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Properties

Label 32.0.237...896.1
Degree $32$
Signature $[0, 16]$
Discriminant $2.379\times 10^{46}$
Root discriminant \(28.14\)
Ramified primes $2,3,17$
Class number $12$ (GRH)
Class group [12] (GRH)
Galois group $C_2^4:C_4$ (as 32T262)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^30 + 32*x^28 - 80*x^26 + 127*x^24 - 80*x^22 - 224*x^20 + 936*x^18 - 2175*x^16 + 3744*x^14 - 3584*x^12 - 5120*x^10 + 32512*x^8 - 81920*x^6 + 131072*x^4 - 131072*x^2 + 65536)
 
gp: K = bnfinit(y^32 - 8*y^30 + 32*y^28 - 80*y^26 + 127*y^24 - 80*y^22 - 224*y^20 + 936*y^18 - 2175*y^16 + 3744*y^14 - 3584*y^12 - 5120*y^10 + 32512*y^8 - 81920*y^6 + 131072*y^4 - 131072*y^2 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 8*x^30 + 32*x^28 - 80*x^26 + 127*x^24 - 80*x^22 - 224*x^20 + 936*x^18 - 2175*x^16 + 3744*x^14 - 3584*x^12 - 5120*x^10 + 32512*x^8 - 81920*x^6 + 131072*x^4 - 131072*x^2 + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 8*x^30 + 32*x^28 - 80*x^26 + 127*x^24 - 80*x^22 - 224*x^20 + 936*x^18 - 2175*x^16 + 3744*x^14 - 3584*x^12 - 5120*x^10 + 32512*x^8 - 81920*x^6 + 131072*x^4 - 131072*x^2 + 65536)
 

\( x^{32} - 8 x^{30} + 32 x^{28} - 80 x^{26} + 127 x^{24} - 80 x^{22} - 224 x^{20} + 936 x^{18} + \cdots + 65536 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(23790908696561643372461609312578223409406672896\) \(\medspace = 2^{96}\cdot 3^{16}\cdot 17^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(28.14\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3}3^{1/2}17^{1/2}\approx 57.1314274283428$
Ramified primes:   \(2\), \(3\), \(17\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{32768}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{10}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{19}-\frac{1}{8}a^{11}+\frac{1}{8}a^{3}$, $\frac{1}{112}a^{20}-\frac{1}{14}a^{18}+\frac{2}{7}a^{16}+\frac{47}{112}a^{12}+\frac{1}{7}a^{10}+\frac{3}{7}a^{8}-\frac{1}{2}a^{6}-\frac{31}{112}a^{4}-\frac{2}{7}a^{2}+\frac{1}{7}$, $\frac{1}{224}a^{21}-\frac{1}{28}a^{19}+\frac{1}{7}a^{17}-\frac{1}{2}a^{15}-\frac{65}{224}a^{13}+\frac{1}{14}a^{11}-\frac{2}{7}a^{9}-\frac{1}{4}a^{7}-\frac{31}{224}a^{5}-\frac{1}{7}a^{3}-\frac{3}{7}a$, $\frac{1}{448}a^{22}-\frac{1}{14}a^{18}-\frac{5}{28}a^{16}-\frac{65}{448}a^{14}+\frac{3}{8}a^{12}-\frac{5}{14}a^{10}+\frac{13}{56}a^{8}+\frac{193}{448}a^{6}+\frac{3}{8}a^{4}-\frac{2}{7}a^{2}+\frac{2}{7}$, $\frac{1}{896}a^{23}-\frac{1}{28}a^{19}-\frac{5}{56}a^{17}+\frac{383}{896}a^{15}+\frac{3}{16}a^{13}+\frac{9}{28}a^{11}-\frac{43}{112}a^{9}+\frac{193}{896}a^{7}+\frac{3}{16}a^{5}-\frac{1}{7}a^{3}+\frac{1}{7}a$, $\frac{1}{23296}a^{24}-\frac{19}{208}a^{18}+\frac{45}{256}a^{16}+\frac{15}{32}a^{14}+\frac{5}{26}a^{12}+\frac{15}{32}a^{10}+\frac{19}{256}a^{8}-\frac{157}{416}a^{6}+\frac{3}{8}a^{4}+\frac{16}{91}$, $\frac{1}{46592}a^{25}-\frac{19}{416}a^{19}+\frac{45}{512}a^{17}-\frac{17}{64}a^{15}-\frac{21}{52}a^{13}+\frac{15}{64}a^{11}+\frac{19}{512}a^{9}-\frac{157}{832}a^{7}-\frac{5}{16}a^{5}-\frac{75}{182}a$, $\frac{1}{93184}a^{26}+\frac{23}{5824}a^{20}+\frac{571}{7168}a^{18}+\frac{201}{896}a^{16}+\frac{31}{104}a^{14}+\frac{337}{896}a^{12}+\frac{1413}{7168}a^{10}-\frac{3595}{11648}a^{8}+\frac{11}{32}a^{6}+\frac{19}{112}a^{4}+\frac{17}{91}a^{2}+\frac{3}{7}$, $\frac{1}{186368}a^{27}+\frac{23}{11648}a^{21}+\frac{571}{14336}a^{19}+\frac{201}{1792}a^{17}-\frac{73}{208}a^{15}+\frac{337}{1792}a^{13}-\frac{5755}{14336}a^{11}-\frac{3595}{23296}a^{9}+\frac{11}{64}a^{7}-\frac{93}{224}a^{5}-\frac{37}{91}a^{3}-\frac{2}{7}a$, $\frac{1}{11554816}a^{28}+\frac{1}{222208}a^{26}+\frac{3}{722176}a^{24}+\frac{751}{722176}a^{22}+\frac{1275}{888832}a^{20}+\frac{2973}{2888704}a^{18}-\frac{71961}{722176}a^{16}+\frac{35281}{111104}a^{14}+\frac{1773025}{11554816}a^{12}+\frac{1161367}{2888704}a^{10}+\frac{14803}{55552}a^{8}+\frac{30053}{180544}a^{6}+\frac{1243}{22568}a^{4}-\frac{27}{217}a^{2}+\frac{1244}{2821}$, $\frac{1}{23109632}a^{29}+\frac{1}{444416}a^{27}+\frac{3}{1444352}a^{25}+\frac{751}{1444352}a^{23}+\frac{1275}{1777664}a^{21}+\frac{2973}{5777408}a^{19}-\frac{71961}{1444352}a^{17}-\frac{75823}{222208}a^{15}+\frac{1773025}{23109632}a^{13}+\frac{1161367}{5777408}a^{11}+\frac{14803}{111104}a^{9}-\frac{150491}{361088}a^{7}-\frac{21325}{45136}a^{5}+\frac{95}{217}a^{3}+\frac{622}{2821}a$, $\frac{1}{4483268608}a^{30}+\frac{23}{560408576}a^{28}-\frac{3}{2694272}a^{26}+\frac{45}{9038848}a^{24}-\frac{3039873}{4483268608}a^{22}+\frac{58955}{21554176}a^{20}-\frac{990237}{70051072}a^{18}+\frac{147580773}{560408576}a^{16}-\frac{171259579}{344866816}a^{14}-\frac{12516013}{140102144}a^{12}-\frac{19337341}{140102144}a^{10}+\frac{866457}{2694272}a^{8}+\frac{455489}{1094548}a^{6}-\frac{1392331}{4378192}a^{4}-\frac{53}{42098}a^{2}+\frac{100882}{273637}$, $\frac{1}{8966537216}a^{31}+\frac{23}{1120817152}a^{29}-\frac{3}{5388544}a^{27}+\frac{45}{18077696}a^{25}-\frac{3039873}{8966537216}a^{23}+\frac{58955}{43108352}a^{21}-\frac{990237}{140102144}a^{19}+\frac{147580773}{1120817152}a^{17}+\frac{173607237}{689733632}a^{15}-\frac{12516013}{280204288}a^{13}-\frac{19337341}{280204288}a^{11}+\frac{866457}{5388544}a^{9}+\frac{455489}{2189096}a^{7}-\frac{1392331}{8756384}a^{5}-\frac{53}{84196}a^{3}+\frac{50441}{273637}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{12}$, which has order $12$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{14177}{140102144} a^{31} - \frac{841741}{2241634304} a^{29} + \frac{31183}{43108352} a^{27} - \frac{20575}{35025536} a^{25} - \frac{30469}{35025536} a^{23} + \frac{897217}{172433408} a^{21} - \frac{6169573}{560408576} a^{19} + \frac{1059733}{70051072} a^{17} - \frac{585987}{21554176} a^{15} - \frac{11507309}{2241634304} a^{13} + \frac{79556065}{560408576} a^{11} - \frac{577719}{1347136} a^{9} + \frac{22725057}{35025536} a^{7} - \frac{761483}{1094548} a^{5} - \frac{11059}{24056} a^{3} + \frac{425238}{273637} a \)  (order $48$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{222405}{2241634304}a^{30}-\frac{193301}{280204288}a^{28}+\frac{125711}{40029184}a^{26}-\frac{148841}{20014592}a^{24}+\frac{25383867}{2241634304}a^{22}-\frac{685593}{140102144}a^{20}-\frac{7088033}{280204288}a^{18}+\frac{3808507}{40029184}a^{16}-\frac{446153403}{2241634304}a^{14}+\frac{23515127}{70051072}a^{12}-\frac{94377995}{280204288}a^{10}-\frac{6363527}{10007296}a^{8}+\frac{4043233}{1250912}a^{6}-\frac{33943919}{4378192}a^{4}+\frac{3120262}{273637}a^{2}-\frac{395977}{39091}$, $\frac{65823}{160116736}a^{30}-\frac{351081}{140102144}a^{28}+\frac{77567}{10007296}a^{26}-\frac{1051461}{70051072}a^{24}+\frac{18098855}{1120817152}a^{22}+\frac{581131}{70051072}a^{20}-\frac{5479013}{70051072}a^{18}+\frac{28971373}{140102144}a^{16}-\frac{461818535}{1120817152}a^{14}+\frac{20160317}{35025536}a^{12}-\frac{3649497}{70051072}a^{10}-\frac{10878107}{4378192}a^{8}+\frac{2688163}{336784}a^{6}-\frac{66392021}{4378192}a^{4}+\frac{4770240}{273637}a^{2}-\frac{2557899}{273637}$, $\frac{299625}{640466944}a^{30}-\frac{842507}{280204288}a^{28}+\frac{1364893}{140102144}a^{26}-\frac{5419619}{280204288}a^{24}+\frac{98009889}{4483268608}a^{22}+\frac{5150721}{560408576}a^{20}-\frac{13828957}{140102144}a^{18}+\frac{20861485}{80058368}a^{16}-\frac{2320201313}{4483268608}a^{14}+\frac{422116341}{560408576}a^{12}-\frac{2526247}{17512768}a^{10}-\frac{15134893}{5003648}a^{8}+\frac{10952297}{1094548}a^{6}-\frac{42688053}{2189096}a^{4}+\frac{6250766}{273637}a^{2}-\frac{3337979}{273637}$, $\frac{20723}{560408576}a^{30}-\frac{95195}{560408576}a^{28}+\frac{201917}{280204288}a^{26}-\frac{1379}{769792}a^{24}+\frac{1122813}{560408576}a^{22}+\frac{150067}{560408576}a^{20}-\frac{1895833}{280204288}a^{18}+\frac{56955}{2501824}a^{16}-\frac{23332293}{560408576}a^{14}+\frac{29837421}{560408576}a^{12}-\frac{13283343}{280204288}a^{10}-\frac{1884443}{10007296}a^{8}+\frac{1796807}{2189096}a^{6}-\frac{857919}{547274}a^{4}+\frac{1819015}{1094548}a^{2}-\frac{41499}{39091}$, $\frac{812437}{1120817152}a^{30}-\frac{1295493}{280204288}a^{28}+\frac{318079}{20014592}a^{26}-\frac{577013}{17512768}a^{24}+\frac{46017323}{1120817152}a^{22}+\frac{1352279}{280204288}a^{20}-\frac{99509}{645632}a^{18}+\frac{61933687}{140102144}a^{16}-\frac{993761227}{1120817152}a^{14}+\frac{370533629}{280204288}a^{12}-\frac{11180257}{20014592}a^{10}-\frac{315767967}{70051072}a^{8}+\frac{71735553}{4378192}a^{6}-\frac{148615359}{4378192}a^{4}+\frac{6589319}{156364}a^{2}-\frac{612681}{21049}$, $\frac{359375}{2241634304}a^{30}-\frac{1116711}{1120817152}a^{28}+\frac{435837}{140102144}a^{26}-\frac{130989}{20014592}a^{24}+\frac{521935}{72310784}a^{22}+\frac{3315475}{1120817152}a^{20}-\frac{568891}{17512768}a^{18}+\frac{3496605}{40029184}a^{16}-\frac{385992865}{2241634304}a^{14}+\frac{289483589}{1120817152}a^{12}-\frac{1111661}{17512768}a^{10}-\frac{4469359}{5003648}a^{8}+\frac{8274417}{2501824}a^{6}-\frac{4128437}{625456}a^{4}+\frac{2090353}{273637}a^{2}-\frac{188546}{39091}$, $\frac{132355}{4483268608}a^{30}-\frac{205321}{1120817152}a^{28}+\frac{71509}{70051072}a^{26}-\frac{19225}{9038848}a^{24}+\frac{13453885}{4483268608}a^{22}-\frac{1798585}{1120817152}a^{20}-\frac{71751}{10777088}a^{18}+\frac{14795759}{560408576}a^{16}-\frac{262699613}{4483268608}a^{14}+\frac{104947573}{1120817152}a^{12}-\frac{1654389}{17512768}a^{10}-\frac{2188639}{8756384}a^{8}+\frac{15056407}{17512768}a^{6}-\frac{9731875}{4378192}a^{4}+\frac{932517}{273637}a^{2}-\frac{851566}{273637}$, $\frac{1530117}{8966537216}a^{31}-\frac{623135}{4483268608}a^{30}-\frac{2815765}{2241634304}a^{29}+\frac{511999}{560408576}a^{28}+\frac{10051}{2501824}a^{27}-\frac{393019}{140102144}a^{26}-\frac{4651453}{560408576}a^{25}+\frac{213557}{40029184}a^{24}+\frac{85045819}{8966537216}a^{23}-\frac{26098017}{4483268608}a^{22}+\frac{7019303}{2241634304}a^{21}-\frac{156867}{40029184}a^{20}-\frac{11697429}{280204288}a^{19}+\frac{4172533}{140102144}a^{18}+\frac{17370503}{160116736}a^{17}-\frac{41141083}{560408576}a^{16}-\frac{1943251803}{8966537216}a^{15}+\frac{642757985}{4483268608}a^{14}+\frac{742156477}{2241634304}a^{13}-\frac{4016085}{20014592}a^{12}-\frac{5462463}{70051072}a^{11}-\frac{239803}{35025536}a^{10}-\frac{23866309}{20014592}a^{9}+\frac{32952091}{35025536}a^{8}+\frac{2655479}{625456}a^{7}-\frac{51129713}{17512768}a^{6}-\frac{72628477}{8756384}a^{5}+\frac{11686293}{2189096}a^{4}+\frac{5475455}{547274}a^{3}-\frac{230564}{39091}a^{2}-\frac{1385024}{273637}a+\frac{720114}{273637}$, $\frac{1643099}{8966537216}a^{31}-\frac{1085547}{1120817152}a^{29}+\frac{874093}{280204288}a^{27}-\frac{3272091}{560408576}a^{25}+\frac{7515763}{1280933888}a^{23}+\frac{2464961}{560408576}a^{21}-\frac{8936719}{280204288}a^{19}+\frac{92009087}{1120817152}a^{17}-\frac{208627827}{1280933888}a^{15}+\frac{60670979}{280204288}a^{13}-\frac{856631}{140102144}a^{11}-\frac{146734781}{140102144}a^{9}+\frac{111955101}{35025536}a^{7}-\frac{51860259}{8756384}a^{5}+\frac{1757115}{273637}a^{3}-\frac{284829}{78182}a+1$, $\frac{6931}{35025536}a^{31}-\frac{3141569}{2241634304}a^{29}+\frac{217865}{43108352}a^{27}-\frac{188057}{17512768}a^{25}+\frac{1917025}{140102144}a^{23}+\frac{32069}{172433408}a^{21}-\frac{26829151}{560408576}a^{19}+\frac{9793673}{70051072}a^{17}-\frac{6075777}{21554176}a^{15}+\frac{974512543}{2241634304}a^{13}-\frac{131114293}{560408576}a^{11}-\frac{1821501}{1347136}a^{9}+\frac{3510845}{673568}a^{7}-\frac{11964697}{1094548}a^{5}+\frac{339055}{24056}a^{3}-\frac{5200631}{547274}a+1$, $\frac{17603}{2241634304}a^{31}+\frac{9885}{280204288}a^{29}-\frac{11497}{43108352}a^{27}+\frac{55365}{70051072}a^{25}-\frac{3213635}{2241634304}a^{23}+\frac{12905}{10777088}a^{21}+\frac{158611}{80058368}a^{19}-\frac{2466775}{280204288}a^{17}+\frac{3221775}{172433408}a^{15}-\frac{1413137}{35025536}a^{13}+\frac{2460469}{80058368}a^{11}+\frac{206035}{10777088}a^{9}-\frac{196593}{673568}a^{7}+\frac{7182295}{8756384}a^{5}-\frac{29889}{21049}a^{3}+\frac{636009}{547274}a-1$, $\frac{1212019}{4483268608}a^{31}+\frac{36745}{70051072}a^{30}-\frac{2018871}{1120817152}a^{29}-\frac{1930121}{560408576}a^{28}+\frac{212371}{35025536}a^{27}+\frac{799023}{70051072}a^{26}-\frac{494083}{40029184}a^{25}-\frac{403177}{17512768}a^{24}+\frac{9380363}{640466944}a^{23}+\frac{263551}{10007296}a^{22}+\frac{4714197}{1120817152}a^{21}+\frac{699695}{80058368}a^{20}-\frac{8407187}{140102144}a^{19}-\frac{283765}{2501824}a^{18}+\frac{92029339}{560408576}a^{17}+\frac{769783}{2501824}a^{16}-\frac{210342059}{640466944}a^{15}-\frac{1538729}{2501824}a^{14}+\frac{548111015}{1120817152}a^{13}+\frac{72683985}{80058368}a^{12}-\frac{5296397}{35025536}a^{11}-\frac{1231153}{5003648}a^{10}-\frac{247805255}{140102144}a^{9}-\frac{536309}{156364}a^{8}+\frac{13646067}{2189096}a^{7}+\frac{8059105}{673568}a^{6}-\frac{54689639}{4378192}a^{5}-\frac{6368748}{273637}a^{4}+\frac{4804789}{312728}a^{3}+\frac{7615022}{273637}a^{2}-\frac{197760}{21049}a-\frac{4460909}{273637}$, $\frac{96031}{689733632}a^{31}-\frac{175319}{320233472}a^{30}-\frac{1068835}{1120817152}a^{29}+\frac{967537}{280204288}a^{28}+\frac{422197}{140102144}a^{27}-\frac{3491}{312728}a^{26}-\frac{3410427}{560408576}a^{25}+\frac{3044849}{140102144}a^{24}+\frac{63428333}{8966537216}a^{23}-\frac{52648095}{2241634304}a^{22}+\frac{1055497}{560408576}a^{21}-\frac{1695459}{140102144}a^{20}-\frac{2087303}{70051072}a^{19}+\frac{4020755}{35025536}a^{18}+\frac{13113409}{160116736}a^{17}-\frac{82325069}{280204288}a^{16}-\frac{1472164077}{8966537216}a^{15}+\frac{1315830431}{2241634304}a^{14}+\frac{67954339}{280204288}a^{13}-\frac{58627545}{70051072}a^{12}-\frac{12469001}{280204288}a^{11}+\frac{7334913}{70051072}a^{10}-\frac{18131965}{20014592}a^{9}+\frac{125613959}{35025536}a^{8}+\frac{15611619}{5003648}a^{7}-\frac{7797731}{673568}a^{6}-\frac{27676431}{4378192}a^{5}+\frac{95589677}{4378192}a^{4}+\frac{2058453}{273637}a^{3}-\frac{6899332}{273637}a^{2}-\frac{2590941}{547274}a+\frac{3717666}{273637}$, $\frac{2317989}{8966537216}a^{31}+\frac{744937}{2241634304}a^{30}-\frac{4321397}{2241634304}a^{29}-\frac{1418713}{560408576}a^{28}+\frac{3779831}{560408576}a^{27}+\frac{2260387}{280204288}a^{26}-\frac{8199277}{560408576}a^{25}-\frac{2278217}{140102144}a^{24}+\frac{23658701}{1280933888}a^{23}+\frac{39628247}{2241634304}a^{22}+\frac{843527}{2241634304}a^{21}+\frac{4065651}{560408576}a^{20}-\frac{36422209}{560408576}a^{19}-\frac{23454919}{280204288}a^{18}+\frac{212531233}{1120817152}a^{17}+\frac{60533901}{280204288}a^{16}-\frac{492157613}{1280933888}a^{15}-\frac{948777335}{2241634304}a^{14}+\frac{1346301693}{2241634304}a^{13}+\frac{358888785}{560408576}a^{12}-\frac{160972641}{560408576}a^{11}-\frac{28848397}{280204288}a^{10}-\frac{8449855}{4519424}a^{9}-\frac{1402399}{564928}a^{8}+\frac{250248639}{35025536}a^{7}+\frac{146436973}{17512768}a^{6}-\frac{2298369}{156364}a^{5}-\frac{34925011}{2189096}a^{4}+\frac{5899223}{312728}a^{3}+\frac{5291386}{273637}a^{2}-\frac{3481350}{273637}a-\frac{2700378}{273637}$, $\frac{845017}{8966537216}a^{31}+\frac{175319}{320233472}a^{30}-\frac{444967}{2241634304}a^{29}-\frac{967537}{280204288}a^{28}+\frac{87705}{560408576}a^{27}+\frac{3491}{312728}a^{26}+\frac{569203}{560408576}a^{25}-\frac{3044849}{140102144}a^{24}-\frac{26052633}{8966537216}a^{23}+\frac{52648095}{2241634304}a^{22}+\frac{12328129}{2241634304}a^{21}+\frac{1695459}{140102144}a^{20}-\frac{1972279}{560408576}a^{19}-\frac{4020755}{35025536}a^{18}-\frac{6224451}{1120817152}a^{17}+\frac{82325069}{280204288}a^{16}+\frac{104032057}{8966537216}a^{15}-\frac{1315830431}{2241634304}a^{14}-\frac{155008109}{2241634304}a^{13}+\frac{58627545}{70051072}a^{12}+\frac{100088277}{560408576}a^{11}-\frac{7334913}{70051072}a^{10}-\frac{4811075}{17512768}a^{9}-\frac{125613959}{35025536}a^{8}-\frac{1282213}{35025536}a^{7}+\frac{7797731}{673568}a^{6}+\frac{8212273}{8756384}a^{5}-\frac{95589677}{4378192}a^{4}-\frac{5569673}{2189096}a^{3}+\frac{6899332}{273637}a^{2}+\frac{1339573}{547274}a-\frac{3444029}{273637}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 37735500852.51004 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 37735500852.51004 \cdot 12}{48\cdot\sqrt{23790908696561643372461609312578223409406672896}}\cr\approx \mathstrut & 0.360879743901538 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^30 + 32*x^28 - 80*x^26 + 127*x^24 - 80*x^22 - 224*x^20 + 936*x^18 - 2175*x^16 + 3744*x^14 - 3584*x^12 - 5120*x^10 + 32512*x^8 - 81920*x^6 + 131072*x^4 - 131072*x^2 + 65536)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^32 - 8*x^30 + 32*x^28 - 80*x^26 + 127*x^24 - 80*x^22 - 224*x^20 + 936*x^18 - 2175*x^16 + 3744*x^14 - 3584*x^12 - 5120*x^10 + 32512*x^8 - 81920*x^6 + 131072*x^4 - 131072*x^2 + 65536, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - 8*x^30 + 32*x^28 - 80*x^26 + 127*x^24 - 80*x^22 - 224*x^20 + 936*x^18 - 2175*x^16 + 3744*x^14 - 3584*x^12 - 5120*x^10 + 32512*x^8 - 81920*x^6 + 131072*x^4 - 131072*x^2 + 65536);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 8*x^30 + 32*x^28 - 80*x^26 + 127*x^24 - 80*x^22 - 224*x^20 + 936*x^18 - 2175*x^16 + 3744*x^14 - 3584*x^12 - 5120*x^10 + 32512*x^8 - 81920*x^6 + 131072*x^4 - 131072*x^2 + 65536);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^4:C_4$ (as 32T262):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 64
The 40 conjugacy class representatives for $C_2^4:C_4$
Character table for $C_2^4:C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), 4.4.9792.1, \(\Q(\zeta_{16})^+\), 4.4.18432.1, 4.4.4352.1, 4.0.18432.2, 4.0.1088.2, 4.0.39168.3, 4.0.2048.2, \(\Q(i, \sqrt{6})\), 4.4.313344.1, 4.4.34816.1, 4.0.34816.1, 4.0.313344.1, \(\Q(\zeta_{12})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), 8.8.392737849344.2, 8.0.98184462336.5, 8.0.4848615424.13, 8.0.392737849344.12, 8.0.98184462336.10, 8.0.392737849344.35, \(\Q(\zeta_{24})\), 8.8.98184462336.2, 8.8.392737849344.1, 8.0.98184462336.11, 8.0.392737849344.42, 8.8.98184462336.1, 8.8.4848615424.1, 8.0.392737849344.37, 8.0.1212153856.10, 8.0.98184462336.37, 8.0.1212153856.9, 8.0.392737849344.40, 8.0.4848615424.10, 8.0.392737849344.24, 8.0.98184462336.12, 8.0.392737849344.45, 8.0.98184462336.46, 8.0.1358954496.4, 8.0.1534132224.8, 8.0.18939904.2, \(\Q(\zeta_{16})\), 8.0.339738624.2, 8.0.95883264.1, 8.0.1534132224.10, 8.0.339738624.1, \(\Q(\zeta_{48})^+\), 8.8.1534132224.1, 8.0.1534132224.4, 8.0.1358954496.3, 16.0.154243018307350441230336.9, 16.16.154243018307350441230336.1, 16.0.154243018307350441230336.7, 16.0.9640188644209402576896.1, 16.0.154243018307350441230336.2, 16.0.154243018307350441230336.4, 16.0.23509071529850699776.10, 16.0.154243018307350441230336.8, 16.0.154243018307350441230336.6, 16.0.154243018307350441230336.3, 16.0.9640188644209402576896.2, 16.0.154243018307350441230336.1, 16.0.154243018307350441230336.5, \(\Q(\zeta_{48})\), 16.0.2353561680715186176.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 32 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.4.0.1}{4} }^{8}$ ${\href{/padicField/7.2.0.1}{2} }^{16}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ R ${\href{/padicField/19.4.0.1}{4} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{16}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{16}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.16.48.1$x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516$$8$$2$$48$$C_4\times C_2^2$$[2, 3, 4]^{2}$
2.16.48.1$x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516$$8$$2$$48$$C_4\times C_2^2$$[2, 3, 4]^{2}$
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(17\) Copy content Toggle raw display 17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$