# Number fields downloaded from the LMFDB on 19 August 2026. # Search link: https://www.lmfdb.org/NumberField/?completions=29.1.20.19a1.3 # Query "{'local_algs': {'$contains': ['29.1.20.19a1.3']}}" returned 8 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. "24.4.270761008401829353605241639483649123576469719409942626953125.2" [-6585670043891, 22227043889439, 8069786480454, -7274032113936, -3470676872911, 156208942105, 402014550880, 16736471880, -17099510555, 9448143015, 3603058165, -567629035, -597431900, -25107040, 20488210, 5496080, -599575, -1885, 2465, 2465, -1304, -4, 6, -4, 1] 270761008401829353605241639483649123576469719409942626953125 "24T1353" [2, 4] "24.4.270761008401829353605241639483649123576469719409942626953125.6" [31122013545, -372292473393, 654364923577, 717193087189, 721074386011, 890153835770, 917999196792, 716490646492, 414475529204, 177543971151, 53296332145, 9616225897, 288484927, -321520216, -40857249, 12951695, 4529857, 300092, -139051, 13841, 4820, 257, 47, -6, 1] 270761008401829353605241639483649123576469719409942626953125 "24T1353" [4] "24.4.270761008401829353605241639483649123576469719409942626953125.8" [338358763849, -687175108716, -277834934826, 287183270484, 56974469259, 97457228755, -6224510120, -77668721620, -42563392655, -19884822660, -5226790060, 868904090, 625240725, 82989010, -10638940, 173855, 1583400, 81490, 15515, 2465, -2174, -4, 6, -4, 1] 270761008401829353605241639483649123576469719409942626953125 "24T1353" [2, 4] "24.4.270761008401829353605241639483649123576469719409942626953125.9" [8710672685155, 18569747459402, -4499980454036, -37892792137502, -33540673060979, -12469668216145, -4731483065509, -4678953132493, -3495074371706, -1562629162232, -428838287440, -64602797057, -1226398054, 1548866817, 240393964, -18808685, -7247986, -679237, -33044, 33972, 800, -88, -6, -7, 1] 270761008401829353605241639483649123576469719409942626953125 "24T1353" [4] "24.4.6769025210045733840131040987091228089411742985248565673828125.1" [-142134894616225, -333094778067275, -303707165704555, -75005259428795, 52566236252030, 10526861697400, -9862870228300, 823833868955, -363153392130, -36121402305, 66919163400, -6862168300, 1179057105, 208438870, -195744705, 28414975, -4281025, -191630, 206480, -25870, 5830, -480, -46, -4, 1] 6769025210045733840131040987091228089411742985248565673828125 "24T1353" NULL "24.4.6769025210045733840131040987091228089411742985248565673828125.3" [56010581875, -99248258750, -286773405625, 1137682488125, -523765574375, -146123238875, 43848203000, 30497763000, -3500169500, 1650665500, -1167334100, -162429725, -58519100, 48632275, 227650, 297395, -237655, 60320, -10005, -1305, 1, -4, 6, -4, 1] 6769025210045733840131040987091228089411742985248565673828125 "24T1353" [2, 4, 4] "24.4.6769025210045733840131040987091228089411742985248565673828125.5" [751938081875, 125130178750, -882473280625, 79569910000, 436357363125, -128610048250, -91930159500, 31324081750, 4226336750, -3653343875, -1875785975, 57825275, 65927150, 19324150, 2873900, 1127520, -52780, -30305, -10005, -1305, 1, -4, 6, -4, 1] 6769025210045733840131040987091228089411742985248565673828125 "24T1353" [2, 4, 20] "24.4.6769025210045733840131040987091228089411742985248565673828125.7" [88163445447261, -234817172836068, 402065305569184, -355090002574042, 178557365193021, -75512246952018, 26369655366134, -12556632345742, 4488374944421, -1291349956173, 306625002524, -86360583312, 11394285106, -2189868478, 105814264, -7907232, -5474434, 859042, -168196, 20223, -1759, 67, 29, -2, 1] 6769025210045733840131040987091228089411742985248565673828125 "24T1353" [2, 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 $1\leq i < n$.