# Number fields downloaded from the LMFDB on 10 June 2026. # Search link: https://www.lmfdb.org/NumberField/?galois_group=20T18 # Query "{'degree': 20, 'galois_label': '20T18'}" returned 18 fields, sorted by degree. # Each entry in the following data list has the form: # [Label, Polynomial, Discriminant, Galois group, Class group] # For more details, see the definitions at the bottom of the file. "20.0.77992289980729574777937412261962890625.1" [29901760, 11605160, 44559375, 21049420, 24667825, 5110798, 5120500, 282975, 502985, -221725, 119035, -58170, 30625, -4515, 5075, -119, 560, 0, 35, 0, 1] 77992289980729574777937412261962890625 "20T18" [2, 2, 270] "20.0.389961449903647873889687061309814453125.1" [9992605, 19930925, 23635780, 18300310, 13887370, 6448680, 3244640, 216755, 44415, -364070, -60326, -81165, 4900, -7980, 3500, -644, 560, -35, 35, 0, 1] 389961449903647873889687061309814453125 "20T18" [2, 2, 540] "20.0.2128822455717240071906250000000000000000.2" [73459824, -51284160, 90306000, -34850760, 37619320, -10183012, 9319680, -2620560, 1645640, -540540, 214505, -43135, 2655, 3760, 635, -1533, 275, 90, -15, -5, 1] 2128822455717240071906250000000000000000 "20T18" [2, 2, 10, 410] "20.0.10644112278586200359531250000000000000000.2" [14016234, 45065640, 139869420, 153976080, -5438420, -113056091, 27800820, 3898470, -2979955, 2547660, -928048, 88030, 38820, -28900, 12470, -4272, 1100, -255, 60, -10, 1] 10644112278586200359531250000000000000000 "20T18" [2, 2, 10, 820] "20.20.4884274415246183319344904689389832005326536704.1" [-58612856, -315124556, 1644305872, -1179787402, -1813914636, 1707884929, 787082462, -826609978, -163802315, 193331394, 14939336, -24570618, -126850, 1744748, -71774, -67464, 4736, 1313, -116, -10, 1] 4884274415246183319344904689389832005326536704 "20T18" [2] "20.0.78148390643938933109518475030237312085224587264.1" [1082598671174, -66338130030, 72454618799, -324133328144, 70623059201, -37574492902, 18977386826, -289614572, 2117576320, 136683460, 139124895, 13556428, 6000753, 728624, 184045, 18596, 4620, 122, 96, -4, 1] 78148390643938933109518475030237312085224587264 "20T18" [2, 2, 2, 6, 6, 5562] "20.20.258866544008047715925279948537661096282306445312.1" [25224604, -321816380, 577682706, 491417950, -1481772709, 182743098, 1107126565, -469380944, -295344111, 194714572, 22954298, -31913924, 1504150, 2366808, -279760, -81202, 12703, 1138, -211, -4, 1] 258866544008047715925279948537661096282306445312 "20T18" [4] "20.0.393644681588957236910214274346592274945338834944.1" [197469154012, -230279722452, 152848977996, -73483265336, 31389769344, -13458733172, 5243487620, -1681413400, 484077264, -141163000, 37812683, -8550529, 1827617, -392560, 79085, -11313, 2263, -278, 55, -1, 1] 393644681588957236910214274346592274945338834944 "20T18" [2, 4, 8] "20.4.393644681588957236910214274346592274945338834944.1" [3619949909, -6711815042, -9802818353, 312500410, 6676113383, 57248456, -466792714, -1104174196, 18475857, 364634002, -60740475, -40542306, 7818605, 2219948, -419598, -65220, 11703, 986, -169, -6, 1] 393644681588957236910214274346592274945338834944 "20T18" [4] "20.0.4141864704128763454804479176602577540516903124992.1" [15554125329, -24323038390, 14977761454, -4594563420, 6843858589, 245640690, 1787066994, 864149074, 321304814, 75736410, 38130743, 7067238, 2612317, 596526, 168540, 23822, 8449, -4, 177, -4, 1] 4141864704128763454804479176602577540516903124992 "20T18" [2, 2, 6, 12, 22248] "20.4.5117380860656444079832785566505699574289404854272.1" [6774058131520, -9459350206144, -15529734627008, -4937110551040, 469017755136, 579288476848, 47447295520, -25051446848, -3607945172, 910761676, 15880149, -29902077, 927943, 1527264, -135227, -15157, -759, 874, -109, -3, 1] 5117380860656444079832785566505699574289404854272 "20T18" [2, 4] "20.0.5117380860656444079832785566505699574289404854272.1" [21376757978, -54399144116, 68366421642, -62448845622, 47528976126, -25919041653, 16546960658, -6211478090, 3311764471, -881542622, 372810412, -70647150, 23877802, -3200544, 888786, -81544, 19018, -1093, 216, -6, 1] 5117380860656444079832785566505699574289404854272 "20T18" [2, 4, 16] "20.4.513006019713288929550454694707222514684527733454667776.1" [284662974336, 8441222075904, 11551268406912, 5938321552896, 2969896724416, 1122667758080, 353271460352, 100368147456, 23314586816, 5143483648, 882147904, 156793344, 23901424, 3205056, 448672, 39680, 9206, -264, 170, -8, 1] 513006019713288929550454694707222514684527733454667776 "20T18" [2] "20.4.513006019713288929550454694707222514684527733454667776.6" [3819758726037376, 669435566788096, 224801039355520, 140593022798336, 8017404276672, 6926897485824, 1209918238208, 75536428032, 73912168896, 437884160, 1563139392, 182050176, 1466736, 7231360, -182240, 83776, 5814, -264, 170, -8, 1] 513006019713288929550454694707222514684527733454667776 "20T18" [3] "20.4.1026012039426577859100909389414445029369055466909335552.5" [17862528270, -27552270696, -10414044912, 18600090204, -819494303, -1354912048, -200770880, -240943472, 71446532, 13608128, 3791408, 779224, -629006, 97552, 16832, 12176, 662, -152, -32, -4, 1] 1026012039426577859100909389414445029369055466909335552 "20T18" [2] "20.4.1026012039426577859100909389414445029369055466909335552.10" [3796415961089083, 671250282787960, 214027887688090, 142262695282424, 6876524892165, 7128978743328, 1164475428104, 84464068512, 72767846382, 742564688, 1533824844, 187691472, 1244202, 7254304, -182168, 83872, 5799, -264, 170, -8, 1] 1026012039426577859100909389414445029369055466909335552 "20T18" [3] "20.20.352193597339274986373593303655705559065832274604727197314842624.2" [186129196859087782, 4737416779644064, -45414639474419396, -637514610703920, 4619658045471117, 33505948247296, -260153771058852, -895470287984, 9055694844272, 12627463952, -204944754828, -76649328, 3069374675, -160752, -30148560, 4448, 186368, -16, -656, 0, 1] 352193597339274986373593303655705559065832274604727197314842624 "20T18" NULL "20.20.18666260658981574277800445093752394630489110554050541457686659072.1" [2300893286748984, 5179746228111552, -10867727001230072, -3019425270314576, 6801945730533617, 52349328823712, -1097427171073028, 5077647112400, 68574729140642, -102900137840, -1921782848456, 628091280, 25314437011, -2239984, -174997424, 8416, 658002, -16, -1276, 0, 1] 18666260658981574277800445093752394630489110554050541457686659072 "20T18" NULL # Label -- # Each (global) number field has a unique label of the form d.r.D.i where # # The discriminant portion of the label can take the form \(a_1\) e \(\epsilon_1\) _ \(a_2\) e \(\epsilon_2\) _ \(\cdots\) _ \(a_k\) e \(\epsilon_k\) to mean the absolute value of the # discriminant equals \(a_1^{\epsilon_1}a_2^{\epsilon_2}\cdots a_k^{\epsilon_k}\). The separators are the letter e and the underscore symbol. #Polynomial (coeffs) -- # A **defining polynomial** of a number field $K$ is an irreducible polynomial $f\in\Q[x]$ such that $K\cong \mathbb{Q}(a)$, where $a$ is a root of $f(x)$. Equivalently, it is a polynomial $f\in \Q[x]$ such that $K \cong \Q[x]/(f)$. # A root \(a \in K\) of the defining polynomial is a generator of \(K\). #Discriminant (disc) -- # The **discriminant** of a number field $K$ is the square of the determinant of the matrix # \[ # \left( \begin{array}{ccc} # \sigma_1(\beta_1) & \cdots & \sigma_1(\beta_n) \\ # \vdots & & \vdots \\ # \sigma_n(\beta_1) & \cdots & \sigma_n(\beta_n) \\ # \end{array} \right) # \] # where $\sigma_1,..., \sigma_n$ are the embeddings of $K$ into the complex numbers $\mathbb{C}$, and $\{\beta_1, \ldots, \beta_n\}$ is an integral basis for the ring of integers of $K$. # The discriminant of $K$ is a non-zero integer divisible exactly by the primes which ramify in $K$. #Galois group (galois_label) -- # Let $K$ be a finite degree $n$ separable extension of a field $F$, and $K^{gal}$ be its # Galois (or normal) closure. # The **Galois group** for $K/F$ is the automorphism group $\Aut(K^{gal}/F)$. # This automorphism group acts on the $n$ embeddings $K\hookrightarrow K^{gal}$ via composition. As a result, we get an injection $\Aut(K^{gal}/F)\hookrightarrow S_n$, which is well-defined up to the labelling of the $n$ embeddings, which corresponds to being well-defined up to conjugation in $S_n$. # We use the notation $\Gal(K/F)$ for $\Aut(K/F)$ when $K=K^{gal}$. # There is a naming convention for Galois groups up to degree $47$. #Class group (class_group) -- # The **ideal class group** of a number field $K$ with ring of integers $O_K$ is the group of equivalence classes of ideals, given by the quotient of the multiplicative group of all fractional ideals of $O_K$ by the subgroup of principal fractional ideals. # Since $K$ is a number field, the ideal class group of $K$ is a finite abelian group, and so has the structure of a product of cyclic groups encoded by a finite list $[a_1,\dots,a_n]$, where the $a_i$ are positive integers with $a_i\mid a_{i+1}$ for $1\leq i < n$.