↑

Properties

Label 32.0.170...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.705\times 10^{54}$
Root discriminant \(49.52\)
Ramified primes $2,5,29,1049$
Class number $138$ (GRH)
Class group [138] (GRH)
Galois group $D_4^2:C_2^3$ (as 32T12882)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^28 + 14*x^24 + 21*x^20 - 255*x^16 + 336*x^12 + 3584*x^8 - 16384*x^4 + 65536)
 
gp: K = bnfinit(y^32 - 4*y^28 + 14*y^24 + 21*y^20 - 255*y^16 + 336*y^12 + 3584*y^8 - 16384*y^4 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 4*x^28 + 14*x^24 + 21*x^20 - 255*x^16 + 336*x^12 + 3584*x^8 - 16384*x^4 + 65536);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 4*x^28 + 14*x^24 + 21*x^20 - 255*x^16 + 336*x^12 + 3584*x^8 - 16384*x^4 + 65536)
 

\( x^{32} - 4x^{28} + 14x^{24} + 21x^{20} - 255x^{16} + 336x^{12} + 3584x^{8} - 16384x^{4} + 65536 \) 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:   \(1705005278424133715816670730943369052160000000000000000\) \(\medspace = 2^{64}\cdot 5^{16}\cdot 29^{8}\cdot 1049^{4}\) 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:  \(49.52\)
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:  $2^{2}5^{1/2}29^{1/2}1049^{1/2}\approx 1560.0256408149194$
Ramified primes:   \(2\), \(5\), \(29\), \(1049\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{18}-\frac{1}{2}a^{10}+\frac{1}{4}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{19}-\frac{1}{2}a^{15}-\frac{1}{4}a^{11}-\frac{3}{8}a^{7}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{20}-\frac{1}{4}a^{16}-\frac{1}{8}a^{12}+\frac{5}{16}a^{8}+\frac{1}{16}a^{4}$, $\frac{1}{32}a^{21}-\frac{1}{8}a^{17}+\frac{7}{16}a^{13}-\frac{11}{32}a^{9}+\frac{1}{32}a^{5}-\frac{1}{2}a$, $\frac{1}{64}a^{22}-\frac{1}{16}a^{18}+\frac{7}{32}a^{14}+\frac{21}{64}a^{10}+\frac{1}{64}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{128}a^{23}-\frac{1}{32}a^{19}+\frac{7}{64}a^{15}+\frac{21}{128}a^{11}+\frac{1}{128}a^{7}-\frac{3}{8}a^{3}$, $\frac{1}{419584}a^{24}+\frac{3023}{104896}a^{20}+\frac{6791}{209792}a^{16}+\frac{141973}{419584}a^{12}+\frac{202561}{419584}a^{8}+\frac{8265}{26224}a^{4}+\frac{16}{1639}$, $\frac{1}{839168}a^{25}+\frac{3023}{209792}a^{21}+\frac{6791}{419584}a^{17}-\frac{277611}{839168}a^{13}+\frac{202561}{839168}a^{9}+\frac{8265}{52448}a^{5}+\frac{8}{1639}a$, $\frac{1}{1678336}a^{26}+\frac{3023}{419584}a^{22}+\frac{6791}{839168}a^{18}+\frac{561557}{1678336}a^{14}+\frac{202561}{1678336}a^{10}-\frac{44183}{104896}a^{6}+\frac{4}{1639}a^{2}$, $\frac{1}{3356672}a^{27}+\frac{3023}{839168}a^{23}+\frac{6791}{1678336}a^{19}-\frac{1116779}{3356672}a^{15}-\frac{1475775}{3356672}a^{11}+\frac{60713}{209792}a^{7}-\frac{1635}{3278}a^{3}$, $\frac{1}{127553536}a^{28}+\frac{1}{1678336}a^{24}-\frac{1312665}{63776768}a^{20}-\frac{13267851}{127553536}a^{16}+\frac{61542801}{127553536}a^{12}-\frac{1188003}{3986048}a^{8}+\frac{8707}{26224}a^{4}+\frac{14208}{31141}$, $\frac{1}{255107072}a^{29}+\frac{1}{3356672}a^{25}-\frac{1312665}{127553536}a^{21}-\frac{13267851}{255107072}a^{17}-\frac{66010735}{255107072}a^{13}-\frac{1188003}{7972096}a^{9}+\frac{8707}{52448}a^{5}-\frac{16933}{62282}a$, $\frac{1}{510214144}a^{30}+\frac{1}{6713344}a^{26}-\frac{1312665}{255107072}a^{22}-\frac{13267851}{510214144}a^{18}-\frac{66010735}{510214144}a^{14}-\frac{1188003}{15944192}a^{10}+\frac{8707}{104896}a^{6}+\frac{45349}{124564}a^{2}$, $\frac{1}{1020428288}a^{31}+\frac{1}{13426688}a^{27}-\frac{1312665}{510214144}a^{23}-\frac{13267851}{1020428288}a^{19}+\frac{444203409}{1020428288}a^{15}-\frac{1188003}{31888384}a^{11}-\frac{96189}{209792}a^{7}-\frac{79215}{249128}a^{3}$ 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

$C_{138}$, which has order $138$ (assuming GRH)

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

Relative class number: $138$ (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:   \( -\frac{4073}{92766208} a^{31} + \frac{2637}{13426688} a^{27} - \frac{28469}{510214144} a^{23} - \frac{3309535}{1020428288} a^{19} + \frac{9523469}{1020428288} a^{15} + \frac{82127}{31888384} a^{11} - \frac{22359}{104896} a^{7} + \frac{70473}{124564} a^{3} \)  (order $8$) 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:   $\frac{43}{610304}a^{28}-\frac{95}{152576}a^{24}-\frac{51}{305152}a^{20}+\frac{4135}{610304}a^{16}-\frac{12261}{610304}a^{12}-\frac{307}{19072}a^{8}+\frac{1683}{2384}a^{4}-\frac{326}{149}$, $\frac{1}{31141}a^{31}-\frac{163}{839168}a^{27}+\frac{41}{1993024}a^{23}+\frac{9737}{7972096}a^{19}-\frac{63893}{15944192}a^{15}-\frac{413285}{15944192}a^{11}+\frac{45127}{209792}a^{7}-\frac{94079}{249128}a^{3}$, $\frac{6827}{255107072}a^{29}-\frac{293}{3356672}a^{25}+\frac{5837}{127553536}a^{21}+\frac{116647}{255107072}a^{17}+\frac{1986331}{255107072}a^{13}+\frac{69049}{7972096}a^{9}+\frac{1769}{6556}a^{5}+\frac{13219}{62282}a$, $\frac{4073}{92766208}a^{31}+\frac{23759}{255107072}a^{29}-\frac{2637}{13426688}a^{27}-\frac{477}{3356672}a^{25}+\frac{28469}{510214144}a^{23}-\frac{95127}{127553536}a^{21}+\frac{3309535}{1020428288}a^{19}+\frac{1010619}{255107072}a^{17}-\frac{9523469}{1020428288}a^{15}-\frac{3093521}{255107072}a^{13}-\frac{82127}{31888384}a^{11}-\frac{762347}{15944192}a^{9}+\frac{22359}{104896}a^{7}+\frac{471}{1639}a^{5}-\frac{70473}{124564}a^{3}-\frac{5185}{62282}a+1$, $\frac{5311}{255107072}a^{30}+\frac{13207}{255107072}a^{29}-\frac{29}{305152}a^{26}-\frac{445}{3356672}a^{25}+\frac{16809}{127553536}a^{22}-\frac{43743}{127553536}a^{21}-\frac{8293}{255107072}a^{18}+\frac{796387}{255107072}a^{17}-\frac{2155609}{255107072}a^{14}-\frac{3227177}{255107072}a^{13}-\frac{169441}{31888384}a^{10}+\frac{301399}{15944192}a^{9}+\frac{8087}{104896}a^{6}+\frac{14605}{52448}a^{5}-\frac{6863}{124564}a^{2}-\frac{20509}{62282}a-1$, $\frac{4073}{92766208}a^{31}-\frac{7907}{510214144}a^{30}-\frac{23759}{255107072}a^{29}-\frac{2637}{13426688}a^{27}+\frac{211}{610304}a^{26}+\frac{477}{3356672}a^{25}+\frac{28469}{510214144}a^{23}-\frac{252341}{255107072}a^{22}+\frac{95127}{127553536}a^{21}+\frac{3309535}{1020428288}a^{19}-\frac{1271711}{510214144}a^{18}-\frac{1010619}{255107072}a^{17}-\frac{9523469}{1020428288}a^{15}+\frac{7647645}{510214144}a^{14}+\frac{3093521}{255107072}a^{13}-\frac{82127}{31888384}a^{11}-\frac{1190999}{31888384}a^{10}+\frac{762347}{15944192}a^{9}+\frac{22359}{104896}a^{7}-\frac{22661}{104896}a^{6}-\frac{471}{1639}a^{5}-\frac{70473}{124564}a^{3}+\frac{137439}{124564}a^{2}+\frac{5185}{62282}a$, $\frac{9897}{1020428288}a^{31}-\frac{995}{255107072}a^{30}+\frac{43}{610304}a^{28}-\frac{33}{1220608}a^{27}-\frac{89}{305152}a^{26}-\frac{95}{152576}a^{24}+\frac{150751}{510214144}a^{23}+\frac{50795}{127553536}a^{22}-\frac{51}{305152}a^{20}+\frac{687133}{1020428288}a^{19}+\frac{292289}{255107072}a^{18}+\frac{4135}{610304}a^{16}+\frac{681481}{1020428288}a^{15}-\frac{4164563}{255107072}a^{14}-\frac{12261}{610304}a^{12}+\frac{992843}{63776768}a^{11}+\frac{49539}{7972096}a^{10}-\frac{307}{19072}a^{8}+\frac{1693}{52448}a^{7}+\frac{15889}{104896}a^{6}+\frac{1683}{2384}a^{4}-\frac{55419}{124564}a^{3}-\frac{151165}{124564}a^{2}-\frac{177}{149}$, $\frac{4073}{92766208}a^{31}-\frac{1153}{15944192}a^{30}+\frac{13137}{255107072}a^{29}-\frac{2637}{13426688}a^{27}+\frac{79}{1678336}a^{26}+\frac{161}{3356672}a^{25}+\frac{28469}{510214144}a^{23}+\frac{159}{181184}a^{22}-\frac{128745}{127553536}a^{21}+\frac{3309535}{1020428288}a^{19}-\frac{31841}{7972096}a^{18}+\frac{1027205}{255107072}a^{17}-\frac{9523469}{1020428288}a^{15}+\frac{117239}{31888384}a^{14}+\frac{1217697}{255107072}a^{13}-\frac{82127}{31888384}a^{11}+\frac{1355253}{31888384}a^{10}-\frac{296453}{7972096}a^{9}+\frac{22359}{104896}a^{7}-\frac{22057}{104896}a^{6}+\frac{635}{4768}a^{5}-\frac{70473}{124564}a^{3}+\frac{3507}{124564}a^{2}+\frac{839}{31141}a+1$, $\frac{621}{46383104}a^{31}-\frac{5311}{255107072}a^{30}-\frac{23759}{255107072}a^{29}-\frac{1325}{63776768}a^{28}-\frac{499}{6713344}a^{27}+\frac{29}{305152}a^{26}+\frac{477}{3356672}a^{25}-\frac{469}{839168}a^{24}-\frac{94759}{255107072}a^{23}-\frac{16809}{127553536}a^{22}+\frac{95127}{127553536}a^{21}+\frac{74021}{31888384}a^{20}+\frac{1327787}{510214144}a^{19}+\frac{8293}{255107072}a^{18}-\frac{1010619}{255107072}a^{17}+\frac{118831}{63776768}a^{16}-\frac{791257}{510214144}a^{15}+\frac{2155609}{255107072}a^{14}+\frac{3093521}{255107072}a^{13}-\frac{1105117}{63776768}a^{12}-\frac{818215}{63776768}a^{11}+\frac{169441}{31888384}a^{10}+\frac{762347}{15944192}a^{9}+\frac{3475}{62282}a^{8}+\frac{5443}{52448}a^{7}-\frac{8087}{104896}a^{6}-\frac{471}{1639}a^{5}+\frac{1007}{6556}a^{4}-\frac{8191}{124564}a^{3}+\frac{6863}{124564}a^{2}+\frac{5185}{62282}a-\frac{5315}{2831}$, $\frac{461}{46383104}a^{31}-\frac{3153}{127553536}a^{30}-\frac{3153}{63776768}a^{29}-\frac{5663}{63776768}a^{28}-\frac{863}{6713344}a^{27}-\frac{15}{76288}a^{26}-\frac{15}{38144}a^{25}-\frac{171}{839168}a^{24}-\frac{16487}{255107072}a^{23}+\frac{16993}{63776768}a^{22}+\frac{16993}{31888384}a^{21}+\frac{26023}{31888384}a^{20}+\frac{522859}{510214144}a^{19}+\frac{150291}{127553536}a^{18}+\frac{150291}{63776768}a^{17}-\frac{306699}{63776768}a^{16}-\frac{3746697}{510214144}a^{15}-\frac{1004477}{127553536}a^{14}-\frac{1004477}{63776768}a^{13}+\frac{1043617}{63776768}a^{12}+\frac{1256599}{63776768}a^{11}+\frac{367597}{31888384}a^{10}+\frac{367597}{15944192}a^{9}+\frac{122625}{3986048}a^{8}+\frac{13927}{209792}a^{7}+\frac{3901}{52448}a^{6}+\frac{3901}{26224}a^{5}-\frac{1967}{6556}a^{4}-\frac{57657}{124564}a^{3}-\frac{72151}{62282}a^{2}-\frac{72151}{31141}a-\frac{1844}{2831}$, $\frac{457}{15944192}a^{31}-\frac{2257}{510214144}a^{30}-\frac{6827}{255107072}a^{29}-\frac{26409}{127553536}a^{28}-\frac{417}{1678336}a^{27}+\frac{345}{610304}a^{26}+\frac{293}{3356672}a^{25}-\frac{461}{1678336}a^{24}+\frac{1305}{3986048}a^{23}-\frac{340471}{255107072}a^{22}-\frac{5837}{127553536}a^{21}+\frac{243169}{63776768}a^{20}-\frac{355}{996512}a^{19}-\frac{708069}{510214144}a^{18}-\frac{116647}{255107072}a^{17}-\frac{772957}{127553536}a^{16}-\frac{312501}{31888384}a^{15}+\frac{12997903}{510214144}a^{14}-\frac{1986331}{255107072}a^{13}+\frac{883287}{127553536}a^{12}+\frac{19167}{2898944}a^{11}-\frac{1675359}{31888384}a^{10}-\frac{69049}{7972096}a^{9}+\frac{1207147}{7972096}a^{8}+\frac{18641}{104896}a^{7}-\frac{15465}{104896}a^{6}-\frac{1769}{6556}a^{5}-\frac{251}{596}a^{4}-\frac{193011}{249128}a^{3}+\frac{109299}{62282}a^{2}+\frac{49063}{62282}a-\frac{84421}{31141}$, $\frac{461}{46383104}a^{31}+\frac{3153}{127553536}a^{30}-\frac{43}{610304}a^{29}-\frac{13207}{127553536}a^{28}-\frac{863}{6713344}a^{27}+\frac{15}{76288}a^{26}+\frac{95}{152576}a^{25}+\frac{445}{1678336}a^{24}-\frac{16487}{255107072}a^{23}-\frac{16993}{63776768}a^{22}+\frac{51}{305152}a^{21}+\frac{43743}{63776768}a^{20}+\frac{522859}{510214144}a^{19}-\frac{150291}{127553536}a^{18}-\frac{4135}{610304}a^{17}-\frac{796387}{127553536}a^{16}-\frac{3746697}{510214144}a^{15}+\frac{1004477}{127553536}a^{14}+\frac{12261}{610304}a^{13}+\frac{3227177}{127553536}a^{12}+\frac{1256599}{63776768}a^{11}-\frac{367597}{31888384}a^{10}+\frac{307}{19072}a^{9}-\frac{301399}{7972096}a^{8}+\frac{13927}{209792}a^{7}-\frac{3901}{52448}a^{6}-\frac{1683}{2384}a^{5}-\frac{14605}{26224}a^{4}-\frac{57657}{124564}a^{3}+\frac{72151}{62282}a^{2}+\frac{326}{149}a+\frac{20509}{31141}$, $\frac{138403}{1020428288}a^{31}-\frac{48805}{255107072}a^{30}+\frac{55}{305152}a^{29}-\frac{409}{15944192}a^{28}-\frac{9085}{13426688}a^{27}+\frac{3667}{3356672}a^{26}-\frac{1101}{839168}a^{25}+\frac{51}{26224}a^{24}-\frac{36203}{510214144}a^{23}-\frac{78371}{127553536}a^{22}+\frac{1611}{1678336}a^{21}-\frac{18519}{7972096}a^{20}+\frac{7411071}{1020428288}a^{19}-\frac{285851}{23191552}a^{18}+\frac{37665}{3356672}a^{17}-\frac{168901}{15944192}a^{16}-\frac{27898925}{1020428288}a^{15}+\frac{10212267}{255107072}a^{14}-\frac{169571}{3356672}a^{13}+\frac{1129875}{15944192}a^{12}+\frac{74379}{31888384}a^{11}-\frac{27003}{1993024}a^{10}+\frac{6029}{209792}a^{9}-\frac{635837}{3986048}a^{8}+\frac{169373}{209792}a^{7}-\frac{6909}{6556}a^{6}+\frac{25837}{26224}a^{5}-\frac{21863}{26224}a^{4}-\frac{248173}{124564}a^{3}+\frac{461177}{124564}a^{2}-\frac{8775}{1639}a+\frac{252545}{31141}$, $\frac{3153}{127553536}a^{31}+\frac{20519}{510214144}a^{30}+\frac{3153}{63776768}a^{29}+\frac{15}{76288}a^{27}-\frac{91}{610304}a^{26}+\frac{15}{38144}a^{25}-\frac{16993}{63776768}a^{23}+\frac{184369}{255107072}a^{22}-\frac{16993}{31888384}a^{21}-\frac{150291}{127553536}a^{19}+\frac{670547}{510214144}a^{18}-\frac{150291}{63776768}a^{17}+\frac{1004477}{127553536}a^{15}-\frac{3629737}{510214144}a^{14}+\frac{1004477}{63776768}a^{13}-\frac{367597}{31888384}a^{11}+\frac{411701}{15944192}a^{10}-\frac{367597}{15944192}a^{9}-\frac{3901}{52448}a^{7}+\frac{14859}{104896}a^{6}-\frac{3901}{26224}a^{5}+\frac{72151}{62282}a^{3}+\frac{6863}{124564}a^{2}+\frac{72151}{31141}a$, $\frac{52385}{1020428288}a^{31}+\frac{8429}{127553536}a^{30}-\frac{43723}{255107072}a^{29}+\frac{19539}{127553536}a^{28}-\frac{265}{1220608}a^{27}+\frac{157}{839168}a^{26}+\frac{609}{3356672}a^{25}-\frac{1077}{1678336}a^{24}+\frac{285223}{510214144}a^{23}-\frac{42685}{63776768}a^{22}+\frac{218035}{127553536}a^{21}-\frac{62043}{63776768}a^{20}+\frac{620789}{1020428288}a^{19}-\frac{3925}{11595776}a^{18}-\frac{195861}{23191552}a^{17}+\frac{1078447}{127553536}a^{16}-\frac{16563391}{1020428288}a^{15}+\frac{1071305}{127553536}a^{14}-\frac{110507}{255107072}a^{13}-\frac{2428893}{127553536}a^{12}+\frac{315079}{63776768}a^{11}-\frac{2495089}{31888384}a^{10}+\frac{1217155}{15944192}a^{9}-\frac{149009}{996512}a^{8}+\frac{2445}{13112}a^{7}-\frac{1717}{26224}a^{6}-\frac{36209}{52448}a^{5}+\frac{4745}{6556}a^{4}-\frac{69145}{124564}a^{3}+\frac{87475}{62282}a^{2}-\frac{35997}{31141}a-\frac{52810}{31141}$ 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:  \( 3793239043715.9175 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 3793239043715.9175 \cdot 138}{8\cdot\sqrt{1705005278424133715816670730943369052160000000000000000}}\cr\approx \mathstrut & 0.295674820148817 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^28 + 14*x^24 + 21*x^20 - 255*x^16 + 336*x^12 + 3584*x^8 - 16384*x^4 + 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^28 + 14*x^24 + 21*x^20 - 255*x^16 + 336*x^12 + 3584*x^8 - 16384*x^4 + 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^28 + 14*x^24 + 21*x^20 - 255*x^16 + 336*x^12 + 3584*x^8 - 16384*x^4 + 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^28 + 14*x^24 + 21*x^20 - 255*x^16 + 336*x^12 + 3584*x^8 - 16384*x^4 + 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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-10}) \), 4.4.46400.1, 4.4.725.1, 4.0.46400.1, 4.0.11600.1, \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(i, \sqrt{10})\), 8.0.551380625.1, 8.0.2258455040000.1, 8.8.141153440000.2, 8.8.2258455040000.1, 8.0.40960000.1, 8.8.2152960000.1, 8.0.34447360000.26, 8.0.34447360000.3, 8.0.134560000.4, 8.0.2152960000.5, 8.0.34447360000.23, 16.0.1186620610969600000000.3, 16.0.5100619167701401600000000.1, 16.16.1305758506931558809600000000.1, 16.0.19924293623833600000000.1, 16.0.1305758506931558809600000000.2, 16.0.1305758506931558809600000000.1, 16.0.5100619167701401600000000.2

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: 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 ${\href{/padicField/3.8.0.1}{8} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.2.0.1}{2} }^{16}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.8.0.1}{8} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ R ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.8.0.1}{8} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.8.0.1}{8} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{8}$

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 $16$$4$$4$$32$
Deg $16$$4$$4$$32$
\(5\) 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 29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(1049\) Copy content Toggle raw display $\Q_{1049}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1049}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1049}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1049}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1049}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1049}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1049}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1049}$$x$$1$$1$$0$Trivial$[\ ]$
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$$2$$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}$