Magma V2.7-1 Tue May 1 2001 04:33:01 on modular [Seed = 1824025117] Type ? for help. Type -D to quit. Loading startup file "/home/was/modsym/init.m" Loading "/home/was/modsym/init-magma.m" C IndexGamma0 R ellap idxG0 CS MS S factormod modcharpoly DC ND Tn factorpadic padiccharpoly ES NS Z fcp qexp F Q charpoly fn x Finding acceptable K for E of conductor 817 Finding mordell Weil group of Elliptic Curve defined by y^2 + y = x^3 + x^2 + x + 6 over Rational Field Finding point on twist by -8 Height bound (6.1312) on point search is too large -- reducing to 0.5000 This means that the computed group may only generate a group of finite index in the actual group. The point is P = (-25/8 : 95/64*sqrtD - 1/2 : 1) Computing first height. H_2() NumeratorIdeal() rhotilde(I, 991298460233611787 + O(3^40) , -991298460233611787 + O(3^40) ) Time: 0.000 Computing class number h Class number h = 1 Raising ideal to the power h Finding generator for the principal ideal I^h I^h is principal. Computing embeddings of canonical power of generator into Qp pi_alpha_v = 1 - q^4 - q^8 - q^9 - q^11 - q^13 + q^14 - q^16 + q^18 + q^19 + q^23 - q^25 + q^26 + q^29 + q^32 - q^33 + q^36 + q^37 - q^39 - q^40 + q^41 + O(q^43) pibar_alpha_v = 1 - q^3 - q^4 + q^8 + q^9 - q^13 + q^14 + q^15 + q^16 + q^17 + q^18 - q^19 + q^20 - q^21 + q^22 + q^23 + q^24 + q^25 + q^27 + q^28 - q^29 - q^31 + q^33 - q^36 - q^37 + q^38 + q^40 + q^41 - q^42 + O(q^43) pi_alpha_v-1 = -q^4 - q^8 - q^9 - q^11 - q^13 + q^14 - q^16 + q^18 + q^19 + q^23 - q^25 + q^26 + q^29 + q^32 - q^33 + q^36 + q^37 - q^39 - q^40 + q^41 + O(q^43) Valuation(pi_alpha_v-1) = 4 pibar_alpha_v-1 = -q^3 - q^4 + q^8 + q^9 - q^13 + q^14 + q^15 + q^16 + q^17 + q^18 - q^19 + q^20 - q^21 + q^22 + q^23 + q^24 + q^25 + q^27 + q^28 - q^29 - q^31 + q^33 - q^36 - q^37 + q^38 + q^40 + q^41 - q^42 + O(q^43) Valuation(pibar_alpha_v-1) = 3 Valuations ok -- now computing p-adic logarithms. lambda(pi_alpha_v) = -q^4 + q^10 - q^11 - q^13 - q^14 + q^15 - q^16 + q^18 + q^19 + q^21 - q^22 + q^23 + q^24 - q^26 + q^27 - q^28 + q^30 + q^31 - q^32 + q^33 + q^35 - q^36 + q^38 + q^39 + q^40 + O(q^43) lambda(pibar_alpha_v) = -q^3 - q^4 + q^6 - q^8 + q^9 - q^10 - q^11 + q^12 + q^14 - q^17 + q^18 + q^19 - q^20 - q^21 - q^22 + q^25 + q^26 - q^27 - q^28 - q^29 - q^30 + q^31 + q^32 - q^34 + q^36 + q^37 - q^38 + q^40 + q^41 + O(q^43) Computing second height. H_2() NumeratorIdeal() rhotilde(I, 991298460233611787 + O(3^40) , -991298460233611787 + O(3^40) ) Time: 0.000 Computing class number h Class number h = 1 Raising ideal to the power h Finding generator for the principal ideal I^h I^h is principal. Computing embeddings of canonical power of generator into Qp pi_alpha_v = 1 + q^2 + q^3 - q^6 + q^7 + q^8 + q^9 + q^10 + q^11 + q^12 + q^13 + q^14 - q^15 + q^16 + q^17 - q^18 + q^22 + q^23 - q^24 - q^25 - q^26 + q^27 + q^28 + q^31 - q^32 - q^36 + O(q^43) pibar_alpha_v = 1 + q^2 + q^4 - q^6 - q^8 + q^9 - q^11 + q^13 + q^15 + q^16 - q^18 + q^19 - q^20 + q^22 + q^23 - q^26 - q^27 + q^28 - q^29 + q^31 + q^33 - q^34 + q^35 + q^37 - q^39 - q^41 + O(q^43) pi_alpha_v-1 = q^2 + q^3 - q^6 + q^7 + q^8 + q^9 + q^10 + q^11 + q^12 + q^13 + q^14 - q^15 + q^16 + q^17 - q^18 + q^22 + q^23 - q^24 - q^25 - q^26 + q^27 + q^28 + q^31 - q^32 - q^36 + O(q^43) Valuation(pi_alpha_v-1) = 2 pibar_alpha_v-1 = q^2 + q^4 - q^6 - q^8 + q^9 - q^11 + q^13 + q^15 + q^16 - q^18 + q^19 - q^20 + q^22 + q^23 - q^26 - q^27 + q^28 - q^29 + q^31 + q^33 - q^34 + q^35 + q^37 - q^39 - q^41 + O(q^43) Valuation(pibar_alpha_v-1) = 2 Valuations ok -- now computing p-adic logarithms. lambda(pi_alpha_v) = q^2 + q^3 + q^4 + q^5 + q^6 + q^7 - q^9 + q^11 + q^13 - q^14 - q^16 + q^17 - q^18 - q^19 + q^23 + q^24 + q^25 - q^26 - q^27 - q^28 + q^29 - q^30 + q^31 + q^34 + q^35 - q^36 + q^38 + q^42 + O(q^43) lambda(pibar_alpha_v) = q^2 - q^4 + q^7 - q^8 - q^9 - q^11 + q^14 + q^17 + q^18 - q^19 - q^20 + q^23 - q^24 - q^25 + q^27 + q^28 - q^29 - q^30 - q^34 + q^35 + q^36 + q^37 - q^39 + q^41 + q^42 + O(q^43) Finding point on twist by -71 Height bound (10.1546) on point search is too large -- reducing to 0.5000 This means that the computed group may only generate a group of finite index in the actual group. The point is P = (-382/71 : 13427/10082*sqrtD - 1/2 : 1) Computing first height. H_2() NumeratorIdeal() rhotilde(I, 1947145412682542934 + O(3^40) , -1947145412682542933 + O(3^40) ) Time: 0.000 Computing class number h Class number h = 7 Raising ideal to the power h Finding generator for the principal ideal I^h