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Properties

Label 32.0.789...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $7.899\times 10^{54}$
Root discriminant \(51.95\)
Ramified primes $2,3,5,101,401$
Class number $210$ (GRH)
Class group [210] (GRH)
Galois group $D_4^2:C_2^3$ (as 32T12882)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 4*x^30 + 2*x^28 - 6*x^26 + 3*x^24 + 6*x^22 - 53*x^20 - 181*x^18 - 427*x^16 - 724*x^14 - 848*x^12 + 384*x^10 + 768*x^8 - 6144*x^6 + 8192*x^4 + 65536*x^2 + 65536)
 
gp: K = bnfinit(y^32 + 4*y^30 + 2*y^28 - 6*y^26 + 3*y^24 + 6*y^22 - 53*y^20 - 181*y^18 - 427*y^16 - 724*y^14 - 848*y^12 + 384*y^10 + 768*y^8 - 6144*y^6 + 8192*y^4 + 65536*y^2 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 4*x^30 + 2*x^28 - 6*x^26 + 3*x^24 + 6*x^22 - 53*x^20 - 181*x^18 - 427*x^16 - 724*x^14 - 848*x^12 + 384*x^10 + 768*x^8 - 6144*x^6 + 8192*x^4 + 65536*x^2 + 65536);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 + 4*x^30 + 2*x^28 - 6*x^26 + 3*x^24 + 6*x^22 - 53*x^20 - 181*x^18 - 427*x^16 - 724*x^14 - 848*x^12 + 384*x^10 + 768*x^8 - 6144*x^6 + 8192*x^4 + 65536*x^2 + 65536)
 

\( x^{32} + 4 x^{30} + 2 x^{28} - 6 x^{26} + 3 x^{24} + 6 x^{22} - 53 x^{20} - 181 x^{18} - 427 x^{16} + \cdots + 65536 \) 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:   \(7898934032955601334170827214092449218560000000000000000\) \(\medspace = 2^{32}\cdot 3^{16}\cdot 5^{16}\cdot 101^{8}\cdot 401^{4}\) 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:  \(51.95\)
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:  $2\cdot 3^{1/2}5^{1/2}101^{1/2}401^{1/2}\approx 1558.8649717021676$
Ramified primes:   \(2\), \(3\), \(5\), \(101\), \(401\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{1}{5}a^{6}-\frac{1}{5}$, $\frac{1}{5}a^{13}-\frac{1}{5}a^{7}-\frac{1}{5}a$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{8}-\frac{1}{5}a^{2}$, $\frac{1}{5}a^{15}-\frac{1}{5}a^{9}-\frac{1}{5}a^{3}$, $\frac{1}{5}a^{16}-\frac{1}{5}a^{10}-\frac{1}{5}a^{4}$, $\frac{1}{10}a^{17}+\frac{2}{5}a^{11}-\frac{1}{2}a^{9}-\frac{1}{10}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{20}a^{18}-\frac{1}{10}a^{14}-\frac{1}{10}a^{12}-\frac{1}{4}a^{10}+\frac{1}{10}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{3}{20}a^{2}-\frac{1}{5}$, $\frac{1}{40}a^{19}+\frac{1}{20}a^{15}+\frac{1}{20}a^{13}-\frac{1}{8}a^{11}+\frac{9}{20}a^{9}+\frac{11}{40}a^{7}+\frac{3}{8}a^{5}-\frac{7}{40}a^{3}-\frac{1}{5}a$, $\frac{1}{80}a^{20}+\frac{1}{40}a^{16}+\frac{1}{40}a^{14}-\frac{1}{16}a^{12}-\frac{11}{40}a^{10}-\frac{29}{80}a^{8}+\frac{3}{16}a^{6}-\frac{7}{80}a^{4}-\frac{1}{10}a^{2}$, $\frac{1}{160}a^{21}+\frac{1}{80}a^{17}-\frac{7}{80}a^{15}-\frac{1}{32}a^{13}+\frac{29}{80}a^{11}+\frac{67}{160}a^{9}-\frac{13}{32}a^{7}-\frac{7}{160}a^{5}+\frac{1}{20}a^{3}-\frac{1}{2}a$, $\frac{1}{1600}a^{22}-\frac{1}{200}a^{20}+\frac{17}{800}a^{18}-\frac{47}{800}a^{16}+\frac{107}{1600}a^{14}-\frac{63}{800}a^{12}+\frac{627}{1600}a^{10}+\frac{679}{1600}a^{8}-\frac{127}{1600}a^{6}+\frac{9}{50}a^{4}-\frac{49}{100}a^{2}-\frac{2}{25}$, $\frac{1}{3200}a^{23}-\frac{1}{400}a^{21}+\frac{17}{1600}a^{19}-\frac{47}{1600}a^{17}+\frac{107}{3200}a^{15}-\frac{63}{1600}a^{13}+\frac{627}{3200}a^{11}+\frac{679}{3200}a^{9}+\frac{1473}{3200}a^{7}+\frac{9}{100}a^{5}+\frac{51}{200}a^{3}-\frac{1}{25}a$, $\frac{1}{6400}a^{24}-\frac{3}{640}a^{20}-\frac{71}{3200}a^{18}-\frac{1}{1280}a^{16}-\frac{11}{128}a^{14}+\frac{259}{6400}a^{12}+\frac{51}{1280}a^{10}+\frac{357}{1280}a^{8}+\frac{109}{800}a^{6}-\frac{29}{80}a^{4}-\frac{1}{20}a^{2}+\frac{1}{25}$, $\frac{1}{12800}a^{25}-\frac{3}{1280}a^{21}-\frac{71}{6400}a^{19}-\frac{1}{2560}a^{17}-\frac{11}{256}a^{15}+\frac{259}{12800}a^{13}+\frac{51}{2560}a^{11}-\frac{923}{2560}a^{9}-\frac{691}{1600}a^{7}-\frac{29}{160}a^{5}+\frac{19}{40}a^{3}-\frac{12}{25}a$, $\frac{1}{25600}a^{26}+\frac{1}{12800}a^{22}-\frac{39}{12800}a^{20}-\frac{197}{25600}a^{18}-\frac{179}{12800}a^{16}-\frac{797}{25600}a^{14}+\frac{2303}{25600}a^{12}-\frac{551}{25600}a^{10}-\frac{19}{640}a^{8}-\frac{19}{1600}a^{6}-\frac{17}{50}a^{4}-\frac{37}{100}a^{2}-\frac{4}{25}$, $\frac{1}{51200}a^{27}+\frac{1}{25600}a^{23}-\frac{39}{25600}a^{21}-\frac{197}{51200}a^{19}-\frac{179}{25600}a^{17}-\frac{797}{51200}a^{15}+\frac{2303}{51200}a^{13}-\frac{551}{51200}a^{11}-\frac{19}{1280}a^{9}+\frac{1581}{3200}a^{7}-\frac{17}{100}a^{5}-\frac{37}{200}a^{3}+\frac{21}{50}a$, $\frac{1}{6656000}a^{28}-\frac{1}{832000}a^{26}+\frac{177}{3328000}a^{24}+\frac{1009}{3328000}a^{22}-\frac{33429}{6656000}a^{20}+\frac{67841}{3328000}a^{18}-\frac{118861}{6656000}a^{16}+\frac{243303}{6656000}a^{14}+\frac{647041}{6656000}a^{12}-\frac{82179}{208000}a^{10}-\frac{107781}{416000}a^{8}+\frac{1247}{104000}a^{6}-\frac{491}{6500}a^{4}+\frac{252}{1625}a^{2}-\frac{61}{1625}$, $\frac{1}{13312000}a^{29}-\frac{1}{1664000}a^{27}+\frac{177}{6656000}a^{25}+\frac{1009}{6656000}a^{23}-\frac{33429}{13312000}a^{21}+\frac{67841}{6656000}a^{19}-\frac{118861}{13312000}a^{17}+\frac{243303}{13312000}a^{15}-\frac{684159}{13312000}a^{13}+\frac{125821}{416000}a^{11}+\frac{308219}{832000}a^{9}+\frac{22047}{208000}a^{7}+\frac{6009}{13000}a^{5}-\frac{1373}{3250}a^{3}-\frac{1361}{3250}a$, $\frac{1}{26624000}a^{30}+\frac{29}{2662400}a^{26}+\frac{69}{2662400}a^{24}-\frac{129}{5324800}a^{22}-\frac{14007}{2662400}a^{20}-\frac{21573}{1064960}a^{18}-\frac{450189}{5324800}a^{16}+\frac{1189}{5324800}a^{14}+\frac{17593}{665600}a^{12}-\frac{84173}{332800}a^{10}-\frac{249}{2080}a^{8}-\frac{1259}{20800}a^{6}-\frac{1663}{5200}a^{4}-\frac{181}{1300}a^{2}-\frac{642}{1625}$, $\frac{1}{53248000}a^{31}+\frac{29}{5324800}a^{27}+\frac{69}{5324800}a^{25}-\frac{129}{10649600}a^{23}-\frac{14007}{5324800}a^{21}-\frac{21573}{2129920}a^{19}-\frac{450189}{10649600}a^{17}+\frac{1189}{10649600}a^{15}+\frac{17593}{1331200}a^{13}-\frac{84173}{665600}a^{11}-\frac{249}{4160}a^{9}-\frac{1259}{41600}a^{7}-\frac{1663}{10400}a^{5}+\frac{1119}{2600}a^{3}+\frac{983}{3250}a$ 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

$C_{210}$, which has order $210$ (assuming GRH)

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

Relative class number: $210$ (assuming GRH)

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:   \( \frac{12509}{53248000} a^{31} + \frac{5221}{13312000} a^{29} - \frac{15371}{26624000} a^{27} - \frac{10487}{26624000} a^{25} + \frac{69687}{53248000} a^{23} - \frac{56073}{26624000} a^{21} - \frac{427377}{53248000} a^{19} - \frac{1294849}{53248000} a^{17} - \frac{2393903}{53248000} a^{15} - \frac{794309}{13312000} a^{13} - \frac{60211}{1664000} a^{11} + \frac{47791}{208000} a^{9} - \frac{57243}{208000} a^{7} - \frac{16747}{26000} a^{5} + \frac{25899}{6500} a^{3} + \frac{11013}{1625} a \)  (order $12$) 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:   $\frac{4531}{26624000}a^{31}+\frac{947}{2662400}a^{29}-\frac{821}{2662400}a^{27}-\frac{667}{2662400}a^{25}+\frac{6729}{5324800}a^{23}-\frac{859}{665600}a^{21}-\frac{35847}{5324800}a^{19}-\frac{94677}{5324800}a^{17}-\frac{196719}{5324800}a^{15}-\frac{30137}{532480}a^{13}-\frac{33871}{665600}a^{11}+\frac{22497}{166400}a^{9}-\frac{5}{52}a^{7}-\frac{151}{208}a^{5}+\frac{109}{40}a^{3}+\frac{9123}{1625}a$, $\frac{461}{4096000}a^{31}+\frac{2003}{5324800}a^{30}+\frac{43}{256000}a^{29}+\frac{217}{266240}a^{28}-\frac{747}{2048000}a^{27}-\frac{1901}{2662400}a^{26}-\frac{459}{2048000}a^{25}-\frac{1777}{2662400}a^{24}+\frac{2239}{4096000}a^{23}+\frac{16457}{5324800}a^{22}-\frac{2431}{2048000}a^{21}-\frac{6251}{2662400}a^{20}-\frac{21289}{4096000}a^{19}-\frac{67999}{5324800}a^{18}-\frac{51613}{4096000}a^{17}-\frac{212983}{5324800}a^{16}-\frac{87611}{4096000}a^{15}-\frac{417689}{5324800}a^{14}-\frac{14639}{512000}a^{13}-\frac{153317}{1331200}a^{12}-\frac{747}{128000}a^{11}-\frac{40591}{332800}a^{10}+\frac{2157}{16000}a^{9}+\frac{26951}{83200}a^{8}-\frac{439}{4000}a^{7}-\frac{4561}{10400}a^{6}-\frac{399}{2000}a^{5}-\frac{10011}{5200}a^{4}+\frac{1083}{500}a^{3}+\frac{76}{13}a^{2}+\frac{428}{125}a+\frac{3789}{325}$, $\frac{27}{204800}a^{30}+\frac{147}{1331200}a^{28}-\frac{461}{1331200}a^{26}+\frac{493}{1331200}a^{24}-\frac{223}{2662400}a^{22}-\frac{617}{332800}a^{20}-\frac{7727}{2662400}a^{18}-\frac{33309}{2662400}a^{16}-\frac{40743}{2662400}a^{14}-\frac{33533}{1331200}a^{12}+\frac{2681}{332800}a^{10}+\frac{14633}{83200}a^{8}-\frac{1653}{10400}a^{6}-\frac{119}{2600}a^{4}+\frac{2901}{1300}a^{2}+\frac{29}{325}$, $\frac{13649}{53248000}a^{31}+\frac{5673}{13312000}a^{29}-\frac{11743}{26624000}a^{27}+\frac{10189}{26624000}a^{25}+\frac{102131}{53248000}a^{23}-\frac{14109}{26624000}a^{21}-\frac{298661}{53248000}a^{19}-\frac{1302437}{53248000}a^{17}-\frac{193223}{4096000}a^{15}-\frac{1044357}{13312000}a^{13}-\frac{344711}{3328000}a^{11}+\frac{117137}{832000}a^{9}-\frac{52417}{104000}a^{7}-\frac{51347}{52000}a^{5}+\frac{25417}{6500}a^{3}+\frac{8677}{1625}a$, $\frac{3069}{53248000}a^{31}-\frac{829}{2662400}a^{30}+\frac{2499}{13312000}a^{29}-\frac{401}{665600}a^{28}+\frac{1501}{26624000}a^{27}+\frac{43}{102400}a^{26}-\frac{703}{26624000}a^{25}+\frac{423}{1331200}a^{24}+\frac{38183}{53248000}a^{23}-\frac{103}{40960}a^{22}-\frac{2757}{26624000}a^{21}+\frac{809}{1331200}a^{20}-\frac{81713}{53248000}a^{19}+\frac{1837}{204800}a^{18}-\frac{275801}{53248000}a^{17}+\frac{82081}{2662400}a^{16}-\frac{828247}{53248000}a^{15}+\frac{167191}{2662400}a^{14}-\frac{372811}{13312000}a^{13}+\frac{5491}{51200}a^{12}-\frac{149933}{3328000}a^{11}+\frac{32277}{332800}a^{10}+\frac{321}{832000}a^{9}-\frac{663}{3200}a^{8}+\frac{707}{52000}a^{7}+\frac{1307}{2600}a^{6}-\frac{1711}{3250}a^{5}+\frac{301}{200}a^{4}+\frac{559}{1000}a^{3}-\frac{274}{65}a^{2}+\frac{3559}{1625}a-\frac{3199}{325}$, $\frac{21}{1024000}a^{30}+\frac{59}{6656000}a^{28}-\frac{927}{6656000}a^{26}-\frac{369}{6656000}a^{24}+\frac{1719}{13312000}a^{22}-\frac{6053}{3328000}a^{20}-\frac{63569}{13312000}a^{18}-\frac{69903}{13312000}a^{16}-\frac{138221}{13312000}a^{14}-\frac{75001}{6656000}a^{12}+\frac{2649}{208000}a^{10}+\frac{14463}{208000}a^{8}+\frac{2463}{104000}a^{6}+\frac{119}{1625}a^{4}+\frac{1673}{1625}a^{2}+\frac{2134}{1625}$, $\frac{3707}{26624000}a^{31}+\frac{567}{1331200}a^{30}+\frac{6189}{13312000}a^{29}+\frac{289}{332800}a^{28}-\frac{3657}{13312000}a^{27}-\frac{721}{665600}a^{26}-\frac{14179}{13312000}a^{25}-\frac{797}{665600}a^{24}+\frac{37469}{26624000}a^{23}+\frac{3157}{1331200}a^{22}-\frac{13813}{6656000}a^{21}-\frac{3279}{665600}a^{20}-\frac{218659}{26624000}a^{19}-\frac{23171}{1331200}a^{18}-\frac{455293}{26624000}a^{17}-\frac{63307}{1331200}a^{16}-\frac{1306671}{26624000}a^{15}-\frac{25041}{266240}a^{14}-\frac{668971}{13312000}a^{13}-\frac{7473}{66560}a^{12}-\frac{118113}{3328000}a^{11}-\frac{193}{4160}a^{10}+\frac{169131}{832000}a^{9}+\frac{9793}{20800}a^{8}+\frac{579}{4000}a^{7}-\frac{309}{1300}a^{6}-\frac{14409}{13000}a^{5}-\frac{9147}{5200}a^{4}+\frac{34087}{13000}a^{3}+\frac{2492}{325}a^{2}+\frac{13369}{1625}a+\frac{4828}{325}$, $\frac{4141}{4096000}a^{31}+\frac{12237}{26624000}a^{30}+\frac{27633}{13312000}a^{29}+\frac{1163}{6656000}a^{28}-\frac{65583}{26624000}a^{27}-\frac{16923}{13312000}a^{26}-\frac{50171}{26624000}a^{25}+\frac{13649}{13312000}a^{24}+\frac{351291}{53248000}a^{23}+\frac{11831}{26624000}a^{22}-\frac{269989}{26624000}a^{21}-\frac{60629}{13312000}a^{20}-\frac{2117421}{53248000}a^{19}-\frac{328241}{26624000}a^{18}-\frac{6062237}{53248000}a^{17}-\frac{1079977}{26624000}a^{16}-\frac{11621939}{53248000}a^{15}-\frac{110323}{2048000}a^{14}-\frac{3865497}{13312000}a^{13}-\frac{542287}{6656000}a^{12}-\frac{201219}{832000}a^{11}-\frac{12513}{832000}a^{10}+\frac{426681}{416000}a^{9}+\frac{5711}{13000}a^{8}-\frac{44891}{52000}a^{7}-\frac{14013}{13000}a^{6}-\frac{219357}{52000}a^{5}+\frac{4479}{13000}a^{4}+\frac{239269}{13000}a^{3}+\frac{13061}{1625}a^{2}+\frac{52458}{1625}a+\frac{5698}{1625}$, $\frac{9987}{26624000}a^{31}+\frac{57}{409600}a^{30}+\frac{9537}{13312000}a^{29}+\frac{327}{665600}a^{28}-\frac{12681}{13312000}a^{27}-\frac{1507}{2662400}a^{26}-\frac{9107}{13312000}a^{25}-\frac{783}{532480}a^{24}+\frac{54117}{26624000}a^{23}+\frac{303}{212992}a^{22}-\frac{26459}{6656000}a^{21}-\frac{7419}{2662400}a^{20}-\frac{323627}{26624000}a^{19}-\frac{49137}{5324800}a^{18}-\frac{1026029}{26624000}a^{17}-\frac{100509}{5324800}a^{16}-\frac{1900863}{26624000}a^{15}-\frac{233547}{5324800}a^{14}-\frac{1143343}{13312000}a^{13}-\frac{14217}{332800}a^{12}-\frac{254019}{3328000}a^{11}-\frac{81}{5200}a^{10}+\frac{314153}{832000}a^{9}+\frac{3379}{16640}a^{8}-\frac{1613}{4000}a^{7}+\frac{357}{2600}a^{6}-\frac{86753}{52000}a^{5}-\frac{387}{325}a^{4}+\frac{42163}{6500}a^{3}+\frac{2029}{650}a^{2}+\frac{7023}{650}a+\frac{3084}{325}$, $\frac{13649}{53248000}a^{31}+\frac{5563}{26624000}a^{30}+\frac{5673}{13312000}a^{29}+\frac{53}{332800}a^{28}-\frac{11743}{26624000}a^{27}-\frac{1393}{2662400}a^{26}+\frac{10189}{26624000}a^{25}-\frac{617}{2662400}a^{24}+\frac{102131}{53248000}a^{23}+\frac{229}{5324800}a^{22}-\frac{14109}{26624000}a^{21}-\frac{5749}{2662400}a^{20}-\frac{298661}{53248000}a^{19}-\frac{31011}{5324800}a^{18}-\frac{1302437}{53248000}a^{17}-\frac{94543}{5324800}a^{16}-\frac{193223}{4096000}a^{15}-\frac{85769}{5324800}a^{14}-\frac{1044357}{13312000}a^{13}-\frac{16179}{665600}a^{12}-\frac{344711}{3328000}a^{11}+\frac{2601}{332800}a^{10}+\frac{117137}{832000}a^{9}+\frac{23007}{83200}a^{8}-\frac{52417}{104000}a^{7}-\frac{4371}{10400}a^{6}-\frac{51347}{52000}a^{5}-\frac{1643}{5200}a^{4}+\frac{25417}{6500}a^{3}+\frac{946}{325}a^{2}+\frac{8677}{1625}a+\frac{5324}{1625}$, $\frac{187}{819200}a^{31}+\frac{16607}{26624000}a^{30}+\frac{3483}{13312000}a^{29}+\frac{981}{832000}a^{28}-\frac{21733}{26624000}a^{27}-\frac{24273}{13312000}a^{26}-\frac{17361}{26624000}a^{25}-\frac{23161}{13312000}a^{24}+\frac{15601}{53248000}a^{23}+\frac{96741}{26624000}a^{22}-\frac{83279}{26624000}a^{21}-\frac{85789}{13312000}a^{20}-\frac{420151}{53248000}a^{19}-\frac{605091}{26624000}a^{18}-\frac{1123127}{53248000}a^{17}-\frac{1715327}{26624000}a^{16}-\frac{1528169}{53248000}a^{15}-\frac{245213}{2048000}a^{14}-\frac{403447}{13312000}a^{13}-\frac{504121}{3328000}a^{12}+\frac{35369}{3328000}a^{11}-\frac{180591}{1664000}a^{10}+\frac{246647}{832000}a^{9}+\frac{266367}{416000}a^{8}-\frac{36237}{104000}a^{7}-\frac{37187}{52000}a^{6}-\frac{29587}{52000}a^{5}-\frac{71427}{26000}a^{4}+\frac{26627}{6500}a^{3}+\frac{36167}{3250}a^{2}+\frac{8851}{1625}a+\frac{32948}{1625}$, $\frac{4033}{10649600}a^{31}+\frac{13633}{26624000}a^{30}+\frac{4707}{6656000}a^{29}+\frac{87}{83200}a^{28}-\frac{24299}{26624000}a^{27}-\frac{3187}{2662400}a^{26}-\frac{471}{2048000}a^{25}-\frac{639}{532480}a^{24}+\frac{175703}{53248000}a^{23}+\frac{11903}{5324800}a^{22}-\frac{66267}{26624000}a^{21}-\frac{14327}{2662400}a^{20}-\frac{666113}{53248000}a^{19}-\frac{99833}{5324800}a^{18}-\frac{2121021}{53248000}a^{17}-\frac{279629}{5324800}a^{16}-\frac{4465467}{53248000}a^{15}-\frac{472763}{5324800}a^{14}-\frac{216897}{1664000}a^{13}-\frac{15279}{133120}a^{12}-\frac{419833}{3328000}a^{11}-\frac{27551}{332800}a^{10}+\frac{107593}{416000}a^{9}+\frac{24991}{41600}a^{8}-\frac{118347}{208000}a^{7}-\frac{11137}{20800}a^{6}-\frac{68931}{52000}a^{5}-\frac{2549}{1040}a^{4}+\frac{93507}{13000}a^{3}+\frac{213}{26}a^{2}+\frac{37971}{3250}a+\frac{23999}{1625}$, $\frac{25659}{53248000}a^{31}-\frac{25731}{26624000}a^{30}+\frac{12521}{13312000}a^{29}-\frac{12787}{6656000}a^{28}-\frac{27741}{26624000}a^{27}+\frac{27277}{13312000}a^{26}-\frac{25177}{26624000}a^{25}+\frac{24169}{13312000}a^{24}+\frac{156737}{53248000}a^{23}-\frac{187849}{26624000}a^{22}-\frac{102283}{26624000}a^{21}+\frac{110391}{13312000}a^{20}-\frac{908567}{53248000}a^{19}+\frac{964479}{26624000}a^{18}-\frac{2708399}{53248000}a^{17}+\frac{2779743}{26624000}a^{16}-\frac{408501}{4096000}a^{15}+\frac{5393441}{26624000}a^{14}-\frac{1828989}{13312000}a^{13}+\frac{1877143}{6656000}a^{12}-\frac{234661}{1664000}a^{11}+\frac{398989}{1664000}a^{10}+\frac{376319}{832000}a^{9}-\frac{337053}{416000}a^{8}-\frac{52289}{104000}a^{7}+\frac{55403}{52000}a^{6}-\frac{48297}{26000}a^{5}+\frac{105873}{26000}a^{4}+\frac{12826}{1625}a^{3}-\frac{26084}{1625}a^{2}+\frac{48981}{3250}a-\frac{51726}{1625}$, $\frac{2343}{53248000}a^{31}-\frac{59}{26624000}a^{30}+\frac{727}{13312000}a^{29}+\frac{3}{3328000}a^{28}-\frac{2097}{26624000}a^{27}-\frac{851}{13312000}a^{26}+\frac{171}{26624000}a^{25}-\frac{119}{1024000}a^{24}-\frac{25451}{53248000}a^{23}+\frac{1207}{26624000}a^{22}-\frac{4487}{2048000}a^{21}-\frac{9963}{13312000}a^{20}-\frac{127139}{53248000}a^{19}-\frac{53617}{26624000}a^{18}-\frac{298483}{53248000}a^{17}+\frac{55011}{26624000}a^{16}-\frac{173661}{53248000}a^{15}-\frac{147243}{26624000}a^{14}+\frac{277977}{13312000}a^{13}-\frac{7921}{1664000}a^{12}+\frac{971}{128000}a^{11}+\frac{74779}{832000}a^{10}+\frac{22003}{832000}a^{9}+\frac{55059}{416000}a^{8}+\frac{10509}{208000}a^{7}+\frac{11637}{104000}a^{6}-\frac{2709}{26000}a^{5}+\frac{7291}{26000}a^{4}+\frac{7041}{13000}a^{3}-\frac{1178}{1625}a^{2}+\frac{5077}{3250}a-\frac{2723}{1625}$, $\frac{30311}{53248000}a^{31}+\frac{13667}{5324800}a^{30}+\frac{14203}{13312000}a^{29}+\frac{1581}{332800}a^{28}-\frac{15553}{26624000}a^{27}-\frac{15909}{2662400}a^{26}+\frac{10459}{26624000}a^{25}-\frac{16069}{2662400}a^{24}+\frac{254261}{53248000}a^{23}+\frac{78417}{5324800}a^{22}-\frac{14819}{26624000}a^{21}-\frac{70353}{2662400}a^{20}-\frac{598691}{53248000}a^{19}-\frac{559303}{5324800}a^{18}-\frac{2547587}{53248000}a^{17}-\frac{1502867}{5324800}a^{16}-\frac{5138189}{53248000}a^{15}-\frac{221993}{409600}a^{14}-\frac{2493227}{13312000}a^{13}-\frac{476017}{665600}a^{12}-\frac{38691}{208000}a^{11}-\frac{16969}{33280}a^{10}+\frac{96011}{416000}a^{9}+\frac{21897}{8320}a^{8}-\frac{218249}{208000}a^{7}-\frac{45999}{20800}a^{6}-\frac{138027}{52000}a^{5}-\frac{22769}{2600}a^{4}+\frac{90959}{13000}a^{3}+\frac{29333}{650}a^{2}+\frac{8029}{650}a+\frac{27551}{325}$ 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:  \( 6738898358917.929 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 6738898358917.929 \cdot 210}{12\cdot\sqrt{7898934032955601334170827214092449218560000000000000000}}\cr\approx \mathstrut & 0.247582876862735 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + 4*x^30 + 2*x^28 - 6*x^26 + 3*x^24 + 6*x^22 - 53*x^20 - 181*x^18 - 427*x^16 - 724*x^14 - 848*x^12 + 384*x^10 + 768*x^8 - 6144*x^6 + 8192*x^4 + 65536*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 + 4*x^30 + 2*x^28 - 6*x^26 + 3*x^24 + 6*x^22 - 53*x^20 - 181*x^18 - 427*x^16 - 724*x^14 - 848*x^12 + 384*x^10 + 768*x^8 - 6144*x^6 + 8192*x^4 + 65536*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 + 4*x^30 + 2*x^28 - 6*x^26 + 3*x^24 + 6*x^22 - 53*x^20 - 181*x^18 - 427*x^16 - 724*x^14 - 848*x^12 + 384*x^10 + 768*x^8 - 6144*x^6 + 8192*x^4 + 65536*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 + 4*x^30 + 2*x^28 - 6*x^26 + 3*x^24 + 6*x^22 - 53*x^20 - 181*x^18 - 427*x^16 - 724*x^14 - 848*x^12 + 384*x^10 + 768*x^8 - 6144*x^6 + 8192*x^4 + 65536*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

$D_4^2:C_2^3$ (as 32T12882):

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 512
The 80 conjugacy class representatives for $D_4^2:C_2^3$
Character table for $D_4^2:C_2^3$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.0.22725.1, 4.4.363600.1, 4.0.40400.1, 4.4.2525.1, \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{15})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 8.8.207086675625.1, 8.0.53014188960000.1, 8.8.654496160000.1, 8.0.2556625625.1, 8.0.12960000.1, 8.0.132204960000.4, 8.0.1632160000.5, 8.0.132204960000.18, 8.8.132204960000.1, 8.0.132204960000.7, 8.0.516425625.1, 16.0.17478151448601600000000.1, 16.0.2810504231086585881600000000.1, 16.0.428365223454745600000000.1, 16.16.2810504231086585881600000000.1, 16.0.2810504231086585881600000000.2, 16.0.2810504231086585881600000000.3, 16.0.42884891221413969140625.1

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: not computed

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 R ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{16}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(3\) Copy content Toggle raw display 3.16.8.1$x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
3.16.8.1$x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(101\) Copy content Toggle raw display 101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(401\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$