Symbol_p_adic_ring() class in the quadratic_forms/genera 

subfolder) to be a list of tuples (one for each Jordan component 

p^m*A at p, where A is a unimodular symmetric matrix with 

coefficients the p-adic integers) of the following form: 

1. If p>2 then return triples of the form [`m`, `n`, `d`] where 

`m` = valuation of the component 

`n` = rank of A 

`d` = det(A) in {1,u} for normalized quadratic non-residue u. 

2. If p=2 then return quintuples of the form [`m`,`n`,`s`, `d`, `o`] where 

`m` = valuation of the component 

`n` = rank of A 

`d` = det(A) in {1,3,5,7} 

`s` = 0 (or 1) if A is even (or odd) 

`o` = oddity of A (= 0 if s = 0) in Z/8Z 

= the trace of the diagonalization of A 

NOTE: The Conway-Sloane convention for describing the prime 'p = 

-1' is not supported here, and neither is the convention for 

including the 'prime' Infinity. See note on p370 of Conway-Sloane 

(3rd ed) for a discussion of this convention. 

INPUT: 

-`p` -- a prime number > 0 

OUTPUT: 

Returns a Conway-Sloane genus symbol at p, which is an 

instance of the Genus_Symbol_p_adic_ring class. 

EXAMPLES:: 

sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3,4]) 

sage: Q.local_genus_symbol(2) 

Genus symbol at 2: [2^-2 4^1 8^1]_6 

sage: Q.local_genus_symbol(3) 

Genus symbol at 3: 1^3 3^-1 

sage: Q.local_genus_symbol(5) 

Genus symbol at 5: 1^4 

""" 

## Check that p is prime and that the form is defined over ZZ. 

if not is_prime(p): 

raise TypeError("Oops! The number " + str(p) + " isn't prime.") 

if not self.base_ring() == IntegerRing(): 

raise TypeError("Oops! The quadratic form is not defined over the integers.") 

## Return the result 

try: 

M = self.Hessian_matrix() 

return LocalGenusSymbol(M, p) 

except Exception: 

raise TypeError("Oops! There is a problem computing the local genus symbol at the prime " + str(p) + " for this form.") 

def CS_genus_symbol_list(self, force_recomputation=False): 

""" 

Returns the list of Conway-Sloane genus symbols in increasing order of primes dividing 2*det. 

EXAMPLES:: 

sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3,4]) 

sage: Q.CS_genus_symbol_list() 

[Genus symbol at 2: [2^-2 4^1 8^1]_6, Genus symbol at 3: 1^3 3^-1] 

""" 

## Try to use the cached list 

if not force_recomputation: 

try: 

return self.__CS_genus_symbol_list 

except AttributeError: 

pass 

## Otherwise recompute and cache the list 

list_of_CS_genus_symbols = [ ] 

for p in prime_divisors(2 * self.det()): 

list_of_CS_genus_symbols.append(self.local_genus_symbol(p)) 

self.__CS_genus_symbol_list = list_of_CS_genus_symbols 

return list_of_CS_genus_symbols