This file contains a table of totally real fields of various degrees with small discriminants. The fields have been found using the method used by Cohen, Diaz y Diaz and Olivier to find totally complex fields of small discriminants (cf. ANTS III), i.e. these fields are Abelian extensions of totally real fields of small degree. For each field, we give the difference between the root discriminant of the field and the corresponding Odlyzko bounds under GRH (express as a percentage), then a polynomial defining this field, then the ground field (for which the field is an Abelian extension), and finally the conductor and the congruence group of this extension. The polynomials have been computed using Stark units and the corresponding function `bnrstark' implemented in PARI. The fields are sorted by increasing degrees and increasing discriminants. DEGREE 4 +1.2% polynomial: x^4 - x^3 - 3*x^2 + x + 1 discriminant: 5^2 * 29 base field: y^2 - 5 conductor: [29, 23; 0, 1] congruence group: Mat([2]) DEGREE 6 +0.4% polynomial: x^6 - x^5 - 7*x^4 + 2*x^3 + 7*x^2 - 2*x - 1 discriminant: 5^3 * 7^4 base field: y^2 - 5 conductor: [7, 0; 0, 7] congruence group: Mat([3]) +4.0% polynomial: x^6 + x^5 - 5*x^4 - 4*x^3 + 6*x^2 + 3*x - 1 discriminant: 13^5 base field: y^2 - 13 conductor: [13, 6; 0, 1] congruence group: Mat([3]) +6.8% polynomial: x^6 - 2*x^5 - 4*x^4 + 5*x^3 + 4*x^2 - 2*x - 1 discriminant: 7^4 * 181 base field: y^3 - y^2 - 2*y + 1 conductor: [181, 37, 79; 0, 1, 0; 0, 0, 1] congruence group: Mat([2]) +7.6% polynomial: x^6 + x^5 - 6*x^4 - 6*x^3 + 8*x^2 + 8*x + 1 discriminant: 3^3 * 7^5 base field: y^2 - 21 conductor: [7, 3; 0, 1] congruence group: Mat([3]) DEGREE 8 +3.1% polynomial: x^8 - 4*x^7 + 14*x^5 - 8*x^4 - 12*x^3 + 7*x^2 + 2*x - 1 discriminant: 2^12 * 41^3 base field: y^2 - 8 conductor: [41, 17; 0, 1] congruence group: Mat([4]) +4.3% polynomial: x^8 - 4*x^7 - x^6 + 17*x^5 - 5*x^4 - 23*x^3 + 6*x^2 + 9*x - 1 discriminant: 5^4 * 19 * 29^2 * 31 base field: y^4 - y^3 - 3*y^2 + y + 1 conductor: [589, 145, 179, 550; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1] congruence group: Mat([2]) +4.9% polynomial: x^8 - 7*x^6 + 14*x^4 - 8*x^2 + 1 discriminant: 2^8 * 3^4 * 5^6 base field: y^2 - 5 conductor: [60, 24; 0, 12] congruence group: [2, 0; 0, 2] +8.0% polynomial: x^8 + x^7 - 7*x^6 - 6*x^5 + 15*x^4 + 10*x^3 - 10*x^2 - 4*x + 1 discriminant: 17^7 base field: y^2 - 17 conductor: [17, 8; 0, 1] congruence group: Mat([4]) +8.7% polynomial: x^8 + 2*x^7 - 7*x^6 - 16*x^5 + 4*x^4 + 18*x^3 + 2*x^2 - 4*x - 1 discriminant: 2^12 * 5^4 * 13^2 base field: y^2 - 40 conductor: [13, 7; 0, 1] congruence group: Mat([4]) +9.0% polynomial: x^8 + 2*x^7 - 12*x^6 - 26*x^5 + 17*x^4 + 36*x^3 - 5*x^2 - 11*x - 1 discriminant: 5^4 * 29^4 base field: y^2 - 5 conductor: [29, 0; 0, 29] congruence group: [2, 0; 0, 2] +9.5% polynomial: x^8 - 2*x^7 - 7*x^6 + 11*x^5 + 14*x^4 - 18*x^3 - 8*x^2 + 9*x - 1 discriminant: 5^4 * 11 * 29^2 * 79 base field: y^4 - y^3 - 3*y^2 + y + 1 conductor: [869, 500, 272, 433; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1] congruence group: Mat([2]) DEGREE 9 +9.9% polynomial: x^9 + 2*x^8 - 9*x^7 - 11*x^6 + 28*x^5 + 18*x^4 - 34*x^3 - 8*x^2 + 13*x - 1 discriminant: 37^2 * 229^3 base field: y^3 - 4*y - 1 conductor: [37, 8, 10; 0, 1, 0; 0, 0, 1] congruence group: Mat([3]) DEGREE 10 +9.1% polynomial: x^10 - x^9 - 10*x^8 + 10*x^7 + 34*x^6 - 34*x^5 - 43*x^4 + 43*x^3 + 12*x^2 - 12*x + 1 discriminant: 3^5 * 11^9 base field: y^2 - 33 conductor: [11, 5; 0, 1] congruence group: Mat([5]) DEGREE 12 +4.1% polynomial: x^12 + x^11 - 12*x^10 - 11*x^9 + 54*x^8 + 43*x^7 - 113*x^6 - 71*x^5 + 110*x^4 + 46*x^3 - 40*x^2 - 8*x + 1 discriminant: 5^9 * 7^10 base field: y^2 - 5 conductor: [35, 14; 0, 7] congruence group: Mat([6]) +5.8% polynomial: x^12 - 3*x^11 - 8*x^10 + 26*x^9 + 25*x^8 - 84*x^7 - 39*x^6 + 122*x^5 + 30*x^4 - 75*x^3 - 8*x^2 + 13*x - 1 discriminant: 3^3 * 103^2 *1327^3 base field: y^4 - y^3 - 4*y^2 + 2*y + 1 conductor: [103, 89, 10, 37; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1] congruence group: Mat([3]) +6.9% polynomial: x^12 - 12*x^10 + x^9 + 54*x^8 - 9*x^7 - 112*x^6 + 27*x^5 + 105*x^4 - 31*x^3 - 36*x^2 + 12*x + 1 discriminant: 3^18 * 5^9 base field: y^2 - 5 conductor: [45, 18; 0, 9] congruence group: Mat([6]) +7.9% polynomial: x^12 - 11*x^10 + 44*x^8 - 78*x^6 + 60*x^4 - 16*x^2 + 1 discriminant: 2^12 * 3^6 * 7^10 base field: y^2 - 12 conductor: [7, 0; 0, 7] congruence group: Mat([6]) DEGREE 16 +5.1% polynomial: x^16 - 2*x^15 - 21*x^14 + 26*x^13 + 178*x^12 - 86*x^11 - 727*x^10 - 78*x^9 + 1373*x^8 + 714*x^7 - 926*x^6 - 742*x^5 + 84*x^4 + 132*x^3 - 8*x^2 - 8*x + 1 discriminant: 2^24 * 3^4 * 5^8 * 13^6 base field: y^2 - 40 conductor: [39, 20; 0, 1] congruence group: Mat([8]) +8.1% polynomial: x^16 - 25*x^14 + 4*x^13 + 221*x^12 - 69*x^11 - 873*x^10 + 404*x^9 + 1560*x^8 - 941*x^7 - 1113*x^6 + 828*x^5 + 159*x^4 - 165*x^3 + 2*x^2 + 9*x - 1 discriminant: 2^8 * 3^4 * 5^8 * 29^8 base field: y^2 - 145 conductor: [12, 11; 0, 1] congruence group: Mat([8]) DEGREE 18 +3.4% polynomial: x^18 - 5*x^17 - 12*x^16 + 84*x^15 + 23*x^14 - 519*x^13 + 223*x^12 + 1437*x^11 - 1153*x^10 - 1709*x^9 + 1895*x^8 + 645*x^7 - 1140*x^6 + 16*x^5 + 262*x^4 - 26*x^3 - 26*x^2 + 2*x + 1 discriminant: 5^9 * 199^8 base field: y^2 - 5 conductor: [199, 61; 0, 1] congruence group: Mat([9]) +8.9% polynomial: x^18 - 30*x^16 + 37*x^15 + 204*x^14 - 294*x^13 - 604*x^12 + 858*x^11 + 984*x^10 - 1201*x^9 - 984*x^8 + 858*x^7 + 604*x^6 - 294*x^5 - 204*x^4 + 37*x^3 + 30*x^2 - 1 discriminant: 2^12 * 3^24 * 13^9 base field: y^2 - 13 conductor: [18, 0; 0, 18] congruence group: [3, 0; 0, 3] +9.4% polynomial: x^18 - 24*x^16 - 3*x^15 + 216*x^14 + 42*x^13 - 943*x^12 - 180*x^11 + 2190*x^10 + 272*x^9 - 2772*x^8 - 54*x^7 + 1819*x^6 - 180*x^5 - 522*x^4 + 110*x^3 + 36*x^2 - 6*x - 1 discriminant: 3^24 * 5^9 * 17^6 base field: y^3 - 12*y - 1 conductor: [5, 1, 4; 0, 1, 0; 0, 0, 1] congruence group: Mat([6]) DEGREE 21 +9.9% polynomial: x^21 - 3*x^20 - 25*x^19 + 101*x^18 + 134*x^17 - 989*x^16 + 338*x^15 + 3951*x^14 - 4581*x^13 - 6201*x^12 + 12873*x^11 + 1110*x^10 - 14632*x^9 + 5944*x^8 + 6584*x^7 - 5079*x^6 - 580*x^5 + 1336*x^4 - 227*x^3 - 73*x^2 + 16*x + 1 discriminant: 29^6 * 1229^7 base field: y^3 - y^2 - 7*y + 6 conductor: [29, 16, 5; 0, 1, 0; 0, 0, 1] congruence group: Mat([7]) DEGREE 22 +5.7% polynomial: x^22 - 2*x^21 - 30*x^20 + 56*x^19 + 353*x^18 - 581*x^17 - 2175*x^16 + 2922*x^15 + 7849*x^14 - 7781*x^13 - 17246*x^12 + 10971*x^11 + 22936*x^10 - 7357*x^9 - 17744*x^8 + 1188*x^7 + 7309*x^6 + 893*x^5 - 1311*x^4 - 330*x^3 + 56*x^2 + 21*x + 1 discriminant: 17^11 * 67^10 base field: y^2 - 17 conductor: [67, 16; 0, 1] congruence group: Mat([11]) DEGREE 24 +4.1% polynomial: x^24 + 12*x^23 + 32*x^22 - 154*x^21 - 949*x^20 - 642*x^19 + 5420*x^18 + 10116*x^17 - 10828*x^16 - 38392*x^15 + 614*x^14 + 68736*x^13 + 28310*x^12 - 63684*x^11 - 42256*x^10 + 29642*x^9 + 26044*x^8 - 6400*x^7 - 7516*x^6 + 692*x^5 + 1070*x^4 - 56*x^3 - 68*x^2 + 4*x + 1 discriminant: 2^36 * 3^12 * 7^22 base field: y^2 - 12 conductor: [14, 0; 0, 14] congruence group: Mat([12]) +4.6% polynomial: x^24 - x^23 - 23*x^22 + 23*x^21 + 230*x^20 - 229*x^19 - 1312*x^18 + 1294*x^17 + 4709*x^16 - 4573*x^15 - 11067*x^14 + 10506*x^13 + 17187*x^12 - 15810*x^11 - 17391*x^10 + 15333*x^9 + 11034*x^8 - 9189*x^7 - 4088*x^6 + 3142*x^5 + 784*x^4 - 528*x^3 - 64*x^2 + 32*x + 1 discriminant: 3^12 * 5^18 * 7^20 base field: y^2 - 21 conductor: [35, 15; 0, 5] congruence group: Mat([12]) +8.3% polynomial: x^24 - 10*x^23 + 9*x^22 + 252*x^21 - 1079*x^20 + 48*x^19 + 9186*x^18 - 17691*x^17 - 17790*x^16 + 94959*x^15 - 60652*x^14 - 171540*x^13 + 288115*x^12 + 19365*x^11 - 377547*x^10 + 253522*x^9 + 125343*x^8 - 229977*x^7 + 74503*x^6 + 39146*x^5 - 39330*x^4 + 12985*x^3 - 1914*x^2 + 96*x + 1 discriminant: 3^18 * 5^12 * 7^22 base field: y^2 - 5 conductor: [21, 0; 0, 21] congruence group: Mat([12]) DEGREE 27 +2.8% polynomial: x^27 + 10*x^26 + 10*x^25 - 218*x^24 - 783*x^23 + 754*x^22 + 7599*x^21 + 5094*x^20 - 31037*x^19 - 43000*x^18 + 68238*x^17 + 135637*x^16 - 91409*x^15 - 242376*x^14 + 86222*x^13 + 272022*x^12 - 72196*x^11 - 195156*x^10 + 58033*x^9 + 84568*x^8 - 34757*x^7 - 17820*x^6 + 11437*x^5 + 146*x^4 - 1325*x^3 + 338*x^2 - 32*x + 1 discriminant: 7^24 * 13^18 base field: y^3 - y^2 - 4*y - 1 conductor: [7, 0, 0; 0, 7, 0; 0, 0, 7] congruence group: Mat([9]) DEGREE 28 +9.8% polynomial: x^28 + 8*x^27 - 14*x^26 - 258*x^25 - 229*x^24 + 3204*x^23 + 6277*x^22 - 19532*x^21 - 56653*x^20 + 59451*x^19 + 271845*x^18 - 61545*x^17 - 790726*x^16 - 156184*x^15 + 1478192*x^14 + 663429*x^13 - 1819825*x^12 - 1115801*x^11 + 1474011*x^10 + 1078469*x^9 - 762365*x^8 - 634729*x^7 + 232274*x^6 + 222088*x^5 - 33063*x^4 - 41782*x^3 - 72*x^2 + 3181*x + 347 discriminant: 5^7 * 29^26 base field: y^2 - 29 conductor: [145, 101; 0, 1] congruence group: Mat([14]) DEGREE 30 +4.2% polynomial: x^30 + 9*x^29 - 221*x^27 - 466*x^26 + 2023*x^25 + 7271*x^24 - 7163*x^23 - 51688*x^22 - 9306*x^21 + 202895*x^20 + 172727*x^19 - 448502*x^18 - 651849*x^17 + 490298*x^16 + 1240538*x^15 - 53663*x^14 - 1287539*x^13 - 450911*x^12 + 686427*x^11 + 455532*x^10 - 143253*x^9 - 179277*x^8 - 9253*x^7 + 29491*x^6 + 7028*x^5 - 1445*x^4 - 683*x^3 - 44*x^2 + 9*x + 1 discriminant: 11^28 * 13^15 base field: y^2 - 13 conductor: [11, 0; 0, 11] congruence group: Mat([15]) DEGREE 34 +6.2% polynomial: x^34 + 10*x^33 - 3*x^32 - 328*x^31 - 639*x^30 + 4492*x^29 + 14057*x^28 - 33202*x^27 - 147378*x^26 + 139410*x^25 + 946821*x^24 - 285576*x^23 - 4082405*x^22 - 134122*x^21 + 12322131*x^20 + 2543908*x^19 - 26496746*x^18 - 7505240*x^17 + 40608593*x^16 + 12095918*x^15 - 43719591*x^14 - 11912396*x^13 + 32100815*x^12 + 7102876*x^11 - 15329167*x^10 - 2304266*x^9 + 4428059*x^8 + 252082*x^7 - 688083*x^6 + 43784*x^5 + 43913*x^4 - 8450*x^3 + 302*x^2 + 20*x - 1 discriminant: 2^51 * 239^16 base field: y^2 - 8 conductor: [239, 99; 0, 1] congruence group: Mat([17]) DEGREE 36 +5.5% polynomial: x^36 - 57*x^34 + 3*x^33 + 1380*x^32 + 75*x^31 - 19249*x^30 - 4577*x^29 + 173997*x^28 + 79989*x^27 - 1075792*x^26 - 757077*x^25 + 4647406*x^24 + 4513144*x^23 - 13982437*x^22 - 18007439*x^21 + 28303408*x^20 + 49262014*x^19 - 34440629*x^18 - 92463710*x^17 + 13165636*x^16 + 116361244*x^15 + 30386884*x^14 - 92119140*x^13 - 57818254*x^12 + 37967867*x^11 + 45731907*x^10 - 561187*x^9 - 17584036*x^8 - 6119368*x^7 + 2254792*x^6 + 1975168*x^5 + 313120*x^4 - 117184*x^3 - 56704*x^2 - 8960*x - 512 discriminant: 3^6 * 19^10 * 2777^12 base field: y^3 - y^2 - 14*y + 23 conductor: [57, 5, 32; 0, 1, 0; 0, 0, 1] congruence group: Mat([12]) +8.9% polynomial: x^36 - 5*x^35 - 47*x^34 + 328*x^33 + 445*x^32 - 7449*x^31 + 9071*x^30 + 67615*x^29 - 202432*x^28 - 147327*x^27 + 1354564*x^26 - 1079934*x^25 - 3857358*x^24 + 7378468*x^23 + 3251005*x^22 - 18932194*x^21 + 8026339*x^20 + 24482530*x^19 - 25361828*x^18 - 14263930*x^17 + 31387055*x^16 - 1849946*x^15 - 21093336*x^14 + 8432951*x^13 + 7730996*x^12 - 5619427*x^11 - 1202915*x^10 + 1817391*x^9 - 93094*x^8 - 300851*x^7 + 57213*x^6 + 22703*x^5 - 6159*x^4 - 640*x^3 + 188*x^2 + 11*x - 1 discriminant: 2^28 * 3^18 * 2351^12 base field: y^3 - y^2 - 23*y + 48 conductor: [12, 8, 4; 0, 2, 0; 0, 0, 2] congruence group: Mat([12]) +9.2% polynomial: x^36 + 6*x^35 - 49*x^34 - 368*x^33 + 761*x^32 + 9326*x^31 + 493*x^30 - 125244*x^29 - 158208*x^28 + 938886*x^27 + 2176986*x^26 - 3582432*x^25 - 14646844*x^24 + 2397542*x^23 + 55612326*x^22 + 36526612*x^21 - 118533021*x^20 - 162354088*x^19 + 118637824*x^18 + 326947232*x^17 + 18211496*x^16 - 358432240*x^15 - 177339239*x^14 + 210730562*x^13 + 191287471*x^12 - 53379620*x^11 - 99261083*x^10 - 4294270*x^9 + 27722460*x^8 + 5588850*x^7 - 4103038*x^6 - 1161258*x^5 + 296017*x^4 + 86894*x^3 - 9823*x^2 - 1690*x + 169 discriminant: 2^54 * 5^30 * 7^24 base field: y^3 - y^2 - 23*y - 13 conductor: [40, 13, 35; 0, 1, 0; 0, 0, 1] congruence group: [6, 0; 0, 2] +9.8% polynomial: x^36 + 9*x^35 - 21*x^34 - 429*x^33 - 489*x^32 + 8205*x^31 + 22887*x^30 - 76317*x^29 - 363060*x^28 + 259808*x^27 + 3161154*x^26 + 1569222*x^25 - 16560132*x^24 - 22521003*x^23 + 51008460*x^22 + 121354695*x^21 - 71196822*x^20 - 375275922*x^19 - 74932234*x^18 + 702028038*x^17 + 524123862*x^16 - 744108411*x^15 - 1020993408*x^14 + 299233731*x^13 + 1032469632*x^12 + 211571850*x^11 - 545661393*x^10 - 309665861*x^9 + 110691582*x^8 + 136817043*x^7 + 16193187*x^6 - 20763366*x^5 - 8123409*x^4 - 358659*x^3 + 262938*x^2 + 28122*x - 1709 discriminant: 2^24 * 3^66 * 5^27 base field: y^3 - 12*y - 14 conductor: [15, 0, 7; 0, 15, 2; 0, 0, 1] congruence group: Mat([12]) DEGREE 40 +2.5% polynomial: x^40 - 63*x^38 + 1804*x^36 - 31102*x^34 + 360576*x^32 - 2974235*x^30 + 18017992*x^28 - 81612654*x^26 + 278850484*x^24 - 720213707*x^22 + 1401055809*x^20 - 2034301536*x^18 + 2172433384*x^16 - 1671146736*x^14 + 901368560*x^12 - 330165440*x^10 + 79355328*x^8 - 12025088*x^6 + 1084160*x^4 - 52224*x^2 + 1024 discriminant: 2^40 * 3^30 * 11^36 base field: y^2 - 12 conductor: [33, 0; 0, 11] congruence group: [10, 0; 0, 2] DEGREE 42 +8.6% polynomial: x^42 + 5*x^41 - 56*x^40 - 313*x^39 + 1165*x^38 + 7774*x^37 - 12037*x^36 - 106837*x^35 + 61208*x^34 + 929721*x^33 - 37405*x^32 - 5496538*x^31 - 1569328*x^30 + 22998336*x^29 + 11508799*x^28 - 69816066*x^27 - 45442382*x^26 + 156075488*x^25 + 118989070*x^24 - 259016042*x^23 - 220318063*x^22 + 319929762*x^21 + 296421321*x^20 - 293337080*x^19 - 293105091*x^18 + 197923181*x^17 + 213344252*x^16 - 96659017*x^15 - 113545798*x^14 + 33223979*x^13 + 43498595*x^12 - 7680992*x^11 - 11671474*x^10 + 1113457*x^9 + 2098168*x^8 - 94439*x^7 - 235037*x^6 + 6261*x^5 + 14462*x^4 - 680*x^3 - 370*x^2 + 40*x - 1 discriminant: 2^21 * 29^13 * 1229^14 base field: y^3 - y^2 - 7*y + 6 conductor: [232, 74, 92; 0, 1, 0; 0, 0, 1] congruence group: Mat([14]) DEGREE 44 +6.9% polynomial: x^44 - 68*x^42 + 2103*x^40 - 39293*x^38 + 496782*x^36 - 4509558*x^34 + 30425066*x^32 - 155761663*x^30 + 612237890*x^28 - 1857079518*x^26 + 4345278300*x^24 - 7799692835*x^22 + 10624439374*x^20 - 10800025336*x^18 + 8000669794*x^16 - 4183742439*x^14 + 1483807058*x^12 - 342091749*x^10 + 49816376*x^8 - 4458448*x^6 + 233160*x^4 - 6365*x^2 + 67 discriminant: 2^22 * 17^22 * 67^21 base field: y^2 - 17 conductor: [268, 16; 0, 1] congruence group: Mat([22]) DEGREE 48 +9.4% polynomial: x^48 + 18*x^47 + 70*x^46 - 627*x^45 - 5676*x^44 - 225*x^43 + 141424*x^42 + 356202*x^41 - 1632810*x^40 - 8217993*x^39 + 6057086*x^38 + 96102240*x^37 + 77560331*x^36 - 672904017*x^35 - 1326147550*x^34 + 2764573458*x^33 + 10011619964*x^32 - 4318879500*x^31 - 47020142551*x^30 - 20600423613*x^29 + 146207270625*x^28 + 164353668732*x^27 - 292277422029*x^26 - 578371893510*x^25 + 301116930311*x^24 + 1278370893150*x^23 + 137485232711*x^22 - 1873932376797*x^21 - 1038831500505*x^20 + 1765305793743*x^19 + 1794184476344*x^18 - 904561350165*x^17 - 1731239171451*x^16 + 20544001917*x^15 + 999427412605*x^14 + 284214255957*x^13 - 325294704114*x^12 - 172969607955*x^11 + 49056839601*x^10 + 45232303368*x^9 - 1120960640*x^8 - 5679508062*x^7 - 394102526*x^6 + 333418170*x^5 + 23797129*x^4 - 8883858*x^3 - 174365*x^2 + 51522*x + 961 discriminant: 3^36 * 5^46 * 19^24 base field: y^2 - 57 conductor: [15, 5; 0, 5] congruence group: Mat([24])