# Number fields downloaded from the LMFDB on 07 August 2026. # Search link: https://www.lmfdb.org/NumberField/?completions=89.1.20.19a1.3 # Query "{'local_algs': {'$contains': ['89.1.20.19a1.3']}}" returned 16 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. "24.4.91810054652229848026521481130092527303208363056182861328125.1" [-91903199858837, -15091490947869, 96703661945176, -13595215964929, -37689124873554, 2324146207637, 10666291776869, 555294801359, -2012872155141, -120260671411, 187723100658, 15522090746, -8957494979, -1041728624, 203615891, 31758582, 320934, -1486656, 40584, 5874, 4273, -449, 6, -4, 1] 91810054652229848026521481130092527303208363056182861328125 "24T1353" NULL "24.4.91810054652229848026521481130092527303208363056182861328125.5" [2006228467125, 1967394528875, 16165180816000, -13972190531300, 1464409611225, 654694955355, -233437373280, 46635223810, 9151935820, 7243837191, -2530983710, -501217985, 128716760, -7372490, -1290046, -2202100, 411200, 164600, -96055, -6321, 4680, -10, -120, 0, 1] 91810054652229848026521481130092527303208363056182861328125 "24T1353" [2, 2, 4] "24.4.91810054652229848026521481130092527303208363056182861328125.6" [14471246844265, -21535626575215, 5468893445760, 12150299345835, -9951527396947, 397551181590, 2708721649225, -959076041810, -144462051275, 125420329119, -2946551855, -9573340425, 1177337010, 532379580, -133024731, -16327735, 7887200, 173055, -278060, 799, 6905, -10, -120, 0, 1] 91810054652229848026521481130092527303208363056182861328125 "24T1353" [2, 2, 4] "24.4.91810054652229848026521481130092527303208363056182861328125.7" [25979013994259, 32252356357608, 47391301297208, 58841762063278, 63217215313641, 56766412919919, 34336140894428, 13985340544188, 4305306446543, 1095953073726, 216820490264, 27608980468, 2308036563, -124652522, -93267574, -20887621, -3317052, -1743022, -124547, -68669, -1196, -827, 63, -2, 1] 91810054652229848026521481130092527303208363056182861328125 "24T1353" [4] "24.4.2295251366305746200663037028252313182580209076404571533203125.2" [-312927910098871, 541180287831136, 114072323164339, 441206727204676, -580910749748916, 51235342679184, 105662214126501, -28596060866796, -6286206651279, 3086068549814, -97275021018, -143840039207, 30807485227, 3477178433, -1893188568, -28337252, 62457032, -598157, -1149493, -5277, 12301, 824, -114, -11, 1] 2295251366305746200663037028252313182580209076404571533203125 "24T1353" [2, 2, 2, 4] "24.4.2295251366305746200663037028252313182580209076404571533203125.3" [-119337300832275, -151975177287750, -94608154678215, 63283022914135, 11151366131449, -43374754905545, 4838910931635, -4539669066765, -4126385140520, -1068771666276, -276744138650, -82118098860, -28707547175, -5462726085, 491077287, 116653690, -34093635, -935550, 1112160, -34087, -10030, 1935, 25, -10, 1] 2295251366305746200663037028252313182580209076404571533203125 "24T1353" [2, 2, 2, 4] "24.4.2295251366305746200663037028252313182580209076404571533203125.4" [-8058266421349, -119801317951219, -313677448742334, -443687608352359, -326471613749334, -90366861809451, 14257456475664, 13899669932829, 2406563134669, -168718807126, -98459485684, -7709179644, 2209626941, 665735486, 6061701, -12083886, -18156, 45034, 25454, 12994, 446, -4, 6, -4, 1] 2295251366305746200663037028252313182580209076404571533203125 "24T1353" [2, 4] "24.4.2295251366305746200663037028252313182580209076404571533203125.8" [3470991058617581, 642150457868791, -1121050409397834, 46785615621511, 150857399860436, -31620921402991, 1399615659494, 5096904048179, -2635907116121, 318568634184, 29403672616, -31114295514, 5667704141, 148582296, -96062239, 9094554, 1275904, 859384, -132076, 6764, 3116, -4, 6, -4, 1] 2295251366305746200663037028252313182580209076404571533203125 "24T1353" [2, 4] "24.4.35863302598527284385359953566442393477815766818821430206298828125.2" [-21580162302394368, -50029902292521984, -29289495523457792, -1094970506171776, -3888629115012928, -2316315944560320, 765727826720400, -442422095928880, 74624141823520, -24849991956160, 3158183643447, -980595596854, 84384989853, -27368555416, 2227739352, -528120304, 30971223, -8671906, 609372, -136934, 6316, -902, 64, -8, 1] 35863302598527284385359953566442393477815766818821430206298828125 "24T1353" [4] "24.4.35863302598527284385359953566442393477815766818821430206298828125.3" [-419768284696857, 293224286414451, 3296208146361217, -4671372579250361, 6757569482264748, -1040206415032347, -42589341925234, 137815195972027, -17379547551176, 9252856332343, -2322137868424, -492175602363, 243319281339, -3438452642, -6202398534, 97710399, 150508008, -3441264, -2254073, 21124, 28961, -133, -216, -2, 1] 35863302598527284385359953566442393477815766818821430206298828125 "24T1353" [2, 4] "24.4.35863302598527284385359953566442393477815766818821430206298828125.4" [18205372928000, 0, 284275397082800, 0, -95369414191688, 0, -12477500025945, 0, 2644385805340, 0, 490016745570, 0, 25447488760, 0, 343628555, 0, -14942655, 0, -699540, 0, -9790, 0, 0, 0, 1] 35863302598527284385359953566442393477815766818821430206298828125 "24T1353" [4] "24.4.35863302598527284385359953566442393477815766818821430206298828125.8" [858450394618182224, -927383945402843232, 139064771037891016, 97786901456032952, -13439138362502051, -4747675198206540, -866046602082015, -8715047372200, 138487686799265, 18528375557180, -2506437023344, -1134655611298, -202726936711, 3640653858, 7747419046, 1059872352, 15768809, -16138302, -1176554, 145552, 20801, 152, -191, -2, 1] 35863302598527284385359953566442393477815766818821430206298828125 "24T1353" [2, 4] "24.4.896582564963182109633998839161059836945394170470535755157470703125.4" [5255995392000, 0, -12699200672000, 0, -71180667348800, 0, -22064899402000, 0, -3595814890500, 0, -431712720525, 0, -36450450625, 0, -1856885765, 0, -52100600, 0, -823250, 0, -7120, 0, 0, 0, 1] 896582564963182109633998839161059836945394170470535755157470703125 "24T1353" NULL "24.4.896582564963182109633998839161059836945394170470535755157470703125.6" [150093087495580416, -92937065600889664, 49846168997904256, 27858329958403216, -7522754457264744, 4771707259687380, 696572665120450, -19654322538325, 87026862121385, 2467042788125, 1688439158893, -33238859092, -159570300332, 877025663, -2267018532, 214136855, 74365635, -2629940, 940100, -3090, -14784, 431, -99, -4, 1] 896582564963182109633998839161059836945394170470535755157470703125 "24T1353" [2, 4, 4] "24.4.896582564963182109633998839161059836945394170470535755157470703125.7" [525709300570411456, 516534814705031808, 17398950398107184, 23743341392067512, 21376347951105196, -8282002593358136, 2380263669327347, -717771526707744, 117396649195528, -22886598332681, 3636688422802, -631563143414, 143167194098, -24039246086, 4152261112, -474785274, 33490368, 342904, -796828, 111376, -12374, 433, 64, -8, 1] 896582564963182109633998839161059836945394170470535755157470703125 "24T1353" [2, 4] "24.4.896582564963182109633998839161059836945394170470535755157470703125.8" [19475512782813222496, -23231218386562315844, 13617369347618628836, -4803003296867383829, 1106353490183569846, -163997645628589366, 12076497815529119, 1597327943149004, -932052046902271, 219160819386589, -25936497075049, -323147684679, 555095871071, -65395748189, -531819484, 871740359, -113806816, 9366649, -129406, -119376, 11696, 476, -104, -9, 1] 896582564963182109633998839161059836945394170470535755157470703125 "24T1353" [2, 4, 4] # 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 $1\leq i < n$.