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Properties

Label 32.0.493...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $4.935\times 10^{51}$
Root discriminant \(41.25\)
Ramified primes $2,3,5,29,769$
Class number $36$ (GRH)
Class group [2, 18] (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 + 7*x^30 + 23*x^28 + 46*x^26 + 55*x^24 + 10*x^22 - 218*x^20 - 924*x^18 - 2271*x^16 - 3696*x^14 - 3488*x^12 + 640*x^10 + 14080*x^8 + 47104*x^6 + 94208*x^4 + 114688*x^2 + 65536)
 
gp: K = bnfinit(y^32 + 7*y^30 + 23*y^28 + 46*y^26 + 55*y^24 + 10*y^22 - 218*y^20 - 924*y^18 - 2271*y^16 - 3696*y^14 - 3488*y^12 + 640*y^10 + 14080*y^8 + 47104*y^6 + 94208*y^4 + 114688*y^2 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 7*x^30 + 23*x^28 + 46*x^26 + 55*x^24 + 10*x^22 - 218*x^20 - 924*x^18 - 2271*x^16 - 3696*x^14 - 3488*x^12 + 640*x^10 + 14080*x^8 + 47104*x^6 + 94208*x^4 + 114688*x^2 + 65536);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 + 7*x^30 + 23*x^28 + 46*x^26 + 55*x^24 + 10*x^22 - 218*x^20 - 924*x^18 - 2271*x^16 - 3696*x^14 - 3488*x^12 + 640*x^10 + 14080*x^8 + 47104*x^6 + 94208*x^4 + 114688*x^2 + 65536)
 

\( x^{32} + 7 x^{30} + 23 x^{28} + 46 x^{26} + 55 x^{24} + 10 x^{22} - 218 x^{20} - 924 x^{18} + \cdots + 65536 \) 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:   \(4935252174511155500627740082045583360000000000000000\) \(\medspace = 2^{32}\cdot 3^{16}\cdot 5^{16}\cdot 29^{8}\cdot 769^{4}\) 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:  \(41.25\)
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:  $2\cdot 3^{1/2}5^{1/2}29^{1/2}769^{1/2}\approx 1156.7454343977329$
Ramified primes:   \(2\), \(3\), \(5\), \(29\), \(769\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{16}-\frac{1}{4}a^{14}-\frac{1}{2}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{19}-\frac{1}{8}a^{17}-\frac{1}{8}a^{15}-\frac{1}{4}a^{13}-\frac{1}{8}a^{11}+\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{20}-\frac{1}{16}a^{18}-\frac{1}{16}a^{16}+\frac{3}{8}a^{14}+\frac{7}{16}a^{12}+\frac{1}{8}a^{10}+\frac{3}{8}a^{8}+\frac{1}{4}a^{6}+\frac{1}{16}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{21}-\frac{1}{32}a^{19}-\frac{1}{32}a^{17}-\frac{5}{16}a^{15}+\frac{7}{32}a^{13}-\frac{7}{16}a^{11}-\frac{5}{16}a^{9}-\frac{3}{8}a^{7}+\frac{1}{32}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{448}a^{22}+\frac{11}{448}a^{20}+\frac{5}{64}a^{18}-\frac{51}{224}a^{16}-\frac{129}{448}a^{14}+\frac{99}{224}a^{12}+\frac{47}{224}a^{10}+\frac{55}{112}a^{8}-\frac{47}{448}a^{6}+\frac{3}{16}a^{4}+\frac{5}{28}a^{2}+\frac{3}{7}$, $\frac{1}{896}a^{23}+\frac{11}{896}a^{21}+\frac{5}{128}a^{19}-\frac{51}{448}a^{17}-\frac{129}{896}a^{15}-\frac{125}{448}a^{13}+\frac{47}{448}a^{11}+\frac{55}{224}a^{9}-\frac{47}{896}a^{7}+\frac{3}{32}a^{5}-\frac{23}{56}a^{3}-\frac{2}{7}a$, $\frac{1}{1792}a^{24}-\frac{1}{1792}a^{22}+\frac{15}{1792}a^{20}-\frac{93}{896}a^{18}-\frac{361}{1792}a^{16}+\frac{313}{896}a^{14}+\frac{21}{128}a^{12}+\frac{165}{448}a^{10}-\frac{223}{1792}a^{8}-\frac{87}{224}a^{6}+\frac{33}{112}a^{4}-\frac{3}{7}a^{2}-\frac{2}{7}$, $\frac{1}{3584}a^{25}-\frac{1}{3584}a^{23}+\frac{15}{3584}a^{21}-\frac{93}{1792}a^{19}-\frac{361}{3584}a^{17}+\frac{313}{1792}a^{15}-\frac{107}{256}a^{13}+\frac{165}{896}a^{11}+\frac{1569}{3584}a^{9}+\frac{137}{448}a^{7}+\frac{33}{224}a^{5}-\frac{3}{14}a^{3}+\frac{5}{14}a$, $\frac{1}{7168}a^{26}-\frac{1}{7168}a^{24}-\frac{1}{7168}a^{22}+\frac{43}{3584}a^{20}+\frac{423}{7168}a^{18}+\frac{9}{3584}a^{16}+\frac{731}{3584}a^{14}-\frac{739}{1792}a^{12}-\frac{831}{7168}a^{10}+\frac{33}{896}a^{8}-\frac{1}{14}a^{6}-\frac{13}{56}a^{4}+\frac{1}{4}a^{2}-\frac{3}{7}$, $\frac{1}{14336}a^{27}-\frac{1}{14336}a^{25}-\frac{1}{14336}a^{23}+\frac{43}{7168}a^{21}+\frac{423}{14336}a^{19}+\frac{9}{7168}a^{17}-\frac{2853}{7168}a^{15}+\frac{1053}{3584}a^{13}+\frac{6337}{14336}a^{11}+\frac{33}{1792}a^{9}+\frac{13}{28}a^{7}-\frac{13}{112}a^{5}+\frac{1}{8}a^{3}-\frac{3}{14}a$, $\frac{1}{493932544}a^{28}+\frac{6143}{493932544}a^{26}-\frac{19575}{70561792}a^{24}-\frac{169525}{246966272}a^{22}-\frac{651487}{70561792}a^{20}-\frac{93285}{2308096}a^{18}+\frac{6339693}{35280896}a^{16}-\frac{266499}{123483136}a^{14}+\frac{1957751}{70561792}a^{12}-\frac{178813}{577024}a^{10}-\frac{51003}{1102528}a^{8}+\frac{28207}{3858848}a^{6}+\frac{14879}{68908}a^{4}+\frac{57824}{120589}a^{2}+\frac{51685}{120589}$, $\frac{1}{987865088}a^{29}+\frac{6143}{987865088}a^{27}-\frac{19575}{141123584}a^{25}-\frac{169525}{493932544}a^{23}-\frac{651487}{141123584}a^{21}-\frac{93285}{4616192}a^{19}+\frac{6339693}{70561792}a^{17}-\frac{266499}{246966272}a^{15}-\frac{68604041}{141123584}a^{13}-\frac{178813}{1154048}a^{11}+\frac{1051525}{2205056}a^{9}-\frac{3830641}{7717696}a^{7}-\frac{54029}{137816}a^{5}+\frac{28912}{120589}a^{3}+\frac{51685}{241178}a$, $\frac{1}{13830111232}a^{30}-\frac{5}{13830111232}a^{28}-\frac{142605}{13830111232}a^{26}-\frac{1360715}{6915055616}a^{24}+\frac{4685663}{13830111232}a^{22}-\frac{138096629}{6915055616}a^{20}+\frac{579591423}{6915055616}a^{18}-\frac{1372295521}{3457527808}a^{16}-\frac{429746287}{13830111232}a^{14}+\frac{1541184177}{3457527808}a^{12}-\frac{16261765}{37581824}a^{10}-\frac{17366907}{54023872}a^{8}+\frac{5821063}{54023872}a^{6}+\frac{5446023}{13505968}a^{4}-\frac{571279}{1688246}a^{2}+\frac{28306}{844123}$, $\frac{1}{27660222464}a^{31}-\frac{5}{27660222464}a^{29}-\frac{142605}{27660222464}a^{27}-\frac{1360715}{13830111232}a^{25}+\frac{4685663}{27660222464}a^{23}-\frac{138096629}{13830111232}a^{21}+\frac{579591423}{13830111232}a^{19}-\frac{1372295521}{6915055616}a^{17}+\frac{13400364945}{27660222464}a^{15}+\frac{1541184177}{6915055616}a^{13}-\frac{16261765}{75163648}a^{11}-\frac{17366907}{108047744}a^{9}+\frac{5821063}{108047744}a^{7}+\frac{5446023}{27011936}a^{5}+\frac{1116967}{3376492}a^{3}-\frac{815817}{1688246}a$ 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

$C_{2}\times C_{18}$, which has order $36$ (assuming GRH)

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

Relative class number: $36$ (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:   \( -\frac{6248719}{27660222464} a^{31} - \frac{31737673}{27660222464} a^{29} - \frac{92766057}{27660222464} a^{27} - \frac{78261681}{13830111232} a^{25} - \frac{128406281}{27660222464} a^{23} + \frac{41958293}{13830111232} a^{21} + \frac{576718011}{13830111232} a^{19} + \frac{962425697}{6915055616} a^{17} + \frac{8458642673}{27660222464} a^{15} + \frac{731569263}{1728763904} a^{13} + \frac{16888657}{75163648} a^{11} - \frac{80327517}{216095488} a^{9} - \frac{267909841}{108047744} a^{7} - \frac{179759883}{27011936} a^{5} - \frac{78017901}{6752984} a^{3} - \frac{18637589}{1688246} a \)  (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:   $\frac{757951}{1728763904}a^{30}+\frac{499187}{216095488}a^{28}+\frac{595015}{108047744}a^{26}+\frac{13456117}{1728763904}a^{24}+\frac{10088485}{1728763904}a^{22}-\frac{18420469}{1728763904}a^{20}-\frac{68509535}{864381952}a^{18}-\frac{213946571}{864381952}a^{16}-\frac{810102017}{1728763904}a^{14}-\frac{924432713}{1728763904}a^{12}-\frac{178632593}{864381952}a^{10}+\frac{210079147}{216095488}a^{8}+\frac{122295737}{27011936}a^{6}+\frac{77099237}{6752984}a^{4}+\frac{55257343}{3376492}a^{2}+\frac{8597649}{844123}$, $\frac{43279}{141123584}a^{30}+\frac{1203183}{987865088}a^{28}+\frac{2511783}{987865088}a^{26}+\frac{11747}{3067904}a^{24}+\frac{74097}{42950656}a^{22}-\frac{513477}{70561792}a^{20}-\frac{23041513}{493932544}a^{18}-\frac{4430277}{35280896}a^{16}-\frac{210873735}{987865088}a^{14}-\frac{539215}{2205056}a^{12}-\frac{3788091}{123483136}a^{10}+\frac{1222605}{2205056}a^{8}+\frac{19723873}{7717696}a^{6}+\frac{1574991}{275632}a^{4}+\frac{3313431}{482356}a^{2}+\frac{565366}{120589}$, $\frac{880169}{13830111232}a^{31}+\frac{4316339}{13830111232}a^{29}+\frac{15146947}{13830111232}a^{27}+\frac{13670289}{6915055616}a^{25}+\frac{24671327}{13830111232}a^{23}-\frac{937717}{6915055616}a^{21}-\frac{79223605}{6915055616}a^{19}-\frac{141178813}{3457527808}a^{17}-\frac{1371423159}{13830111232}a^{15}-\frac{551037071}{3457527808}a^{13}-\frac{7383437}{75163648}a^{11}+\frac{15862059}{216095488}a^{9}+\frac{77295271}{108047744}a^{7}+\frac{54230713}{27011936}a^{5}+\frac{27012871}{6752984}a^{3}+\frac{7695753}{1688246}a-1$, $\frac{43279}{141123584}a^{30}+\frac{1203183}{987865088}a^{28}+\frac{2511783}{987865088}a^{26}+\frac{11747}{3067904}a^{24}+\frac{74097}{42950656}a^{22}-\frac{513477}{70561792}a^{20}-\frac{23041513}{493932544}a^{18}-\frac{4430277}{35280896}a^{16}-\frac{210873735}{987865088}a^{14}-\frac{539215}{2205056}a^{12}-\frac{3788091}{123483136}a^{10}+\frac{1222605}{2205056}a^{8}+\frac{19723873}{7717696}a^{6}+\frac{1574991}{275632}a^{4}+\frac{3313431}{482356}a^{2}+\frac{685955}{120589}$, $\frac{33149}{150327296}a^{31}+\frac{1366635}{864381952}a^{29}+\frac{3771283}{864381952}a^{27}+\frac{22579741}{3457527808}a^{25}+\frac{20102953}{3457527808}a^{23}-\frac{19707637}{3457527808}a^{21}-\frac{86545335}{1728763904}a^{19}-\frac{311558167}{1728763904}a^{17}-\frac{1323475333}{3457527808}a^{15}-\frac{1603546409}{3457527808}a^{13}-\frac{452470477}{1728763904}a^{11}+\frac{125809905}{216095488}a^{9}+\frac{330920285}{108047744}a^{7}+\frac{57579827}{6752984}a^{5}+\frac{96333555}{6752984}a^{3}+\frac{8245859}{844123}a$, $\frac{145883}{3457527808}a^{30}+\frac{110269}{864381952}a^{28}+\frac{107193}{1728763904}a^{26}-\frac{1261401}{3457527808}a^{24}-\frac{604469}{3457527808}a^{22}-\frac{9083817}{3457527808}a^{20}-\frac{2789951}{864381952}a^{18}-\frac{15993297}{1728763904}a^{16}+\frac{20715855}{3457527808}a^{14}+\frac{98652055}{3457527808}a^{12}+\frac{46473701}{864381952}a^{10}+\frac{11076603}{108047744}a^{8}+\frac{1473753}{6752984}a^{6}+\frac{4112475}{13505968}a^{4}-\frac{2366223}{3376492}a^{2}-\frac{1327791}{844123}$, $\frac{2329539}{27660222464}a^{31}+\frac{1378753}{3457527808}a^{30}+\frac{1235889}{27660222464}a^{29}+\frac{8532065}{3457527808}a^{28}-\frac{12236135}{27660222464}a^{27}+\frac{23177597}{3457527808}a^{26}-\frac{9345761}{13830111232}a^{25}+\frac{8725289}{864381952}a^{24}-\frac{340393}{258506752}a^{23}+\frac{28078947}{3457527808}a^{22}-\frac{19039711}{13830111232}a^{21}-\frac{2351841}{216095488}a^{20}-\frac{61652931}{13830111232}a^{19}-\frac{148420051}{1728763904}a^{18}+\frac{7964253}{6915055616}a^{17}-\frac{61697439}{216095488}a^{16}+\frac{1074602099}{27660222464}a^{15}-\frac{2020599543}{3457527808}a^{14}+\frac{343887987}{6915055616}a^{13}-\frac{1210775227}{1728763904}a^{12}+\frac{111624183}{1728763904}a^{11}-\frac{79248973}{216095488}a^{10}+\frac{4134463}{108047744}a^{9}+\frac{205589005}{216095488}a^{8}+\frac{12509587}{108047744}a^{7}+\frac{267930261}{54023872}a^{6}-\frac{1789093}{13505968}a^{5}+\frac{45234843}{3376492}a^{4}-\frac{6952663}{3376492}a^{3}+\frac{36446545}{1688246}a^{2}-\frac{2977265}{1688246}a+\frac{12666264}{844123}$, $\frac{11082493}{27660222464}a^{31}+\frac{294905}{493932544}a^{30}+\frac{62325827}{27660222464}a^{29}+\frac{1442283}{493932544}a^{28}+\frac{179250195}{27660222464}a^{27}+\frac{3632567}{493932544}a^{26}+\frac{144672887}{13830111232}a^{25}+\frac{2677047}{246966272}a^{24}+\frac{241503163}{27660222464}a^{23}+\frac{3904051}{493932544}a^{22}-\frac{82129675}{13830111232}a^{21}-\frac{3228965}{246966272}a^{20}-\frac{1088601273}{13830111232}a^{19}-\frac{25919731}{246966272}a^{18}-\frac{1852968303}{6915055616}a^{17}-\frac{39982779}{123483136}a^{16}-\frac{16134233699}{27660222464}a^{15}-\frac{313043775}{493932544}a^{14}-\frac{2650537427}{3457527808}a^{13}-\frac{93497969}{123483136}a^{12}-\frac{702375227}{1728763904}a^{11}-\frac{37473875}{123483136}a^{10}+\frac{157389639}{216095488}a^{9}+\frac{18784931}{15435392}a^{8}+\frac{505144863}{108047744}a^{7}+\frac{1450803}{241178}a^{6}+\frac{346689025}{27011936}a^{5}+\frac{1805000}{120589}a^{4}+\frac{75018975}{3376492}a^{3}+\frac{5371209}{241178}a^{2}+\frac{33304127}{1688246}a+\frac{1904358}{120589}$, $\frac{1761481}{3951460352}a^{31}-\frac{17749}{70561792}a^{30}+\frac{9618355}{3951460352}a^{29}-\frac{82585}{246966272}a^{28}+\frac{19477259}{3951460352}a^{27}+\frac{21891}{15435392}a^{26}+\frac{12283965}{1975730176}a^{25}+\frac{10875}{3067904}a^{24}+\frac{15184311}{3951460352}a^{23}+\frac{133667}{21475328}a^{22}-\frac{27378413}{1975730176}a^{21}+\frac{589425}{70561792}a^{20}-\frac{153476441}{1975730176}a^{19}+\frac{1692615}{123483136}a^{18}-\frac{241851577}{987865088}a^{17}-\frac{142635}{35280896}a^{16}-\frac{1605627367}{3951460352}a^{15}-\frac{79359079}{493932544}a^{14}-\frac{380367903}{987865088}a^{13}-\frac{3149517}{10080256}a^{12}-\frac{1088219}{10737664}a^{11}-\frac{6386441}{15435392}a^{10}+\frac{67183047}{61741568}a^{9}-\frac{4993}{9844}a^{8}+\frac{65670453}{15435392}a^{7}-\frac{2703475}{7717696}a^{6}+\frac{1315206}{120589}a^{5}+\frac{42081}{39376}a^{4}+\frac{12574829}{964712}a^{3}+\frac{2139363}{241178}a^{2}+\frac{764177}{241178}a+\frac{1388297}{120589}$, $\frac{2825145}{13830111232}a^{30}+\frac{14200167}{13830111232}a^{28}+\frac{36874039}{13830111232}a^{26}+\frac{25736187}{6915055616}a^{24}+\frac{41611615}{13830111232}a^{22}-\frac{27459039}{6915055616}a^{20}-\frac{253898213}{6915055616}a^{18}-\frac{392691539}{3457527808}a^{16}-\frac{3047234727}{13830111232}a^{14}-\frac{433692059}{1728763904}a^{12}-\frac{66197359}{864381952}a^{10}+\frac{27564231}{54023872}a^{8}+\frac{63344629}{27011936}a^{6}+\frac{74195865}{13505968}a^{4}+\frac{29273345}{3376492}a^{2}+\frac{5252413}{844123}$, $\frac{5599709}{27660222464}a^{31}+\frac{1835871}{6915055616}a^{30}+\frac{36376927}{27660222464}a^{29}+\frac{10377469}{6915055616}a^{28}+\frac{99896983}{27660222464}a^{27}+\frac{1182187}{300654592}a^{26}+\frac{84625481}{13830111232}a^{25}+\frac{21767531}{3457527808}a^{24}+\frac{140337507}{27660222464}a^{23}+\frac{29796969}{6915055616}a^{22}-\frac{62544185}{13830111232}a^{21}-\frac{20545327}{3457527808}a^{20}-\frac{613901357}{13830111232}a^{19}-\frac{185482091}{3457527808}a^{18}-\frac{1083268997}{6915055616}a^{17}-\frac{291587167}{1728763904}a^{16}-\frac{8979972499}{27660222464}a^{15}-\frac{2425520737}{6915055616}a^{14}-\frac{3082890535}{6915055616}a^{13}-\frac{778032323}{1728763904}a^{12}-\frac{378527287}{1728763904}a^{11}-\frac{154405113}{864381952}a^{10}+\frac{207140769}{432190976}a^{9}+\frac{124150499}{216095488}a^{8}+\frac{289871801}{108047744}a^{7}+\frac{168527637}{54023872}a^{6}+\frac{102322545}{13505968}a^{5}+\frac{107526539}{13505968}a^{4}+\frac{41854653}{3376492}a^{3}+\frac{22485343}{1688246}a^{2}+\frac{9298846}{844123}a+\frac{8892253}{844123}$, $\frac{2134225}{3951460352}a^{31}+\frac{11623}{141123584}a^{30}+\frac{10645183}{3951460352}a^{29}+\frac{499713}{987865088}a^{28}+\frac{23956479}{3951460352}a^{27}+\frac{1355125}{987865088}a^{26}+\frac{17357923}{1975730176}a^{25}+\frac{3347}{1533952}a^{24}+\frac{24003543}{3951460352}a^{23}+\frac{71789}{42950656}a^{22}-\frac{25058255}{1975730176}a^{21}-\frac{3951}{2205056}a^{20}-\frac{183403477}{1975730176}a^{19}-\frac{7906111}{493932544}a^{18}-\frac{283274667}{987865088}a^{17}-\frac{514331}{8820224}a^{16}-\frac{2047160047}{3951460352}a^{15}-\frac{118839991}{987865088}a^{14}-\frac{301921549}{493932544}a^{13}-\frac{10986757}{70561792}a^{12}-\frac{1086175}{5368832}a^{11}-\frac{9514923}{123483136}a^{10}+\frac{16438279}{15435392}a^{9}+\frac{400013}{2205056}a^{8}+\frac{40526187}{7717696}a^{7}+\frac{6503373}{7717696}a^{6}+\frac{25729657}{1929424}a^{5}+\frac{760735}{275632}a^{4}+\frac{16942515}{964712}a^{3}+\frac{2145063}{482356}a^{2}+\frac{2767711}{241178}a+\frac{615861}{120589}$, $\frac{20901}{432190976}a^{31}-\frac{837245}{864381952}a^{30}-\frac{107549}{150327296}a^{29}-\frac{21152155}{3457527808}a^{28}-\frac{11890619}{3457527808}a^{27}-\frac{56021513}{3457527808}a^{26}-\frac{21052103}{3457527808}a^{25}-\frac{85170669}{3457527808}a^{24}-\frac{6235725}{864381952}a^{23}-\frac{35849741}{1728763904}a^{22}-\frac{10679209}{3457527808}a^{21}+\frac{75322391}{3457527808}a^{20}+\frac{3955443}{216095488}a^{19}+\frac{350727331}{1728763904}a^{18}+\frac{188985521}{1728763904}a^{17}+\frac{52618009}{75163648}a^{16}+\frac{142391289}{432190976}a^{15}+\frac{619477443}{432190976}a^{14}+\frac{1677897077}{3457527808}a^{13}+\frac{5994201025}{3457527808}a^{12}+\frac{743058929}{1728763904}a^{11}+\frac{773925743}{864381952}a^{10}+\frac{5499821}{108047744}a^{9}-\frac{493295095}{216095488}a^{8}-\frac{156995051}{108047744}a^{7}-\frac{324179039}{27011936}a^{6}-\frac{158728383}{27011936}a^{5}-\frac{443477095}{13505968}a^{4}-\frac{543009}{36701}a^{3}-\frac{89040191}{1688246}a^{2}-\frac{24138399}{1688246}a-\frac{30794402}{844123}$, $\frac{3194637}{27660222464}a^{31}+\frac{6977841}{3457527808}a^{30}+\frac{21149691}{27660222464}a^{29}+\frac{1052385}{108047744}a^{28}+\frac{61755227}{27660222464}a^{27}+\frac{42339649}{1728763904}a^{26}+\frac{53061811}{13830111232}a^{25}+\frac{130576305}{3457527808}a^{24}+\frac{97985595}{27660222464}a^{23}+\frac{97036697}{3457527808}a^{22}-\frac{12145247}{13830111232}a^{21}-\frac{140360431}{3457527808}a^{20}-\frac{364394257}{13830111232}a^{19}-\frac{13113721}{37581824}a^{18}-\frac{637405443}{6915055616}a^{17}-\frac{1880336127}{1728763904}a^{16}-\frac{5535039731}{27660222464}a^{15}-\frac{7402599779}{3457527808}a^{14}-\frac{518279773}{1728763904}a^{13}-\frac{9348942647}{3457527808}a^{12}-\frac{360796197}{1728763904}a^{11}-\frac{980710093}{864381952}a^{10}+\frac{57512797}{216095488}a^{9}+\frac{825330035}{216095488}a^{8}+\frac{166346041}{108047744}a^{7}+\frac{133618003}{6752984}a^{6}+\frac{116630181}{27011936}a^{5}+\frac{685018225}{13505968}a^{4}+\frac{13371059}{1688246}a^{3}+\frac{64861533}{844123}a^{2}+\frac{15170691}{1688246}a+\frac{52234683}{844123}$, $\frac{27862945}{27660222464}a^{31}+\frac{25055183}{13830111232}a^{30}+\frac{158652455}{27660222464}a^{29}+\frac{111682093}{13830111232}a^{28}+\frac{414236615}{27660222464}a^{27}+\frac{252900117}{13830111232}a^{26}+\frac{305270831}{13830111232}a^{25}+\frac{7934881}{300654592}a^{24}+\frac{485921479}{27660222464}a^{23}+\frac{9824663}{601309184}a^{22}-\frac{315362987}{13830111232}a^{21}-\frac{279902855}{6915055616}a^{20}-\frac{118465779}{601309184}a^{19}-\frac{1998618431}{6915055616}a^{18}-\frac{4472073007}{6915055616}a^{17}-\frac{2971292259}{3457527808}a^{16}-\frac{35784758111}{27660222464}a^{15}-\frac{21490576641}{13830111232}a^{14}-\frac{2728120399}{1728763904}a^{13}-\frac{6198689827}{3457527808}a^{12}-\frac{1253122905}{1728763904}a^{11}-\frac{113847425}{216095488}a^{10}+\frac{962918745}{432190976}a^{9}+\frac{725589647}{216095488}a^{8}+\frac{613760207}{54023872}a^{7}+\frac{870286211}{54023872}a^{6}+\frac{102824641}{3376492}a^{5}+\frac{132459057}{3376492}a^{4}+\frac{319891559}{6752984}a^{3}+\frac{173991857}{3376492}a^{2}+\frac{27813291}{844123}a+\frac{31357524}{844123}$ 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:  \( 1024801800652.4087 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 1024801800652.4087 \cdot 36}{12\cdot\sqrt{4935252174511155500627740082045583360000000000000000}}\cr\approx \mathstrut & 0.258216716320674 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + 7*x^30 + 23*x^28 + 46*x^26 + 55*x^24 + 10*x^22 - 218*x^20 - 924*x^18 - 2271*x^16 - 3696*x^14 - 3488*x^12 + 640*x^10 + 14080*x^8 + 47104*x^6 + 94208*x^4 + 114688*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 + 7*x^30 + 23*x^28 + 46*x^26 + 55*x^24 + 10*x^22 - 218*x^20 - 924*x^18 - 2271*x^16 - 3696*x^14 - 3488*x^12 + 640*x^10 + 14080*x^8 + 47104*x^6 + 94208*x^4 + 114688*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 + 7*x^30 + 23*x^28 + 46*x^26 + 55*x^24 + 10*x^22 - 218*x^20 - 924*x^18 - 2271*x^16 - 3696*x^14 - 3488*x^12 + 640*x^10 + 14080*x^8 + 47104*x^6 + 94208*x^4 + 114688*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 + 7*x^30 + 23*x^28 + 46*x^26 + 55*x^24 + 10*x^22 - 218*x^20 - 924*x^18 - 2271*x^16 - 3696*x^14 - 3488*x^12 + 640*x^10 + 14080*x^8 + 47104*x^6 + 94208*x^4 + 114688*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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), 4.4.104400.1, 4.0.6525.1, 4.4.725.1, 4.0.11600.1, \(\Q(i, \sqrt{15})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), 8.8.103476640000.1, 8.0.404205625.1, 8.8.32740655625.1, 8.0.8381607840000.1, 8.0.12960000.1, 8.0.10899360000.2, 8.0.134560000.4, 8.8.10899360000.1, 8.0.10899360000.14, 8.0.42575625.1, 8.0.10899360000.6, 16.0.118796048409600000000.1, 16.0.10707415025689600000000.1, 16.0.70251349983549465600000000.1, 16.16.70251349983549465600000000.1, 16.0.70251349983549465600000000.2, 16.0.70251349983549465600000000.3, 16.0.1071950530754844140625.1

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 ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.2.0.1}{2} }^{16}$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ R ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.8.0.1}{8} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$

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 3.8.4.1$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}$
\(5\) Copy content Toggle raw display 5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(29\) Copy content Toggle raw display 29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(769\) 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$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$