Coverage for local/lib/python2.7/site-packages/sage/tests/parigp.py : 100%

r""" 

This file is meant to catch errors in the PARI/GP package which are not 

caught by any other tests. 

Check that :trac:`9876` has been fixed, this test comes from PARI's 

self-test :pari:`rnfkummer` but was modified such that the answer is 

canonical:: 

sage: pari('addprimes([31438243, 238576291, 18775387483, 24217212463267, 1427657500359111961, 135564809928627323997297867381959])') 

[31438243, 238576291, 18775387483, 24217212463267, 1427657500359111961, 135564809928627323997297867381959] 

sage: pari('K = bnfinit(y^4-52*y^2+26,1); pol = rnfkummer(bnrinit(K,3,1),Mat(5)); L = rnfinit(K, pol); polredabs(polredbest(L.polabs))') # long time 

x^20 - 112*x^18 + 5108*x^16 - 123460*x^14 + 1724337*x^12 - 14266996*x^10 + 69192270*x^8 - 188583712*x^6 + 260329852*x^4 - 141461008*x^2 + 19860776 

Check that :trac:`10195` (PARI bug 1153) has been fixed:: 

sage: print(gp.eval("mathnf([0,0,0,0,0,0,0,0,0,13;0,0,0,0,0,0,0,0,23,6;0,0,0,0,0,0,0,23,-4,-7;0,0,0,0,0,0,17,-3,5,-5;0,0,0,0,0,56,16,-16,-15,-17;0,0,0,0,57,24,-16,-25,2,-21;0,0,0,114,9,56,51,-52,25,-55;0,0,113,-31,-11,24,0,28,34,-16;0,50,3,2,16,-6,-2,7,-19,-21;118,43,51,23,37,-52,18,38,51,28],0)")) 

[787850171872400 32189386376004 356588299060422 742392731867995 282253457851430 665185047494955 664535243562463 744564809133574 113975061998590 527459013372200] 

[0 12 6 11 5 3 7 6 6 0] 

[0 0 3 1 2 1 1 0 0 0] 

[0 0 0 1 0 0 0 0 0 0] 

[0 0 0 0 1 0 0 0 0 0] 

[0 0 0 0 0 1 0 0 0 0] 

[0 0 0 0 0 0 1 0 0 0] 

[0 0 0 0 0 0 0 1 0 0] 

[0 0 0 0 0 0 0 0 1 0] 

[0 0 0 0 0 0 0 0 0 1] 

Check that :trac:`11604` (PARI bug 1154) has been fixed:: 

sage: A = Matrix(ZZ,4,4,[32982266684193100, 1368614777139719, 224591013270052693, 276460184982223238,1368614777139719, 56791380087354, 9319512049770279, 11471848267545007,224591013270052693, 9319512049770279,1529340971891522140, 1882541434053596358,276460184982223238, 11471848267545007, 1882541434053596358, 2317313350044091414]) 

sage: pari(A).qfminim(2,0) 

[0, 0, [;]] 

Check :trac:`13314`, the following should not give a 

Segmentation Fault:: 

sage: x = polygen(ComplexField(128)) 

sage: p = x^84 + (16*x^4 - 1)^20 * (2^48*x^4 - 2049^4) 

sage: len(pari(p).polroots(precision=128)) 

84 

Check that the optional PARI databases work:: 

sage: gp.ellinit('"299998a1"') # optional -- database_pari 

[1, 0, 1, 110, -3660, ...] 

sage: E = EllipticCurve("1728ba1") 

sage: gp(E).ellidentify() # optional -- database_pari 

[["1728ba1", [0, 0, 0, -6, 6], [[1, 1]]], [1, 0, 0, 0]] 

sage: pari("ellmodulareqn(211)") # optional -- database_pari 

[x^212 + (-y^7 + 5207*y^6 - 10241606*y^5 + 9430560101*y^4 - 4074860204015*y^3 + 718868274900397*y^2 - 34897101275826114*y + 104096378056356968)*x^211... 

The following requires the modular polynomials up to degree 223, while 

only those up to degree 199 come standard in Sage:: 

sage: p = next_prime(2^328) 

sage: E = EllipticCurve(GF(p), [6,1]) 

sage: E.cardinality() # long time (108s on sage.math, 2013), optional -- database_pari 

546812681195752981093125556779405341338292357723293496548601032930284335897180749997402596957976244 

Create a number field with Galois group `A4`. Group `A4` corresponds to 

transitive group `(12,3)` in GAP:: 

sage: R.<x> = PolynomialRing(ZZ) 

sage: pol = pari("galoisgetpol(12,3)[1]") # optional -- database_pari 

sage: K.<a> = NumberField(R(pol)) # optional -- database_pari 

sage: factor(K.discriminant()) # optional -- database_pari 

163^8 

sage: [F.degree() for F,a,b in K.subfields()] # optional -- database_pari 

[1, 3, 4, 4, 4, 4, 6, 6, 6, 12] 

sage: sorted([12/H.cardinality() for H in AlternatingGroup(4).subgroups()]) 

[1, 3, 4, 4, 4, 4, 6, 6, 6, 12] 

"""