# Number fields downloaded from the LMFDB on 27 July 2026. # Search link: https://www.lmfdb.org/NumberField/?completions=109.1.20.19a1.3 # Query "{'local_algs': {'$contains': ['109.1.20.19a1.3']}}" returned 20 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. "24.4.7937670270439849675741558348430291472575696480274200439453125.1" [-28803949486872709, 26007923032387921, -46488436134411758, 10940492801655277, 3270771528761191, -1275716187976979, -49731181757494, 43709580488417, 1067500702067, -1777336978324, -199348343379, 136201021381, 4639762577, -5857285238, 476072951, 146129816, -29238869, -1314528, 595607, -50169, -4924, 1011, -48, -3, 1] 7937670270439849675741558348430291472575696480274200439453125 "24T1353" [2] "24.4.7937670270439849675741558348430291472575696480274200439453125.3" [-557334739690193, -884202385961979, 1645956357578605, 39193205099073, -1624823723192944, 1369547075868283, -464202423303746, 37908911704435, 16629918291242, -3526824626556, -416576528428, 225699820351, -30035184260, -561722527, 758033746, -44809022, -18135546, 4101970, -103228, -50756, 5232, 191, -10, -7, 1] 7937670270439849675741558348430291472575696480274200439453125 "24T1353" [2] "24.4.198441756760996241893538958710757286814392412006855010986328125.1" [-43788301427225, 63713167016750, -37058252983075, 35395870489275, -24477246719700, -5700030099325, 15635409017400, -6598619973625, 18717638250, 931546438375, -437711604180, 103071532200, -8141707720, -4971008140, 2075133195, -636706975, 115591720, -19567110, 2635055, -272065, 32594, -1880, 209, -5, 1] 198441756760996241893538958710757286814392412006855010986328125 "24T1353" NULL "24.4.198441756760996241893538958710757286814392412006855010986328125.4" [25870416406358929, 1614204526794420, 10994425979664316, 1441034178181325, 762235631605796, 664506903596229, -295898798851140, 136568401104246, -57112338853215, 12307692947266, -2881514159386, 394516859000, 8589863531, -887782745, 1390598856, -588128056, 27898590, -726439, 1017560, -21919, 2534, -1720, -4, 0, 1] 198441756760996241893538958710757286814392412006855010986328125 "24T1353" [2, 2] "24.4.3100652449390566279586546229855582606474881437607109546661376953125.1" [-12243302508029584947, -7069968264645204397, 1807623391985900651, 1624646840245764032, 128927288240805416, -31548707766373451, 6505464300663649, 656059631472823, -618357410982019, -51292150331142, 1948975349062, -353321674938, 328478108269, 52399337618, -2717296001, -450020679, -76160579, -9808513, 1053984, 158842, 3228, -147, -129, -8, 1] 3100652449390566279586546229855582606474881437607109546661376953125 "24T1353" NULL "24.4.3100652449390566279586546229855582606474881437607109546661376953125.2" [-97343601424442475, 144706889504174735, -77837845170326738, 220719460432342, 23737920905602591, -14584396630946395, 4332524947644800, -504355550480534, -85935470940599, 51960327427103, -10776819849125, 1233676621485, 18570656718, -37029689892, 9115230369, -1237460435, 80487860, 4410739, -649511, -154273, 12895, 2570, -248, -8, 1] 3100652449390566279586546229855582606474881437607109546661376953125 "24T1353" NULL "24.4.3100652449390566279586546229855582606474881437607109546661376953125.3" [-138227932264867, 1275542802740324, -1618865338234484, 1022073252974979, -429046905921579, 132621568642227, -31152601913539, 6058909229774, -1042831991924, 128787890109, 946807313, -6093135791, 1688745126, -291002611, 54820101, -9016213, 1377786, 131264, -116504, 2279, 773, 854, -119, -6, 1] 3100652449390566279586546229855582606474881437607109546661376953125 "24T1353" NULL "24.4.3100652449390566279586546229855582606474881437607109546661376953125.6" [1539376879689673, 1128448284182756, -773374619523099, -163900066389289, -34948090904564, -10830692287662, -12470926761144, -951472296959, 1061941855001, 264536772046, -50043794302, -9890668734, 3702224901, -279645234, -161302319, 32762768, 3255941, -1075364, 24186, 13671, -597, 41, -19, -4, 1] 3100652449390566279586546229855582606474881437607109546661376953125 "24T1353" NULL "24.4.77516311234764156989663655746389565161872035940177738666534423828125.1" [-526693993977600, -7242594366566400, -7970914560545600, -14251912534362400, -12074325776481600, -10323209578181600, -5203181801343800, -2178231732896650, -575516846950175, -80289385905850, -325999515690, 3257072992890, 564886906010, -95001237860, -24980880090, 389603410, 12191610, -52795400, 2423260, 228200, -36164, 4324, 26, -1, 1] 77516311234764156989663655746389565161872035940177738666534423828125 "24T1353" NULL "24.4.77516311234764156989663655746389565161872035940177738666534423828125.2" [-156414394013519, -147220000013204, 141232682009801, 70895246532636, -56632692617989, -58884808349206, -19502917205716, -2081058435416, 303196438954, 9787960309, -53028192729, -17573498949, -2840137354, -334047504, -58221914, -10570166, -2160816, -257131, -36951, 2289, 1, -4, 6, -4, 1] 77516311234764156989663655746389565161872035940177738666534423828125 "24T1353" NULL "24.4.77516311234764156989663655746389565161872035940177738666534423828125.5" [128639378702231, 123118816711296, 81660544543551, -115542441171739, 7413167545761, 11095491646169, -2871158445966, 1090144100834, -872862194796, 502337228434, -177590708604, 42847272051, -7905503604, 1274179371, -225945664, 39556209, -6534441, 764744, -36951, 2289, 1, -4, 6, -4, 1] 77516311234764156989663655746389565161872035940177738666534423828125 "24T1353" NULL "24.4.77516311234764156989663655746389565161872035940177738666534423828125.8" [264495221253438445, 474137879628295090, 480945466938287990, 378979505633068470, 224526598105176305, 112913570686439785, 43107261720991995, 12436896913663470, 2730610794749210, 462660530553940, 44959677718280, -4307072744615, -1888734115400, -209785570025, -20115830640, -1787464860, 361677480, 62282400, -3287795, -551700, 43724, 3268, -332, -11, 1] 77516311234764156989663655746389565161872035940177738666534423828125 "24T1353" NULL "24.4.1211192363043189952963494621037336955654250561565277166664600372314453125.5" [613116592600078125, 0, 63149113723696375, 0, -7199514724704670, 0, 183163137913575, 0, 38406857938650, 0, -1856266269475, 0, -37017062175, 0, 3043866965, 0, 5986825, 0, -1814850, 0, 16350, 0, 0, 0, 1] 1211192363043189952963494621037336955654250561565277166664600372314453125 "24T1353" [4] "24.4.1211192363043189952963494621037336955654250561565277166664600372314453125.6" [1034553768331546475, 827626342716637805, 650857501854281055, 711676780139306705, 274147047781904980, 34369213753089725, -8643679244966650, -10114745628055450, -2500699679307025, 279974977306050, 151550967164950, 7393793313620, -2064320544855, -309619963080, -16119328205, 4472357200, 809984450, -31340225, -13123600, 27250, 113470, 1086, -539, -4, 1] 1211192363043189952963494621037336955654250561565277166664600372314453125 "24T1353" NULL "24.4.1211192363043189952963494621037336955654250561565277166664600372314453125.7" [32717178984424509504, 29487604013892516000, -26759151775733541280, -16395190145863156920, -1641152941333342264, -96335762207475134, -231843662150173315, -68216971899770040, -5382532219292770, 232636356206209, 16789597854924, 3284311801605, 2472619323385, 119927852710, -31140205664, -3314452609, -510772970, 12487020, 4859175, -92261, 104704, -2015, 500, -10, 1] 1211192363043189952963494621037336955654250561565277166664600372314453125 "24T1353" NULL "24.4.1211192363043189952963494621037336955654250561565277166664600372314453125.8" [777019087232248674575, -220230876669439407034, 141472039110471732990, -22438363614405112672, 2216757763306056151, -385128811285350640, -45579398797207878, 5105977496091715, 3434957990181416, 530594726847237, 83412644565090, 9477419064504, -127075389375, -93433926183, -12334220651, -2923221605, -64580422, -6559960, -1200746, 247873, -930, 401, -40, -12, 1] 1211192363043189952963494621037336955654250561565277166664600372314453125 "24T1353" NULL "24.4.30279809076079748824087365525933423891356264039131929166615009307861328125.2" [2271403140498000, 0, 80286843623000, 0, 4520800192684405, 0, 1145957171176100, 0, 122627097345300, 0, 7233058852900, 0, 236206858600, 0, 2810042345, 0, -68321200, 0, -2616000, 0, -10900, 0, 0, 0, 1] 30279809076079748824087365525933423891356264039131929166615009307861328125 "24T1353" NULL "24.4.30279809076079748824087365525933423891356264039131929166615009307861328125.4" [1178915603470686505, 907059044865183450, 2345716619408079065, 1910970795246895600, 937304416391101555, 294114815359314515, 83856726566419600, 13391522375867970, 2047220323717975, 45471115514965, 50536588219815, -20356427375050, -2491469565005, 31076049875, -43113578235, -5622369875, 983351675, 127486400, -10627500, -689425, 80006, 0, -327, 0, 1] 30279809076079748824087365525933423891356264039131929166615009307861328125 "24T1353" NULL "24.4.30279809076079748824087365525933423891356264039131929166615009307861328125.5" [10069108374601211445, -389400971404249110, -7612128467148681691, -1934235095941169054, 1798798175900669701, 1003641385636169600, 149502475065329320, -9977135306495559, -6138271022337721, -980914470669021, 72891074608505, 103983139272220, 26822906943269, 2651575997586, -30353054894, -22183903385, 54280540, 206603326, 144869, -1553796, 1890, 6125, -186, -9, 1] 30279809076079748824087365525933423891356264039131929166615009307861328125 "24T1353" NULL "24.4.30279809076079748824087365525933423891356264039131929166615009307861328125.6" [102513143473323231709, -33978910528075722209, -19782820377631184533, 5030120600016639407, 738562956788278941, -227488047481772008, 69931212171066273, 9886343624744886, -5008136828513789, -946929364183972, 47307452100241, 46791472406149, 1318676901773, -1037927849712, -18311679996, 8613838618, 649290837, -57122136, -12895796, 330222, 132859, -2914, -338, -8, 1] 30279809076079748824087365525933423891356264039131929166615009307861328125 "24T1353" 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 $1\leq i < n$.