# Number fields downloaded from the LMFDB on 16 May 2026.
# Search link: https://www.lmfdb.org/NumberField/?degree=43
# Query "{'degree': 43}" returned 31 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.



"43.1.17193642429484970947547009316647777700494014568443353175841873665979443.1"	[-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, 1]	-17193642429484970947547009316647777700494014568443353175841873665979443	"43T10"	NULL
"43.1.17493904304575564092260553260976286616122110509580830730313660732011571.1"	[-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, 1]	-17493904304575564092260553260976286616122110509580830730313660732011571	"43T10"	NULL
"43.1.73637590561926486479405912723338063733861614147899437889774600865607334016376111104.1"	[-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, 1]	-73637590561926486479405912723338063733861614147899437889774600865607334016376111104	"43T10"	NULL
"43.1.74958156254086214703443254449187449178282319802617704397438354084969927035548860416.1"	[-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, 1]	-74958156254086214703443254449187449178282319802617704397438354084969927035548860416	"43T10"	NULL
"43.1.76278721946245792796543050878464477850730861202878156857131538565555284161188593664.1"	[-2, -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, 1]	-76278721946245792796543050878464477850730861202878156857131538565555284161188593664	"43T10"	NULL
"43.1.76278721946245942927480596140901011555290620195994458453535643977522773548439024487.1"	[-4, -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, 1]	-76278721946245942927480596140901011555290620195994458453535643977522773548439024487	"43T10"	NULL
"43.1.76278721946245942927480596175036834622703025457335970905102107304332520054721609728.1"	[-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, 1]	-76278721946245942927480596175036834622703025457335970905102107304332520054721609728	"43T8"	NULL
"43.1.77599287638405671151517937900886220067123731112054237412765860523695113073894359040.1"	[-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, 1]	-77599287638405671151517937900886220067123731112054237412765860523695113073894359040	"43T10"	NULL
"43.1.78919853330565399375555279626735605511544436766772503920429613743057706093067108352.1"	[-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, 1]	-78919853330565399375555279626735605511544436766772503920429613743057706093067108352	"43T10"	NULL
"43.3.49281526270717699300509198722877092539271399739749264767815294437558226291076362607004621.1"	[-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, 1]	49281526270717699300509198722877092539271399739749264767815294437558226291076362607004621	"43T10"	NULL
"43.1.49281602549439645563795899570503535096009803731586754262094262078677622576674184527609856.1"	[-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, 1]	-49281602549439645563795899570503535096009803731586754262094262078677622576674184527609856	"43T10"	NULL
"43.1.1897738148225932665352052841514539485919791655231439979873933030555398665105804881388324651.1"	[-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, 1]	-1897738148225932665352052841514539485919791655231439979873933030555398665105804881388324651	"43T10"	NULL
"43.1.1897738149546498357511630934614335915196820327679981380134385490248583145691162007028057899.1"	[-3, -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, 1]	-1897738149546498357511630934614335915196820327679981380134385490248583145691162007028057899	"43T10"	NULL
"43.1.1897738149546498357511781065551881211769177099652145634592199538219151884468397900561073963.1"	[-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, 1]	-1897738149546498357511781065551881211769177099652145634592199538219151884468397900561073963	"43T8"	NULL
"43.1.1947019675817216056829634037641788571828352280680706931518322895195722202712552030367074091.1"	[-3, 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, 1]	-1947019675817216056829634037641788571828352280680706931518322895195722202712552030367074091	"43T10"	NULL
"43.1.3354773669271577653568285803640546404693410239170973401358900578361902104169371242332059336704.1"	[-4, -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, 1]	-3354773669271577653568285803640546404693410239170973401358900578361902104169371242332059336704	"43T10"	NULL
"43.43.9952594992767664919302480397055915291864099597331865326873171797085952964991380209309055259529.1"	[-1, -54, -370, 12893, 90620, -1134285, -2599666, 27505221, 37041429, -294221159, -287529242, 1732796449, 1310830451, -6269987218, -3755010451, 14834334408, 7121607714, -23896748020, -9297003415, 27001558938, 8590544711, -21895326590, -5730420663, 12962901258, 2797394685, -5671315301, -1007873658, 1846875875, 268953093, -448758019, -53056499, 81126975, 7671648, -10810701, -798942, 1042773, 58076, -70521, -2786, 3160, 79, -84, -1, 1]	9952594992767664919302480397055915291864099597331865326873171797085952964991380209309055259529	"43T1"	[]
"43.1.11615818670173463204840555036546350439901529918004080992211006956821939425490704861593631716403.1"	[-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, 1]	-11615818670173463204840555036546350439901529918004080992211006956821939425490704861593631716403	"43T10"	NULL
"43.1.11615818670249741926786783635700464005809046848845495205636184619664502520703084303881154330624.1"	[-2, 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, 1]	-11615818670249741926786783635700464005809046848845495205636184619664502520703084303881154330624	"43T10"	NULL
"43.1.11617716408323009703198049473838535290845779191322444325813440848297619565283220181726993794859.1"	[-3, 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, 1]	-11617716408323009703198049473838535290845779191322444325813440848297619565283220181726993794859	"43T10"	NULL
"43.1.335477317645632815204821422239306587903706814127360732280247849979340897599838242672095300419584.1"	[-4, -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, 1]	-335477317645632815204821422239306587903706814127360732280247849979340897599838242672095300419584	"43T10"	NULL
"43.1.42669778834265525646357377125943811028661226892061887227006364333992726275526826083130013680402432.1"	[-2, 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, 1]	-42669778834265525646357377125943811028661226892061887227006364333992726275526826083130013680402432	"43T10"	NULL
"43.3.170679115337062026306707562240488543267018465011509144916187967841691937460987908046922232801004493.1"	[-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, 1]	170679115337062026306707562240488543267018465011509144916187967841691937460987908046922232801004493	"43T10"	NULL
"43.1.3772838381179050124528639896195900262054652258977602756862847748815326440308126620948314666748046875.1"	[-5, -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, 1]	-3772838381179050124528639896195900262054652258977602756862847748815326440308126620948314666748046875	"43T10"	NULL
"43.1.3943517496516112150835347458453582447751155694936658911095683494357512392337557882171078773215030811.1"	[-5, -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, 1]	-3943517496516112150835347458453582447751155694936658911095683494357512392337557882171078773215030811	"43T10"	NULL
"43.1.3943517496516112150835347458453732578688700991509015683067847748815326440308126620948314666748046875.1"	[-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, 1]	-3943517496516112150835347458453732578688700991509015683067847748815326440308126620948314666748046875	"43T8"	NULL
"43.1.3943517496565393677106065157771585550778608351568190864096409045741449797284696939192468796554047003.1"	[-5, 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, 1]	-3943517496565393677106065157771585550778608351568190864096409045741449797284696939192468796554047003	"43T10"	NULL
"43.1.4114196611853174177142055020711564895322749724040428609272847748815326440308126620948314666748046875.1"	[-5, 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, 1]	-4114196611853174177142055020711564895322749724040428609272847748815326440308126620948314666748046875	"43T10"	NULL
"43.43.444677695956607074780919035502815976195331208356891344496189409566167816104341684988715963441514628066825276961.1"	[1403424452501, 7715035819499, -29312399288701, -166814479770181, 160575454035341, 1249295360564226, -40649514655288, -4579013283800046, -2183897418754987, 9075115953805640, 7728522779787490, -9886946345787043, -12486138055465655, 5604720542659268, 11681897146129224, -902655412698460, -7011360351157045, -937302717934781, 2859558999201013, 795734675389219, -819692515707425, -325504806681996, 168116844068603, 86521587673786, -24791417254787, -16218499586789, 2604819705111, 2218265633870, -189102381409, -224649876009, 8734312688, 16894280750, -185951871, -936601673, -4029555, 37567316, 410572, -1053196, -12392, 19424, 177, -210, -1, 1]	444677695956607074780919035502815976195331208356891344496189409566167816104341684988715963441514628066825276961	"43T1"	NULL
"43.43.101554487341926844216969846221080781237895685553160643066505060271306560421136844000042773102578137815468250505123323781432009.1"	[18221685305593614631, 193966087122261586736, -90731765675438007935, -2063406363097082700573, 1408273844297690430983, 6100770874409784770241, -5304980067316890480275, -8188607772821299311768, 8799946070456643867879, 5701308150866478483031, -8037383143222291539220, -1981830512581480780535, 4502348802434562518439, 139804498689687778120, -1650628224018564426381, 166866840849449475306, 413102772817597404709, -79441906767691419891, -72594260758345520281, 19125104314160845818, 9129779342869086290, -2983438147701667253, -832294062829303477, 325598895562519548, 55430418074111539, -25775660913572833, -2705517116526302, 1509193690295722, 96629348183472, -65969924993323, -2510570612339, 2155113502777, 47086072273, -52222121579, -636827636, 921810336, 6368083, -11463758, -49892, 94630, 301, -462, -1, 1]	101554487341926844216969846221080781237895685553160643066505060271306560421136844000042773102578137815468250505123323781432009	"43T1"	NULL
"43.43.162686032778208990102858628859785420567496242104134005559503199497609643882923419981647276367075859293620549051195773051892887390454194801.1"	[2572343484535669027372727, 14571491290875996179048369, -7933826886293481520543865, -118832831442545361067982885, -33622183448420036732904682, 294222510747058121054570076, 170563929616560831037152772, -263326021640166500214931210, -192818992256047972290970619, 119306453339061961678830425, 107534691527795878017440861, -30515963297503545387850753, -36228120784944419229616219, 4203376282667672263555078, 8070410654031859605939559, -134666416459554320685647, -1247516677692517201034941, -60838518194479293931722, 137573117593988043652238, 13140703542377826450494, -10999588529245966977339, -1445787294046167541816, 643326429466437698421, 103748981858304966660, -27621776964130665961, -5201925841676982029, 869728040966016241, 187329847291648022, -19967338988758228, -4897798025442393, 330473143424644, 93052767268909, -3868686195114, -1275092913940, 31035129471, 12386889944, -161379172, -82668446, 487362, 358276, -645, -903, 0, 1]	162686032778208990102858628859785420567496242104134005559503199497609643882923419981647276367075859293620549051195773051892887390454194801	"43T1"	NULL


# Label --
#    Each (global) number field has a unique label of the form d.r.D.i where
#    <ul>
#    <li>\(d\) is the degree;
#    <li>\(r\) is the real signature;  the full signature is therefore \([r,(d-r)/2]\);
#    <li>\(D\) is the absolute value of the discriminant;
#    <li>\(i\) is the index, counting from 1.  This is in case there is more than one
#      field with the same signature and absolute value of the
#      discriminant: for example <a href="/NumberField/4.0.1008.1">4.0.1008.1</a> and <a href="/NumberField/4.0.1008.2">4.0.1008.2</a>.
#    </ul>
#    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 $1\leq i < n$.