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Properties

Label 32.0.156...000.7
Degree $32$
Signature $[0, 16]$
Discriminant $1.569\times 10^{59}$
Root discriminant \(70.77\)
Ramified primes $2,5,7$
Class number $108800$ (GRH)
Class group [80, 1360] (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 64*x^30 + 1856*x^28 + 32256*x^26 + 374431*x^24 + 3063248*x^22 + 18166288*x^20 + 79134848*x^18 + 253925217*x^16 + 596716576*x^14 + 1012551072*x^12 + 1211673344*x^10 + 986220191*x^8 + 516827632*x^6 + 159875504*x^4 + 25087872*x^2 + 1442401)
 
gp: K = bnfinit(y^32 + 64*y^30 + 1856*y^28 + 32256*y^26 + 374431*y^24 + 3063248*y^22 + 18166288*y^20 + 79134848*y^18 + 253925217*y^16 + 596716576*y^14 + 1012551072*y^12 + 1211673344*y^10 + 986220191*y^8 + 516827632*y^6 + 159875504*y^4 + 25087872*y^2 + 1442401, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 64*x^30 + 1856*x^28 + 32256*x^26 + 374431*x^24 + 3063248*x^22 + 18166288*x^20 + 79134848*x^18 + 253925217*x^16 + 596716576*x^14 + 1012551072*x^12 + 1211673344*x^10 + 986220191*x^8 + 516827632*x^6 + 159875504*x^4 + 25087872*x^2 + 1442401);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 64*x^30 + 1856*x^28 + 32256*x^26 + 374431*x^24 + 3063248*x^22 + 18166288*x^20 + 79134848*x^18 + 253925217*x^16 + 596716576*x^14 + 1012551072*x^12 + 1211673344*x^10 + 986220191*x^8 + 516827632*x^6 + 159875504*x^4 + 25087872*x^2 + 1442401)
 

\( x^{32} + 64 x^{30} + 1856 x^{28} + 32256 x^{26} + 374431 x^{24} + 3063248 x^{22} + 18166288 x^{20} + \cdots + 1442401 \) 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:   \(156938077449417789520626992646455296000000000000000000000000\) \(\medspace = 2^{96}\cdot 5^{24}\cdot 7^{16}\) 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:  \(70.77\)
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:  $2^{3}5^{3/4}7^{1/2}\approx 70.7728215461241$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(560=2^{4}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(363,·)$, $\chi_{560}(391,·)$, $\chi_{560}(407,·)$, $\chi_{560}(533,·)$, $\chi_{560}(279,·)$, $\chi_{560}(153,·)$, $\chi_{560}(27,·)$, $\chi_{560}(197,·)$, $\chi_{560}(169,·)$, $\chi_{560}(477,·)$, $\chi_{560}(433,·)$, $\chi_{560}(307,·)$, $\chi_{560}(181,·)$, $\chi_{560}(183,·)$, $\chi_{560}(449,·)$, $\chi_{560}(69,·)$, $\chi_{560}(461,·)$, $\chi_{560}(463,·)$, $\chi_{560}(211,·)$, $\chi_{560}(349,·)$, $\chi_{560}(97,·)$, $\chi_{560}(99,·)$, $\chi_{560}(491,·)$, $\chi_{560}(111,·)$, $\chi_{560}(83,·)$, $\chi_{560}(377,·)$, $\chi_{560}(379,·)$, $\chi_{560}(281,·)$, $\chi_{560}(253,·)$, $\chi_{560}(559,·)$, $\chi_{560}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{14}-\frac{1}{3}a^{12}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{15}-\frac{1}{3}a^{13}-\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{18}+\frac{1}{3}a^{14}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{19}+\frac{1}{3}a^{15}+\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{627}a^{20}+\frac{40}{627}a^{18}+\frac{53}{627}a^{16}+\frac{130}{627}a^{14}+\frac{34}{627}a^{12}+\frac{20}{57}a^{10}-\frac{2}{19}a^{8}-\frac{1}{19}a^{6}-\frac{3}{19}a^{4}-\frac{214}{627}a^{2}+\frac{167}{627}$, $\frac{1}{753027}a^{21}+\frac{90119}{753027}a^{19}-\frac{30148}{251009}a^{17}+\frac{305479}{753027}a^{15}-\frac{170510}{753027}a^{13}-\frac{17707}{68457}a^{11}+\frac{3646}{22819}a^{9}-\frac{25007}{68457}a^{7}+\frac{31930}{68457}a^{5}+\frac{99134}{251009}a^{3}-\frac{232868}{753027}a$, $\frac{1}{753027}a^{22}+\frac{4}{68457}a^{20}+\frac{23897}{251009}a^{18}+\frac{2614}{39633}a^{16}+\frac{168172}{753027}a^{14}-\frac{245219}{753027}a^{12}-\frac{3560}{22819}a^{10}-\frac{32213}{68457}a^{8}+\frac{28327}{68457}a^{6}+\frac{59501}{251009}a^{4}+\frac{217507}{753027}a^{2}+\frac{5}{209}$, $\frac{1}{753027}a^{23}+\frac{122599}{753027}a^{19}+\frac{13058}{753027}a^{17}+\frac{10191}{251009}a^{15}+\frac{4017}{13211}a^{13}-\frac{7418}{68457}a^{11}+\frac{3784}{22819}a^{9}-\frac{4105}{22819}a^{7}-\frac{215077}{753027}a^{5}+\frac{61429}{251009}a^{3}-\frac{278171}{753027}a$, $\frac{1}{753027}a^{24}+\frac{97}{753027}a^{20}-\frac{117851}{753027}a^{18}+\frac{3379}{39633}a^{16}-\frac{133733}{753027}a^{14}+\frac{6829}{251009}a^{12}+\frac{155}{3603}a^{10}+\frac{15308}{68457}a^{8}+\frac{313363}{753027}a^{6}+\frac{87851}{251009}a^{4}-\frac{55893}{251009}a^{2}-\frac{35}{209}$, $\frac{1}{753027}a^{25}-\frac{24693}{251009}a^{19}+\frac{17318}{251009}a^{17}-\frac{146134}{753027}a^{15}+\frac{244372}{753027}a^{13}+\frac{31918}{68457}a^{11}+\frac{1332}{22819}a^{9}-\frac{121383}{251009}a^{7}+\frac{80458}{753027}a^{5}+\frac{101371}{753027}a^{3}-\frac{128719}{753027}a$, $\frac{1}{753027}a^{26}+\frac{383}{753027}a^{20}+\frac{1666}{68457}a^{18}+\frac{11739}{251009}a^{16}+\frac{45027}{251009}a^{14}-\frac{129302}{753027}a^{12}-\frac{12818}{68457}a^{10}-\frac{2484}{251009}a^{8}-\frac{117707}{753027}a^{6}+\frac{259903}{753027}a^{4}-\frac{83340}{251009}a^{2}-\frac{305}{627}$, $\frac{1}{753027}a^{27}-\frac{109018}{753027}a^{19}+\frac{12009}{251009}a^{17}+\frac{18833}{39633}a^{15}+\frac{54899}{251009}a^{13}-\frac{31099}{68457}a^{11}+\frac{347419}{753027}a^{9}-\frac{62332}{251009}a^{7}+\frac{279637}{753027}a^{5}+\frac{4919}{68457}a^{3}-\frac{95352}{251009}a$, $\frac{1}{753027}a^{28}+\frac{91}{251009}a^{20}-\frac{10045}{68457}a^{18}-\frac{124975}{753027}a^{16}+\frac{105341}{251009}a^{14}-\frac{140321}{753027}a^{12}-\frac{207443}{753027}a^{10}-\frac{371950}{753027}a^{8}-\frac{21283}{251009}a^{6}+\frac{25336}{68457}a^{4}-\frac{330493}{753027}a^{2}-\frac{20}{209}$, $\frac{1}{753027}a^{29}-\frac{114100}{753027}a^{19}-\frac{32645}{753027}a^{17}+\frac{385}{68457}a^{15}-\frac{9252}{251009}a^{13}-\frac{82410}{251009}a^{11}+\frac{8707}{39633}a^{9}+\frac{231490}{753027}a^{7}+\frac{2485}{68457}a^{5}+\frac{102565}{251009}a^{3}+\frac{82212}{251009}a$, $\frac{1}{753027}a^{30}-\frac{5}{753027}a^{20}+\frac{4331}{251009}a^{18}+\frac{9018}{251009}a^{16}-\frac{255946}{753027}a^{14}-\frac{133135}{753027}a^{12}-\frac{5905}{13211}a^{10}+\frac{231490}{753027}a^{8}+\frac{2485}{68457}a^{6}+\frac{102565}{251009}a^{4}-\frac{72830}{753027}a^{2}+\frac{10}{33}$, $\frac{1}{753027}a^{31}-\frac{12810}{251009}a^{19}+\frac{76852}{753027}a^{17}+\frac{267413}{753027}a^{15}+\frac{269360}{753027}a^{13}-\frac{18475}{251009}a^{11}-\frac{170956}{753027}a^{9}-\frac{1646}{3603}a^{7}-\frac{195236}{753027}a^{5}+\frac{53045}{251009}a^{3}-\frac{61041}{251009}a$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{80}\times C_{1360}$, which has order $108800$ (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:   \( -1 \)  (order $2$) 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:   $\frac{31}{251009}a^{24}+\frac{1488}{251009}a^{22}+\frac{31248}{251009}a^{20}+\frac{19840}{13211}a^{18}+\frac{2884449}{251009}a^{16}+\frac{14593056}{251009}a^{14}+\frac{49401248}{251009}a^{12}+\frac{10084608}{22819}a^{10}+\frac{14604480}{22819}a^{8}+\frac{142263296}{251009}a^{6}+\frac{3664896}{13211}a^{4}+\frac{14917632}{251009}a^{2}+\frac{721}{209}$, $\frac{31}{753027}a^{30}+\frac{620}{251009}a^{28}+\frac{50284}{753027}a^{26}+\frac{269776}{251009}a^{24}+\frac{8633249}{753027}a^{22}+\frac{5845636}{68457}a^{20}+\frac{343241924}{753027}a^{18}+\frac{23277968}{13211}a^{16}+\frac{3710154527}{753027}a^{14}+\frac{2473074636}{251009}a^{12}+\frac{3458303940}{251009}a^{10}+\frac{9784079216}{753027}a^{8}+\frac{5881174049}{753027}a^{6}+\frac{2058970508}{753027}a^{4}+\frac{359062633}{753027}a^{2}+\frac{19556}{627}$, $\frac{20}{753027}a^{30}+\frac{400}{251009}a^{28}+\frac{32464}{753027}a^{26}+\frac{523328}{753027}a^{24}+\frac{5597249}{753027}a^{22}+\frac{3805444}{68457}a^{20}+\frac{224797444}{753027}a^{18}+\frac{15379088}{13211}a^{16}+\frac{2482237727}{753027}a^{14}+\frac{5054347108}{753027}a^{12}+\frac{7253298919}{753027}a^{10}+\frac{7096077196}{753027}a^{8}+\frac{4490567909}{753027}a^{6}+\frac{563430036}{251009}a^{4}+\frac{328302233}{753027}a^{2}+\frac{20260}{627}$, $\frac{17}{753027}a^{31}+\frac{1031}{753027}a^{29}+\frac{9393}{251009}a^{27}+\frac{458731}{753027}a^{25}+\frac{1650982}{251009}a^{23}+\frac{1132788}{22819}a^{21}+\frac{202541248}{753027}a^{19}+\frac{13975536}{13211}a^{17}+\frac{2273882705}{753027}a^{15}+\frac{1554329869}{251009}a^{13}+\frac{2242750193}{251009}a^{11}+\frac{6601963627}{753027}a^{9}+\frac{4176097906}{753027}a^{7}+\frac{520906460}{251009}a^{5}+\frac{297464896}{753027}a^{3}+\frac{20638768}{753027}a$, $\frac{271}{753027}a^{23}+\frac{12466}{753027}a^{21}+\frac{249320}{753027}a^{19}+\frac{2843215}{753027}a^{17}+\frac{1852490}{68457}a^{15}+\frac{95401348}{753027}a^{13}+\frac{468624}{1201}a^{11}+\frac{17724560}{22819}a^{9}+\frac{21795520}{22819}a^{7}+\frac{500543360}{753027}a^{5}+\frac{165667328}{753027}a^{3}+\frac{5657856}{251009}a$, $\frac{1}{753027}a^{27}+\frac{18}{251009}a^{25}+\frac{432}{251009}a^{23}+\frac{552}{22819}a^{21}+\frac{5040}{22819}a^{19}+\frac{18144}{13211}a^{17}+\frac{78336}{13211}a^{15}+\frac{13387177}{753027}a^{13}+\frac{2480062}{68457}a^{11}+\frac{1851980}{39633}a^{9}+\frac{7099040}{251009}a^{7}-\frac{9012640}{753027}a^{5}-\frac{20753824}{753027}a^{3}-\frac{6666944}{753027}a$, $\frac{17}{753027}a^{31}+\frac{1054}{753027}a^{29}+\frac{29512}{753027}a^{27}+\frac{164424}{251009}a^{25}+\frac{5480800}{753027}a^{23}+\frac{3878720}{68457}a^{21}+\frac{21720832}{68457}a^{19}+\frac{1551488}{1201}a^{17}+\frac{4587008}{1201}a^{15}+\frac{321090560}{39633}a^{13}+\frac{477048832}{39633}a^{11}+\frac{274664697}{22819}a^{9}+\frac{173476226}{22819}a^{7}+\frac{186886948}{68457}a^{5}+\frac{31554640}{68457}a^{3}+\frac{1664048}{68457}a$, $\frac{16}{251009}a^{28}+\frac{896}{251009}a^{26}+\frac{22400}{251009}a^{24}+\frac{329728}{251009}a^{22}+\frac{9523298}{753027}a^{20}+\frac{63016784}{753027}a^{18}+\frac{26668784}{68457}a^{16}+\frac{967938560}{753027}a^{14}+\frac{204960151}{68457}a^{12}+\frac{3654443960}{753027}a^{10}+\frac{1338063656}{251009}a^{8}+\frac{950975616}{251009}a^{6}+\frac{401467889}{251009}a^{4}+\frac{254181016}{753027}a^{2}+\frac{1352}{57}$, $\frac{31}{251009}a^{28}+\frac{1736}{251009}a^{26}+\frac{43400}{251009}a^{24}+\frac{638848}{251009}a^{22}+\frac{18449138}{753027}a^{20}+\frac{122004944}{753027}a^{18}+\frac{566847184}{753027}a^{16}+\frac{1860968960}{753027}a^{14}+\frac{4286197904}{753027}a^{12}+\frac{357411968}{39633}a^{10}+\frac{2383018112}{251009}a^{8}+\frac{82884608}{13211}a^{6}+\frac{592720769}{251009}a^{4}+\frac{317314072}{753027}a^{2}+\frac{15256}{627}$, $\frac{47}{753027}a^{28}+\frac{2632}{753027}a^{26}+\frac{65707}{753027}a^{24}+\frac{964112}{753027}a^{22}+\frac{279697}{22819}a^{20}+\frac{1834152}{22819}a^{18}+\frac{277809029}{753027}a^{16}+\frac{896627872}{753027}a^{14}+\frac{672492437}{251009}a^{12}+\frac{1032557048}{251009}a^{10}+\frac{3129218408}{753027}a^{8}+\frac{654570880}{251009}a^{6}+\frac{700990195}{753027}a^{4}+\frac{42418824}{251009}a^{2}+\frac{9496}{627}$, $\frac{28}{753027}a^{30}+\frac{560}{251009}a^{28}+\frac{45668}{753027}a^{26}+\frac{744016}{753027}a^{24}+\frac{8097236}{753027}a^{22}+\frac{1881712}{22819}a^{20}+\frac{344828183}{753027}a^{18}+\frac{24602396}{13211}a^{16}+\frac{126469109}{22819}a^{14}+\frac{8982594124}{753027}a^{12}+\frac{4549730143}{251009}a^{10}+\frac{1279272652}{68457}a^{8}+\frac{9246847630}{753027}a^{6}+\frac{3504606632}{753027}a^{4}+\frac{647580191}{753027}a^{2}+\frac{34876}{627}$, $\frac{14}{753027}a^{31}+\frac{27}{22819}a^{29}+\frac{8546}{251009}a^{27}+\frac{440780}{753027}a^{25}+\frac{5041505}{753027}a^{23}+\frac{40428302}{753027}a^{21}+\frac{77807336}{251009}a^{19}+\frac{981271096}{753027}a^{17}+\frac{1001258037}{251009}a^{15}+\frac{6631771042}{753027}a^{13}+\frac{3451549853}{251009}a^{11}+\frac{335415449}{22819}a^{9}+\frac{7693547503}{753027}a^{7}+\frac{3192688858}{753027}a^{5}+\frac{674736728}{753027}a^{3}+\frac{16393578}{251009}a$, $\frac{37}{753027}a^{31}+\frac{2309}{753027}a^{29}+\frac{65224}{753027}a^{27}+\frac{1102672}{753027}a^{25}+\frac{12429674}{753027}a^{23}+\frac{32839421}{251009}a^{21}+\frac{564260549}{753027}a^{19}+\frac{2363547578}{753027}a^{17}+\frac{2415932525}{251009}a^{15}+\frac{16132225496}{753027}a^{13}+\frac{8527939174}{251009}a^{11}+\frac{28015903735}{753027}a^{9}+\frac{6709657789}{251009}a^{7}+\frac{2896375851}{251009}a^{5}+\frac{1893422303}{753027}a^{3}+\frac{11898160}{68457}a$, $\frac{14}{753027}a^{31}+\frac{797}{753027}a^{29}+\frac{1852}{68457}a^{27}+\frac{308894}{753027}a^{25}+\frac{3092675}{753027}a^{23}+\frac{7180256}{251009}a^{21}+\frac{107139589}{753027}a^{19}+\frac{128173123}{251009}a^{17}+\frac{331374017}{251009}a^{15}+\frac{1826287442}{753027}a^{13}+\frac{771777889}{251009}a^{11}+\frac{1919007073}{753027}a^{9}+\frac{939467951}{753027}a^{7}+\frac{73794877}{251009}a^{5}+\frac{19937152}{753027}a^{3}+\frac{296740}{39633}a$, $\frac{31}{753027}a^{30}+\frac{636}{251009}a^{28}+\frac{2788}{39633}a^{26}+\frac{292176}{251009}a^{24}+\frac{3207297}{251009}a^{22}+\frac{73800245}{753027}a^{20}+\frac{135252841}{251009}a^{18}+\frac{1614553940}{753027}a^{16}+\frac{421681417}{68457}a^{14}+\frac{3165134091}{251009}a^{12}+\frac{1228494268}{68457}a^{10}+\frac{12868895144}{753027}a^{8}+\frac{7759265057}{753027}a^{6}+\frac{2740782175}{753027}a^{4}+\frac{504648193}{753027}a^{2}+\frac{10661}{209}$ 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:  \( 185908723307.47064 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 185908723307.47064 \cdot 108800}{2\cdot\sqrt{156938077449417789520626992646455296000000000000000000000000}}\cr\approx \mathstrut & 0.150630319702821 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + 64*x^30 + 1856*x^28 + 32256*x^26 + 374431*x^24 + 3063248*x^22 + 18166288*x^20 + 79134848*x^18 + 253925217*x^16 + 596716576*x^14 + 1012551072*x^12 + 1211673344*x^10 + 986220191*x^8 + 516827632*x^6 + 159875504*x^4 + 25087872*x^2 + 1442401)
 
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 + 64*x^30 + 1856*x^28 + 32256*x^26 + 374431*x^24 + 3063248*x^22 + 18166288*x^20 + 79134848*x^18 + 253925217*x^16 + 596716576*x^14 + 1012551072*x^12 + 1211673344*x^10 + 986220191*x^8 + 516827632*x^6 + 159875504*x^4 + 25087872*x^2 + 1442401, 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 + 64*x^30 + 1856*x^28 + 32256*x^26 + 374431*x^24 + 3063248*x^22 + 18166288*x^20 + 79134848*x^18 + 253925217*x^16 + 596716576*x^14 + 1012551072*x^12 + 1211673344*x^10 + 986220191*x^8 + 516827632*x^6 + 159875504*x^4 + 25087872*x^2 + 1442401);
 
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 + 64*x^30 + 1856*x^28 + 32256*x^26 + 374431*x^24 + 3063248*x^22 + 18166288*x^20 + 79134848*x^18 + 253925217*x^16 + 596716576*x^14 + 1012551072*x^12 + 1211673344*x^10 + 986220191*x^8 + 516827632*x^6 + 159875504*x^4 + 25087872*x^2 + 1442401);
 
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

$C_2\times C_4^2$ (as 32T36):

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)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$

Intermediate fields

\(\Q(\sqrt{35}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}, \sqrt{35})\), 4.0.2048.2, 4.0.2508800.1, \(\Q(\sqrt{10}, \sqrt{14})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{7})\), 4.0.51200.2, 4.0.100352.5, \(\Q(\sqrt{7}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{14})\), 4.0.12544000.2, 4.0.256000.4, 4.0.12544000.1, 4.0.256000.2, 4.4.392000.1, 4.4.8000.1, 4.4.6125.1, \(\Q(\zeta_{20})^+\), 8.0.25176309760000.40, 8.8.98344960000.1, 8.0.25176309760000.66, 8.0.2621440000.1, 8.0.40282095616.1, 8.0.6294077440000.7, 8.0.25176309760000.41, 8.0.629407744000000.21, 8.0.629407744000000.70, 8.8.2458624000000.1, 8.8.9604000000.1, 8.0.157351936000000.83, 8.0.65536000000.1, 8.8.153664000000.1, \(\Q(\zeta_{40})^+\), 8.0.629407744000000.44, 8.0.629407744000000.69, 8.8.2458624000000.2, 8.8.153664000000.2, 16.0.633846573131471257600000000.9, 16.0.396154108207169536000000000000.15, 16.16.6044831973376000000000000.1, 16.0.24759631762948096000000000000.2, 16.0.68719476736000000000000.2, 16.0.396154108207169536000000000000.12, 16.0.24759631762948096000000000000.3

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{8}$ R R ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{8}$

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 Deg $32$$8$$4$$96$
\(5\) Copy content Toggle raw display 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
\(7\) Copy content Toggle raw display 7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$