# Number fields downloaded from the LMFDB on 28 July 2026. # Search link: https://www.lmfdb.org/NumberField/?degree=46 # Query "{'degree': 46}" returned 81 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. "46.0.1755511210260049172778020908173078657717675374080672665297567056535308458607.1" [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1] -1755511210260049172778020908173078657717675374080672665297567056535308458607 "46T1" [695] "46.2.30928409945245615376155082830035810589532701226571507444404380354493327009821.1" [-1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 30928409945245615376155082830035810589532701226571507444404380354493327009821 "46T56" NULL "46.0.128682699210806627582755979490032711472673123289336814628063820138372779450600259584.1" [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] -128682699210806627582755979490032711472673123289336814628063820138372779450600259584 "46T6" NULL "46.0.1288062448421334108846325065368965062091432904338961405982013309614570671408084459427.1" [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] -1288062448421334108846325065368965062091432904338961405982013309614570671408084459427 "46T6" NULL "46.0.3516370336176915030779886015601767871077707157889593350075735586626118367196692091787.1" [1, -12, 210, 220, 7073, -3718, 98956, -17303, 724724, -192764, 3336047, -774410, 10388534, -2477645, 23456774, -5363639, 39899588, -8963564, 52902122, -11469872, 55884245, -11732918, 47923922, -9654035, 33775844, -6515312, 19758077, -3619962, 9645429, -1670119, 3942124, -638602, 1347421, -202420, 383674, -52612, 90115, -11094, 17196, -1841, 2591, -232, 295, -20, 23, -1, 1] -3516370336176915030779886015601767871077707157889593350075735586626118367196692091787 "46T1" NULL "46.46.165269405800315006446654642733283089940652236420810887453559572571427563258244528313989.1" [1, -48, 48, 4276, -4276, -114634, 114634, 1431196, -1431196, -10162529, 10162529, 45908941, -45908941, -141714824, 141714824, 313942891, -313942891, -516962354, 516962354, 649220446, -649220446, -633580634, 633580634, 486968926, -486968926, -297415766, 297415766, 145057650, -145057650, -56562010, 56562010, 17581994, -17581994, -4324189, 4324189, 830207, -830207, -121731, 121731, 13159, -13159, -988, 988, 46, -46, -1, 1] 165269405800315006446654642733283089940652236420810887453559572571427563258244528313989 "46T1" [] "46.0.2628364239485334411197227624389642263541714240820003570059378902799863859937013225488384.1" [1, 0, 276, 0, 12650, 0, 230230, 0, 2220075, 0, 13123110, 0, 51895935, 0, 145422675, 0, 300540195, 0, 471435600, 0, 573166440, 0, 548354040, 0, 417225900, 0, 254186856, 0, 124403620, 0, 48903492, 0, 15380937, 0, 3838380, 0, 749398, 0, 111930, 0, 12341, 0, 946, 0, 45, 0, 1] -2628364239485334411197227624389642263541714240820003570059378902799863859937013225488384 "46T1" NULL "46.2.17455927136175424851782795394947398572755562200353461405376235401663658494141675420261489.1" [-1, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 17455927136175424851782795394947398572755562200353461405376235401663658494141675420261489 "46T56" NULL "46.46.123533119255810717326269698346313186386460569318540167792790808431593601417039621597954048.1" [-47, 0, 4324, 0, -118910, 0, 1545830, 0, -11593725, 0, 56071470, 0, -187623765, 0, 455657715, 0, -830905245, 0, 1166182800, 0, -1282801080, 0, 1120549560, 0, -784384692, 0, 442473416, 0, -201619660, 0, 74144004, 0, -21906183, 0, 5154396, 0, -951938, 0, 134890, 0, -14147, 0, 1034, 0, -47, 0, 1] 123533119255810717326269698346313186386460569318540167792790808431593601417039621597954048 "46T1" NULL "46.46.445262221645814097378614331350194306504897709623466822770698795048558574688434600830078125.1" [-1, -12, 342, 352, -19657, -6578, 445588, 114257, -5308732, -1297868, 38414233, 9203272, -183867242, -43115073, 615043968, 140457727, -1492308292, -329962748, 2694891598, 573064052, -3686881837, -748176098, 3866596078, 742002577, -3130570732, -562037616, 1962978291, 325518500, -952979825, -143761015, 356957890, 48099584, -102471229, -12065268, 22311838, 2234528, -3629323, -298794, 431244, 27871, -36151, -1712, 2017, 62, -67, -1, 1] 445262221645814097378614331350194306504897709623466822770698795048558574688434600830078125 "46T1" NULL "46.2.557190287166858506148646530757915616540761669704958558941746593061657502340770471174209536.1" [-4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 557190287166858506148646530757915616540761669704958558941746593061657502340770471174209536 "46T56" NULL "46.0.1079468720061366162593727471597924323721357543409917117883493186123315004681540942348419072.1" [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] -1079468720061366162593727471597924323721357543409917117883493186123315004681540942348419072 "46T6" NULL "46.2.1079468720061366162593727471597924323721357543409917117883493186123315004681540942348419072.1" [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 1079468720061366162593727471597924323721357543409917117883493186123315004681540942348419072 "46T6" NULL "46.2.1096924647197541587445510266556877778401440441409917117883493186123315004681540942348419072.1" [-2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 1096924647197541587445510266556877778401440441409917117883493186123315004681540942348419072 "46T56" NULL "46.0.20927324417353262576794873573459132405730192352302940670222843367282253010356426239013671875.1" [6643838879, -6643838832, 6643838832, -6643834508, 6643834508, -6643715598, 6643715598, -6642169768, 6642169768, -6630576043, 6630576043, -6574504573, 6574504573, -6386880808, 6386880808, -5931223093, 5931223093, -5100317848, 5100317848, -3934135048, 3934135048, -2651333968, 2651333968, -1530784408, 1530784408, -746399716, 746399716, -303926300, 303926300, -102306640, 102306640, -28162636, 28162636, -6256453, 6256453, -1102057, 1102057, -150119, 150119, -15229, 15229, -1082, 1082, -48, 48, -1, 1] -20927324417353262576794873573459132405730192352302940670222843367282253010356426239013671875 "46T1" NULL "46.0.1022258356729509596871646965938036242035921334072080382372452419278261598416849846089945644983.1" [8388608, -50331648, 1019215872, 161480704, 20306198528, -3073900544, 162748039168, -19502399488, 692213350400, -84719697920, 1816862654464, -217643319296, 3213707960320, -377834643456, 4064622286848, -466369885184, 3831228318848, -427289762048, 2773773179200, -299245739680, 1577302832552, -163773410984, 716241923902, -71169036677, 262854128576, -24835843890, 78601898736, -7010057107, 19242860452, -1605788227, 3862036333, -298356295, 634011776, -44739144, 84584992, -5356876, 9061304, -502716, 764082, -35837, 49088, -1838, 2272, -61, 68, -1, 1] -1022258356729509596871646965938036242035921334072080382372452419278261598416849846089945644983 "46T1" NULL "46.2.12391693186733410362886430147153084686853607607137680458329824915608696755953133106231689453125.1" [-5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 12391693186733410362886430147153084686853607607137680458329824915608696755953133106231689453125 "46T6" NULL "46.0.22048317286260592124444353227775948209084132414256608507828666339058160374378508639437661929472.1" [8388608, 0, 1157627904, 0, 26528972800, 0, 241413652480, 0, 1163958681600, 0, 3440144547840, 0, 6802103992320, 0, 9530420428800, 0, 9848101109760, 0, 7724000870400, 0, 4695379476480, 0, 2246058147840, 0, 854478643200, 0, 260287340544, 0, 63694653440, 0, 12519293952, 0, 1968759936, 0, 245656320, 0, 23980736, 0, 1790880, 0, 98728, 0, 3784, 0, 90, 0, 1] -22048317286260592124444353227775948209084132414256608507828666339058160374378508639437661929472 "46T1" NULL "46.46.22048317286260592124444353227775948209084132414256608507828666339058160374378508639437661929472.1" [-8388608, 0, 1157627904, 0, -26528972800, 0, 241413652480, 0, -1163958681600, 0, 3440144547840, 0, -6802103992320, 0, 9530420428800, 0, -9848101109760, 0, 7724000870400, 0, -4695379476480, 0, 2246058147840, 0, -854478643200, 0, 260287340544, 0, -63694653440, 0, 12519293952, 0, -1968759936, 0, 245656320, 0, -23980736, 0, 1790880, 0, -98728, 0, 3784, 0, -90, 0, 1] 22048317286260592124444353227775948209084132414256608507828666339058160374378508639437661929472 "46T1" NULL "46.46.48046142766286951052967407399087703375688302701387777971505263706078295125591942766227445314201.1" [22705043, -416969619, 416969619, 17719200877, -17719200877, -231653143443, 231653143443, 1389267094637, -1389267094637, -4689183798163, 4689183798163, 10009615633517, -10009615633517, -14582606492563, 14582606492563, 15279377517677, -15279377517677, -11947725550483, 11947725550483, 7159013444717, -7159013444717, -3349693002643, 3349693002643, 1240077995117, -1240077995117, -366341854099, 366341854099, 86750923885, -86750923885, -16478342035, 16478342035, 2502522989, -2502522989, -301468435, 301468435, 28412909, -28412909, -2049107, 2049107, 109133, -109133, -4043, 4043, 93, -93, -1, 1] 48046142766286951052967407399087703375688302701387777971505263706078295125591942766227445314201 "46T1" NULL "46.0.749085428928598873917583984529112970694334716941605800990955391911147726327162798519362537914368.1" [3, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] -749085428928598873917583984529112970694334716941605800990955391911147726327162798519362537914368 "46T56" NULL "46.0.1036270912454247829848884601705469565826954223470060599867947317935733537595789906053570110685184.1" [394264576, 0, 18136170496, 0, 249372344320, 0, 1620920238080, 0, 6078450892800, 0, 14698799431680, 0, 24592222126080, 0, 29861984010240, 0, 27227103068160, 0, 19106738995200, 0, 10508706447360, 0, 4589770997760, 0, 1606419849216, 0, 453092777984, 0, 103229265920, 0, 18980865024, 0, 2803991424, 0, 329881344, 0, 30462016, 0, 2158240, 0, 113176, 0, 4136, 0, 94, 0, 1] -1036270912454247829848884601705469565826954223470060599867947317935733537595789906053570110685184 "46T1" NULL "46.46.1036270912454247829848884601705469565826954223470060599867947317935733537595789906053570110685184.1" [-394264576, 0, 18136170496, 0, -249372344320, 0, 1620920238080, 0, -6078450892800, 0, 14698799431680, 0, -24592222126080, 0, 29861984010240, 0, -27227103068160, 0, 19106738995200, 0, -10508706447360, 0, 4589770997760, 0, -1606419849216, 0, 453092777984, 0, -103229265920, 0, 18980865024, 0, -2803991424, 0, 329881344, 0, -30462016, 0, 2158240, 0, -113176, 0, 4136, 0, -94, 0, 1] 1036270912454247829848884601705469565826954223470060599867947317935733537595789906053570110685184 "46T1" NULL "46.2.2198572730507716078972973592834389965222171489600660772808467944071535302949868457350342936384821.1" [-1, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 2198572730507716078972973592834389965222171489600660772808467944071535302949868457350342936384821 "46T56" NULL "46.2.2198573809976436140308455840261067288915834777310563784825797656919642572984790479136920131622197.1" [-2, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 2198573809976436140308455840261067288915834777310563784825797656919642572984790479136920131622197 "46T56" NULL "46.0.2726879904444547310266403727163556673049075149201429809891173702168707134366554981450412880362299.1" [2726249, 8659371, -12752729, -25416074, 45005646, -68525839, 77496743, 60480343, 68541629, 138359819, 66181028, 243869890, 88850041, -2906153, 374031026, 52705940, 97270295, 277327722, -44044110, 226793510, 147048808, -79205588, 203991966, 67898048, -49045994, 102351606, 24664884, -17522906, 31840228, 5969903, -3933649, 6340570, 914794, -562441, 812047, 87297, -50535, 65894, 4990, -2737, 3247, 155, -81, 88, 2, -1, 1] -2726879904444547310266403727163556673049075149201429809891173702168707134366554981450412880362299 "46T1" NULL "46.0.33445485745541531749546267414718227385437022890768869389066429180265423777079649185104842349209411.1" [1523302345994551, -3020416683060232, 6584077168581835, -9315770272382043, 12777846887095319, -14502253394888869, 15643485715141214, -15106564909634047, 13862890922775304, -11787888134809172, 9562510933179307, -7317027854923049, 5363969908946175, -3747360102999732, 2517231470674524, -1621717831102939, 1007440894372913, -602707519775443, 348417978801750, -194491671575647, 105060081565914, -54891520145448, 27776633354903, -13607127733128, 6457810749477, -2967568274611, 1320756058230, -568897301282, 237111330579, -95529053696, 37178279250, -13956961576, 5047798326, -1755193762, 585518431, -186820582, 56828720, -16402546, 4465637, -1137429, 268455, -57853, 11104, -1813, 234, -21, 1] -33445485745541531749546267414718227385437022890768869389066429180265423777079649185104842349209411 "46T1" NULL 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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 $1\leq i < n$.