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Properties

Label 32.0.103...616.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.034\times 10^{48}$
Root discriminant \(31.66\)
Ramified primes $2,3,41,113$
Class number $18$ (GRH)
Class group [3, 6] (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^31 + 8*x^30 - 16*x^29 + 25*x^28 - 24*x^27 + 24*x^26 - 20*x^25 - 36*x^23 + 152*x^22 - 256*x^21 + 425*x^20 - 584*x^19 + 296*x^18 + 4*x^17 + 113*x^16 + 8*x^15 + 1184*x^14 - 4672*x^13 + 6800*x^12 - 8192*x^11 + 9728*x^10 - 4608*x^9 - 10240*x^7 + 24576*x^6 - 49152*x^5 + 102400*x^4 - 131072*x^3 + 131072*x^2 - 131072*x + 65536)
 
gp: K = bnfinit(y^32 - 4*y^31 + 8*y^30 - 16*y^29 + 25*y^28 - 24*y^27 + 24*y^26 - 20*y^25 - 36*y^23 + 152*y^22 - 256*y^21 + 425*y^20 - 584*y^19 + 296*y^18 + 4*y^17 + 113*y^16 + 8*y^15 + 1184*y^14 - 4672*y^13 + 6800*y^12 - 8192*y^11 + 9728*y^10 - 4608*y^9 - 10240*y^7 + 24576*y^6 - 49152*y^5 + 102400*y^4 - 131072*y^3 + 131072*y^2 - 131072*y + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 4*x^31 + 8*x^30 - 16*x^29 + 25*x^28 - 24*x^27 + 24*x^26 - 20*x^25 - 36*x^23 + 152*x^22 - 256*x^21 + 425*x^20 - 584*x^19 + 296*x^18 + 4*x^17 + 113*x^16 + 8*x^15 + 1184*x^14 - 4672*x^13 + 6800*x^12 - 8192*x^11 + 9728*x^10 - 4608*x^9 - 10240*x^7 + 24576*x^6 - 49152*x^5 + 102400*x^4 - 131072*x^3 + 131072*x^2 - 131072*x + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 4*x^31 + 8*x^30 - 16*x^29 + 25*x^28 - 24*x^27 + 24*x^26 - 20*x^25 - 36*x^23 + 152*x^22 - 256*x^21 + 425*x^20 - 584*x^19 + 296*x^18 + 4*x^17 + 113*x^16 + 8*x^15 + 1184*x^14 - 4672*x^13 + 6800*x^12 - 8192*x^11 + 9728*x^10 - 4608*x^9 - 10240*x^7 + 24576*x^6 - 49152*x^5 + 102400*x^4 - 131072*x^3 + 131072*x^2 - 131072*x + 65536)
 

\( x^{32} - 4 x^{31} + 8 x^{30} - 16 x^{29} + 25 x^{28} - 24 x^{27} + 24 x^{26} - 20 x^{25} - 36 x^{23} + \cdots + 65536 \) 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:   \(1033818800357475102568684143000990752577071087616\) \(\medspace = 2^{64}\cdot 3^{16}\cdot 41^{8}\cdot 113^{4}\) 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:  \(31.66\)
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:  $2^{2}3^{1/2}41^{1/2}113^{1/2}\approx 471.5760808183553$
Ramified primes:   \(2\), \(3\), \(41\), \(113\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{4}a^{14}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{6}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{16}-\frac{1}{4}a^{13}-\frac{1}{2}a^{10}-\frac{1}{4}a^{7}-\frac{7}{16}a^{4}$, $\frac{1}{16}a^{17}-\frac{1}{2}a^{11}-\frac{1}{4}a^{8}-\frac{7}{16}a^{5}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{18}-\frac{1}{2}a^{12}-\frac{1}{4}a^{9}-\frac{7}{16}a^{6}+\frac{1}{4}a^{3}$, $\frac{1}{16}a^{19}-\frac{1}{4}a^{10}-\frac{7}{16}a^{7}+\frac{1}{4}a^{4}-\frac{1}{2}a$, $\frac{1}{16}a^{20}-\frac{1}{4}a^{11}-\frac{7}{16}a^{8}+\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{21}-\frac{1}{32}a^{17}-\frac{1}{8}a^{14}-\frac{1}{8}a^{12}-\frac{1}{4}a^{11}+\frac{9}{32}a^{9}+\frac{1}{8}a^{8}-\frac{3}{8}a^{6}-\frac{9}{32}a^{5}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{22}+\frac{1}{64}a^{18}-\frac{1}{16}a^{15}-\frac{1}{16}a^{13}+\frac{1}{8}a^{12}-\frac{1}{2}a^{11}-\frac{23}{64}a^{10}-\frac{1}{16}a^{9}-\frac{3}{16}a^{7}-\frac{23}{64}a^{6}+\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{128}a^{23}+\frac{1}{128}a^{19}-\frac{1}{32}a^{18}-\frac{1}{32}a^{16}-\frac{1}{32}a^{14}+\frac{1}{16}a^{13}+\frac{41}{128}a^{11}+\frac{15}{32}a^{10}+\frac{1}{8}a^{9}-\frac{3}{32}a^{8}+\frac{41}{128}a^{7}-\frac{9}{32}a^{6}+\frac{1}{16}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}$, $\frac{1}{256}a^{24}-\frac{7}{256}a^{20}-\frac{1}{64}a^{19}-\frac{1}{64}a^{17}-\frac{1}{64}a^{15}+\frac{1}{32}a^{14}+\frac{41}{256}a^{12}+\frac{23}{64}a^{11}+\frac{1}{16}a^{10}-\frac{3}{64}a^{9}-\frac{31}{256}a^{8}+\frac{23}{64}a^{7}+\frac{1}{32}a^{6}-\frac{1}{16}a^{5}+\frac{3}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{512}a^{25}-\frac{7}{512}a^{21}+\frac{3}{128}a^{20}-\frac{1}{128}a^{18}-\frac{1}{32}a^{17}-\frac{1}{128}a^{16}+\frac{1}{64}a^{15}-\frac{1}{8}a^{14}+\frac{41}{512}a^{13}-\frac{41}{128}a^{12}+\frac{5}{32}a^{11}+\frac{61}{128}a^{10}+\frac{225}{512}a^{9}+\frac{11}{128}a^{8}+\frac{1}{64}a^{7}+\frac{15}{32}a^{6}-\frac{15}{32}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{5120}a^{26}-\frac{1}{1280}a^{24}+\frac{1}{640}a^{23}+\frac{9}{5120}a^{22}-\frac{1}{256}a^{21}+\frac{7}{1280}a^{20}-\frac{3}{1280}a^{19}-\frac{1}{40}a^{18}-\frac{21}{1280}a^{17}-\frac{3}{640}a^{16}-\frac{3}{64}a^{15}+\frac{617}{5120}a^{14}-\frac{21}{256}a^{13}+\frac{619}{1280}a^{12}-\frac{105}{256}a^{11}-\frac{623}{5120}a^{10}+\frac{191}{1280}a^{9}-\frac{151}{1280}a^{8}-\frac{99}{640}a^{7}+\frac{17}{40}a^{6}-\frac{15}{32}a^{5}-\frac{27}{80}a^{4}+\frac{11}{40}a^{3}-\frac{2}{5}a^{2}+\frac{2}{5}$, $\frac{1}{10240}a^{27}-\frac{1}{2560}a^{25}+\frac{1}{1280}a^{24}+\frac{9}{10240}a^{23}-\frac{1}{512}a^{22}+\frac{7}{2560}a^{21}+\frac{77}{2560}a^{20}-\frac{1}{80}a^{19}-\frac{21}{2560}a^{18}-\frac{3}{1280}a^{17}-\frac{3}{128}a^{16}+\frac{617}{10240}a^{15}-\frac{21}{512}a^{14}+\frac{619}{2560}a^{13}-\frac{105}{512}a^{12}-\frac{1903}{10240}a^{11}-\frac{1089}{2560}a^{10}-\frac{151}{2560}a^{9}-\frac{379}{1280}a^{8}-\frac{23}{80}a^{7}-\frac{15}{64}a^{6}-\frac{7}{160}a^{5}-\frac{29}{80}a^{4}+\frac{3}{10}a^{3}-\frac{1}{4}a^{2}-\frac{3}{10}a$, $\frac{1}{2293760}a^{28}-\frac{11}{573440}a^{27}+\frac{1}{286720}a^{26}+\frac{89}{143360}a^{25}-\frac{4471}{2293760}a^{24}+\frac{5}{7168}a^{23}+\frac{1847}{286720}a^{22}-\frac{8861}{573440}a^{21}+\frac{4241}{143360}a^{20}+\frac{7031}{573440}a^{19}-\frac{8793}{286720}a^{18}+\frac{113}{8960}a^{17}-\frac{5591}{2293760}a^{16}+\frac{431}{20480}a^{15}-\frac{5419}{57344}a^{14}+\frac{8687}{81920}a^{13}-\frac{124991}{2293760}a^{12}+\frac{12083}{71680}a^{11}-\frac{10929}{35840}a^{10}+\frac{26591}{71680}a^{9}+\frac{419}{2240}a^{8}-\frac{341}{1120}a^{7}-\frac{1033}{4480}a^{6}-\frac{11}{224}a^{5}-\frac{241}{560}a^{4}-\frac{249}{1120}a^{3}-\frac{3}{35}a^{2}-\frac{43}{140}a-\frac{199}{560}$, $\frac{1}{4587520}a^{29}-\frac{17}{573440}a^{27}-\frac{1}{286720}a^{26}-\frac{2743}{4587520}a^{25}+\frac{279}{229376}a^{24}+\frac{273}{81920}a^{23}-\frac{2677}{1146880}a^{22}-\frac{807}{143360}a^{21}-\frac{26073}{1146880}a^{20}+\frac{685}{114688}a^{19}-\frac{303}{28672}a^{18}-\frac{58839}{4587520}a^{17}-\frac{20761}{1146880}a^{16}+\frac{289}{573440}a^{15}+\frac{79073}{1146880}a^{14}+\frac{74749}{917504}a^{13}+\frac{113863}{229376}a^{12}-\frac{7267}{71680}a^{11}+\frac{49711}{143360}a^{10}-\frac{10439}{35840}a^{9}-\frac{11}{448}a^{8}-\frac{681}{8960}a^{7}-\frac{93}{560}a^{6}-\frac{9}{140}a^{5}-\frac{1073}{2240}a^{4}+\frac{191}{560}a^{3}-\frac{67}{280}a^{2}-\frac{151}{1120}a+\frac{107}{280}$, $\frac{1}{211658997760}a^{30}+\frac{8699}{105829498880}a^{29}-\frac{10923}{52914749440}a^{28}-\frac{55279}{1889812480}a^{27}-\frac{1823313}{30236999680}a^{26}-\frac{55653543}{105829498880}a^{25}+\frac{16121887}{52914749440}a^{24}-\frac{53452033}{52914749440}a^{23}-\frac{13262617}{3779624960}a^{22}-\frac{21535173}{3112632320}a^{21}+\frac{86307243}{13228687360}a^{20}-\frac{74738773}{6614343680}a^{19}+\frac{5183540233}{211658997760}a^{18}-\frac{1892741887}{105829498880}a^{17}+\frac{477832011}{52914749440}a^{16}-\frac{14377399}{622526464}a^{15}+\frac{3348841897}{211658997760}a^{14}-\frac{6050051447}{105829498880}a^{13}-\frac{13135916631}{52914749440}a^{12}-\frac{2401429377}{6614343680}a^{11}-\frac{194399689}{3307171840}a^{10}-\frac{31941589}{97269760}a^{9}-\frac{12753041}{59056640}a^{8}-\frac{57050797}{206698240}a^{7}-\frac{1779657}{103349120}a^{6}-\frac{51178157}{103349120}a^{5}+\frac{1845089}{7382080}a^{4}+\frac{1195667}{3691040}a^{3}-\frac{9384191}{51674560}a^{2}+\frac{8731809}{25837280}a-\frac{2329967}{12918640}$, $\frac{1}{423317995520}a^{31}+\frac{3}{54132736}a^{29}+\frac{4203}{26457374720}a^{28}-\frac{12312599}{423317995520}a^{27}-\frac{3208253}{105829498880}a^{26}-\frac{41543407}{105829498880}a^{25}-\frac{156940673}{105829498880}a^{24}+\frac{21846471}{6614343680}a^{23}+\frac{584132363}{105829498880}a^{22}+\frac{93143943}{7559249920}a^{21}-\frac{12326463}{1150320640}a^{20}-\frac{2236834379}{84663599104}a^{19}+\frac{227616731}{21165899776}a^{18}-\frac{901330683}{105829498880}a^{17}-\frac{1979489851}{105829498880}a^{16}+\frac{173101137}{423317995520}a^{15}+\frac{7756402319}{105829498880}a^{14}+\frac{22206553323}{105829498880}a^{13}-\frac{1982054257}{5291474944}a^{12}+\frac{172213997}{3307171840}a^{11}-\frac{1715089}{472453120}a^{10}+\frac{67881517}{413396480}a^{9}+\frac{83802111}{206698240}a^{8}+\frac{47205517}{206698240}a^{7}+\frac{80913831}{206698240}a^{6}+\frac{124569}{875840}a^{5}-\frac{18928597}{51674560}a^{4}+\frac{3985041}{103349120}a^{3}+\frac{10828623}{25837280}a^{2}+\frac{1766133}{3691040}a+\frac{1098613}{6459320}$ 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

$C_{3}\times C_{6}$, which has order $18$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{1593236667}{423317995520} a^{31} - \frac{1971102443}{211658997760} a^{30} + \frac{209909977}{13228687360} a^{29} - \frac{1902864259}{52914749440} a^{28} + \frac{16602969107}{423317995520} a^{27} - \frac{381405713}{12450529280} a^{26} + \frac{333853813}{7559249920} a^{25} - \frac{54278569}{6225264640} a^{24} - \frac{679321413}{52914749440} a^{23} - \frac{16448134747}{105829498880} a^{22} + \frac{8844704229}{26457374720} a^{21} - \frac{11905849059}{26457374720} a^{20} + \frac{385813198163}{423317995520} a^{19} - \frac{24537871407}{30236999680} a^{18} - \frac{6243208907}{52914749440} a^{17} - \frac{19043198463}{105829498880} a^{16} + \frac{73902225651}{423317995520} a^{15} + \frac{12898997451}{42331799552} a^{14} + \frac{129265098757}{26457374720} a^{13} - \frac{534848452331}{52914749440} a^{12} + \frac{66945785597}{6614343680} a^{11} - \frac{12736096263}{826792960} a^{10} + \frac{958595899}{71895040} a^{9} + \frac{232102257}{82679296} a^{8} + \frac{289011}{66080} a^{7} - \frac{946273007}{29528320} a^{6} + \frac{263104557}{6079360} a^{5} - \frac{3052709039}{25837280} a^{4} + \frac{4238847323}{20669824} a^{3} - \frac{1872027245}{10334912} a^{2} + \frac{281827617}{1291864} a - \frac{2113804967}{12918640} \)  (order $24$) 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:   $\frac{3703794851}{211658997760}a^{31}-\frac{1197894699}{26457374720}a^{30}+\frac{402671777}{5291474944}a^{29}-\frac{1826811157}{10582949888}a^{28}+\frac{5852074109}{30236999680}a^{27}-\frac{7750748149}{52914749440}a^{26}+\frac{804498087}{3779624960}a^{25}-\frac{634377313}{13228687360}a^{24}-\frac{45209289}{661434368}a^{23}-\frac{7697360423}{10582949888}a^{22}+\frac{6171099447}{3779624960}a^{21}-\frac{14402412217}{6614343680}a^{20}+\frac{184857860015}{42331799552}a^{19}-\frac{213875554313}{52914749440}a^{18}-\frac{14379874553}{26457374720}a^{17}-\frac{2612486103}{3779624960}a^{16}+\frac{204972245843}{211658997760}a^{15}+\frac{4717923387}{3112632320}a^{14}+\frac{303292561377}{13228687360}a^{13}-\frac{2613800806821}{52914749440}a^{12}+\frac{11630827107}{236226560}a^{11}-\frac{6106489071}{82679296}a^{10}+\frac{4716296847}{71895040}a^{9}+\frac{152085309}{12158720}a^{8}+\frac{5652523}{320960}a^{7}-\frac{7967541093}{51674560}a^{6}+\frac{1097168817}{5167456}a^{5}-\frac{1036063411}{1845520}a^{4}+\frac{51644954237}{51674560}a^{3}-\frac{11379333309}{12918640}a^{2}+\frac{3387108433}{3229660}a-\frac{10443159413}{12918640}$, $\frac{1799647}{188981248}a^{31}-\frac{310992513}{12450529280}a^{30}+\frac{259816279}{6225264640}a^{29}-\frac{5020445981}{52914749440}a^{28}+\frac{284665253}{2645737472}a^{27}-\frac{3400326277}{42331799552}a^{26}+\frac{12481414041}{105829498880}a^{25}-\frac{301016447}{10582949888}a^{24}-\frac{420888395}{10582949888}a^{23}-\frac{10512546859}{26457374720}a^{22}+\frac{47742830757}{52914749440}a^{21}-\frac{3951500387}{3307171840}a^{20}+\frac{15876079967}{6614343680}a^{19}-\frac{95774979077}{42331799552}a^{18}-\frac{31759430351}{105829498880}a^{17}-\frac{19174548567}{52914749440}a^{16}+\frac{6234469639}{10582949888}a^{15}+\frac{188632755543}{211658997760}a^{14}+\frac{263991035493}{21165899776}a^{13}-\frac{1447754989061}{52914749440}a^{12}+\frac{35847690341}{1322868736}a^{11}-\frac{26860607581}{661434368}a^{10}+\frac{528688227}{14379008}a^{9}+\frac{554453469}{82679296}a^{8}+\frac{81473739}{8986880}a^{7}-\frac{8792961057}{103349120}a^{6}+\frac{12103225489}{103349120}a^{5}-\frac{3167776905}{10334912}a^{4}+\frac{14298116207}{25837280}a^{3}-\frac{25164534561}{51674560}a^{2}+\frac{14906636993}{25837280}a-\frac{5861479037}{12918640}$, $\frac{86053767}{60473999360}a^{31}-\frac{936425449}{211658997760}a^{30}+\frac{95797407}{13228687360}a^{29}-\frac{861504649}{52914749440}a^{28}+\frac{1755989797}{84663599104}a^{27}-\frac{3092894923}{211658997760}a^{26}+\frac{1144517489}{52914749440}a^{25}-\frac{171568623}{21165899776}a^{24}-\frac{28365143}{3112632320}a^{23}-\frac{6462130833}{105829498880}a^{22}+\frac{606055461}{3779624960}a^{21}-\frac{161201243}{755924992}a^{20}+\frac{176132612729}{423317995520}a^{19}-\frac{19276787111}{42331799552}a^{18}-\frac{60735001}{3112632320}a^{17}-\frac{390168731}{15118499840}a^{16}+\frac{60024717849}{423317995520}a^{15}+\frac{45766733629}{211658997760}a^{14}+\frac{10142962463}{5291474944}a^{13}-\frac{263356548817}{52914749440}a^{12}+\frac{32937956903}{6614343680}a^{11}-\frac{5991199759}{826792960}a^{10}+\frac{529966057}{71895040}a^{9}+\frac{296118779}{413396480}a^{8}+\frac{1332943}{2246720}a^{7}-\frac{2969730447}{206698240}a^{6}+\frac{436079395}{20669824}a^{5}-\frac{1302397577}{25837280}a^{4}+\frac{10523281317}{103349120}a^{3}-\frac{4719598507}{51674560}a^{2}+\frac{666767179}{6459320}a-\frac{1222491493}{12918640}$, $\frac{1260740877}{423317995520}a^{31}-\frac{771759291}{105829498880}a^{30}+\frac{652754147}{52914749440}a^{29}-\frac{3231703}{115032064}a^{28}+\frac{12900234421}{423317995520}a^{27}-\frac{614315511}{26457374720}a^{26}+\frac{1801431529}{52914749440}a^{25}-\frac{698807429}{105829498880}a^{24}-\frac{275068287}{26457374720}a^{23}-\frac{2600209665}{21165899776}a^{22}+\frac{13976962731}{52914749440}a^{21}-\frac{2303695729}{6614343680}a^{20}+\frac{298937238613}{423317995520}a^{19}-\frac{8341185787}{13228687360}a^{18}-\frac{5805861133}{52914749440}a^{17}-\frac{13573768727}{105829498880}a^{16}+\frac{12799815657}{84663599104}a^{15}+\frac{342967319}{1556316160}a^{14}+\frac{20277898267}{5291474944}a^{13}-\frac{210159930461}{26457374720}a^{12}+\frac{923533807}{118113280}a^{11}-\frac{1950305491}{165358592}a^{10}+\frac{531651849}{51674560}a^{9}+\frac{6827361}{3039680}a^{8}+\frac{342542001}{103349120}a^{7}-\frac{1050992897}{41339648}a^{6}+\frac{44228627}{1291864}a^{5}-\frac{2377634877}{25837280}a^{4}+\frac{16528970589}{103349120}a^{3}-\frac{902819441}{6459320}a^{2}+\frac{1080898583}{6459320}a-\frac{160747935}{1291864}$, $\frac{639327637}{52914749440}a^{31}-\frac{719748529}{21165899776}a^{30}+\frac{1480018437}{26457374720}a^{29}-\frac{6709201987}{52914749440}a^{28}+\frac{7979155817}{52914749440}a^{27}-\frac{11500911637}{105829498880}a^{26}+\frac{2123229043}{13228687360}a^{25}-\frac{2505789399}{52914749440}a^{24}-\frac{1600015567}{26457374720}a^{23}-\frac{962549259}{1889812480}a^{22}+\frac{32559058039}{26457374720}a^{21}-\frac{4288025391}{2645737472}a^{20}+\frac{33981766085}{10582949888}a^{19}-\frac{340862707893}{105829498880}a^{18}-\frac{4335471839}{13228687360}a^{17}-\frac{3542832187}{10582949888}a^{16}+\frac{9964222559}{10582949888}a^{15}+\frac{8253618627}{6225264640}a^{14}+\frac{105911116971}{6614343680}a^{13}-\frac{1986261188911}{52914749440}a^{12}+\frac{122852880581}{3307171840}a^{11}-\frac{22625033019}{413396480}a^{10}+\frac{85976117699}{1653585920}a^{9}+\frac{18675399}{2431744}a^{8}+\frac{227479991}{25837280}a^{7}-\frac{11742605717}{103349120}a^{6}+\frac{1659592653}{10334912}a^{5}-\frac{5204449963}{12918640}a^{4}+\frac{19633104401}{25837280}a^{3}-\frac{2476188067}{3691040}a^{2}+\frac{1258536387}{1614830}a-\frac{8387150967}{12918640}$, $\frac{399914983}{15118499840}a^{31}-\frac{3694808717}{52914749440}a^{30}+\frac{209203251}{1793720320}a^{29}-\frac{175036579}{661434368}a^{28}+\frac{31854563761}{105829498880}a^{27}-\frac{2379296011}{10582949888}a^{26}+\frac{34767732689}{105829498880}a^{25}-\frac{1060504167}{13228687360}a^{24}-\frac{365718797}{3307171840}a^{23}-\frac{14590940907}{13228687360}a^{22}+\frac{16663064469}{6614343680}a^{21}-\frac{88512072877}{26457374720}a^{20}+\frac{708457655681}{105829498880}a^{19}-\frac{334667639447}{52914749440}a^{18}-\frac{84972968991}{105829498880}a^{17}-\frac{279550197}{287580160}a^{16}+\frac{7456724039}{4601282560}a^{15}+\frac{7602764171}{3112632320}a^{14}+\frac{3673729968577}{105829498880}a^{13}-\frac{2020223687509}{26457374720}a^{12}+\frac{125548406677}{1653585920}a^{11}-\frac{375031901609}{3307171840}a^{10}+\frac{84808255551}{826792960}a^{9}+\frac{11160357}{607936}a^{8}+\frac{1028267493}{41339648}a^{7}-\frac{3060075623}{12918640}a^{6}+\frac{8468943359}{25837280}a^{5}-\frac{8837806329}{10334912}a^{4}+\frac{39919379779}{25837280}a^{3}-\frac{3519622697}{2583728}a^{2}+\frac{41643826713}{25837280}a-\frac{8193613333}{6459320}$, $\frac{1776376131}{211658997760}a^{31}-\frac{894936793}{42331799552}a^{30}+\frac{1892063689}{52914749440}a^{29}-\frac{4293287647}{52914749440}a^{28}+\frac{2693547501}{30236999680}a^{27}-\frac{14524115553}{211658997760}a^{26}+\frac{2618396861}{26457374720}a^{25}-\frac{74621069}{3779624960}a^{24}-\frac{92649839}{3112632320}a^{23}-\frac{3676351307}{10582949888}a^{22}+\frac{40255472063}{52914749440}a^{21}-\frac{3850440209}{3779624960}a^{20}+\frac{433891325131}{211658997760}a^{19}-\frac{390281104337}{211658997760}a^{18}-\frac{425146223}{1556316160}a^{17}-\frac{2385585957}{6614343680}a^{16}+\frac{12198808969}{30236999680}a^{15}+\frac{141424346759}{211658997760}a^{14}+\frac{145095091889}{13228687360}a^{13}-\frac{1214229803169}{52914749440}a^{12}+\frac{151573864049}{6614343680}a^{11}-\frac{510629373}{14764160}a^{10}+\frac{49612214213}{1653585920}a^{9}+\frac{2575741997}{413396480}a^{8}+\frac{951084513}{103349120}a^{7}-\frac{16728201}{230690}a^{6}+\frac{10200467717}{103349120}a^{5}-\frac{490671099}{1845520}a^{4}+\frac{4798959293}{10334912}a^{3}-\frac{21097192657}{51674560}a^{2}+\frac{1583790969}{3229660}a-\frac{206707703}{561680}$, $\frac{2190043}{661434368}a^{31}-\frac{168678915}{21165899776}a^{30}+\frac{703213023}{52914749440}a^{29}-\frac{1628131909}{52914749440}a^{28}+\frac{434619471}{13228687360}a^{27}-\frac{155656391}{6225264640}a^{26}+\frac{1991733901}{52914749440}a^{25}-\frac{21593897}{3112632320}a^{24}-\frac{62413213}{5291474944}a^{23}-\frac{1807213973}{13228687360}a^{22}+\frac{1088960663}{3779624960}a^{21}-\frac{493235817}{1322868736}a^{20}+\frac{10245871767}{13228687360}a^{19}-\frac{14490849003}{21165899776}a^{18}-\frac{1008937997}{7559249920}a^{17}-\frac{8482514049}{52914749440}a^{16}+\frac{142150735}{755924992}a^{15}+\frac{29791850313}{105829498880}a^{14}+\frac{9718691959}{2300641280}a^{13}-\frac{459585140717}{52914749440}a^{12}+\frac{27683039159}{3307171840}a^{11}-\frac{21302270009}{1653585920}a^{10}+\frac{18680036749}{1653585920}a^{9}+\frac{540559763}{206698240}a^{8}+\frac{22121691}{6079360}a^{7}-\frac{582560397}{20669824}a^{6}+\frac{110516447}{3039680}a^{5}-\frac{2590902249}{25837280}a^{4}+\frac{4511328621}{25837280}a^{3}-\frac{3883244543}{25837280}a^{2}+\frac{2355820153}{12918640}a-\frac{1759661413}{12918640}$, $\frac{1063437}{1314652160}a^{31}-\frac{497513111}{211658997760}a^{30}+\frac{59445701}{15118499840}a^{29}-\frac{94833143}{10582949888}a^{28}+\frac{2278416069}{211658997760}a^{27}-\frac{1653196723}{211658997760}a^{26}+\frac{1174306353}{105829498880}a^{25}-\frac{87908847}{26457374720}a^{24}-\frac{2844493}{622526464}a^{23}-\frac{1773262331}{52914749440}a^{22}+\frac{650914667}{7559249920}a^{21}-\frac{1540738253}{13228687360}a^{20}+\frac{9636322897}{42331799552}a^{19}-\frac{9772349255}{42331799552}a^{18}-\frac{118270063}{6225264640}a^{17}-\frac{120922021}{13228687360}a^{16}+\frac{2191337149}{42331799552}a^{15}+\frac{19681790861}{211658997760}a^{14}+\frac{116993343637}{105829498880}a^{13}-\frac{139729652283}{52914749440}a^{12}+\frac{3559101095}{1322868736}a^{11}-\frac{1836213479}{472453120}a^{10}+\frac{6040257211}{1653585920}a^{9}+\frac{40080443}{82679296}a^{8}+\frac{69154083}{206698240}a^{7}-\frac{49690427}{6459320}a^{6}+\frac{1190016963}{103349120}a^{5}-\frac{1471731077}{51674560}a^{4}+\frac{553463859}{10334912}a^{3}-\frac{496119615}{10334912}a^{2}+\frac{1431524013}{25837280}a-\frac{117432207}{2583728}$, $\frac{417517259}{84663599104}a^{31}-\frac{536856643}{42331799552}a^{30}+\frac{1114154519}{52914749440}a^{29}-\frac{27768003}{575160320}a^{28}+\frac{22656073551}{423317995520}a^{27}-\frac{8418290589}{211658997760}a^{26}+\frac{1614145153}{26457374720}a^{25}-\frac{1479876479}{105829498880}a^{24}-\frac{1032661177}{52914749440}a^{23}-\frac{22005329343}{105829498880}a^{22}+\frac{12027354447}{26457374720}a^{21}-\frac{3133111649}{5291474944}a^{20}+\frac{517195091919}{423317995520}a^{19}-\frac{14064494869}{12450529280}a^{18}-\frac{2435619409}{13228687360}a^{17}-\frac{3713280059}{15118499840}a^{16}+\frac{28838976387}{84663599104}a^{15}+\frac{20859684047}{42331799552}a^{14}+\frac{20016009673}{3112632320}a^{13}-\frac{91295578081}{6614343680}a^{12}+\frac{12588934551}{944906240}a^{11}-\frac{34161049063}{1653585920}a^{10}+\frac{15486271219}{826792960}a^{9}+\frac{220971771}{59056640}a^{8}+\frac{27721317}{5167456}a^{7}-\frac{9162235507}{206698240}a^{6}+\frac{5986468193}{103349120}a^{5}-\frac{2003380237}{12918640}a^{4}+\frac{4132781393}{14764160}a^{3}-\frac{12501986173}{51674560}a^{2}+\frac{3791477889}{12918640}a-\frac{748865619}{3229660}$, $\frac{700562339}{423317995520}a^{31}-\frac{2905251}{661434368}a^{30}+\frac{787772439}{105829498880}a^{29}-\frac{884997991}{52914749440}a^{28}+\frac{8047103291}{423317995520}a^{27}-\frac{219073153}{15118499840}a^{26}+\frac{2139586767}{105829498880}a^{25}-\frac{493922101}{105829498880}a^{24}-\frac{2522713}{389079040}a^{23}-\frac{1438421203}{21165899776}a^{22}+\frac{8390327879}{52914749440}a^{21}-\frac{5779170559}{26457374720}a^{20}+\frac{178771432411}{423317995520}a^{19}-\frac{1807082141}{4601282560}a^{18}-\frac{246647013}{6225264640}a^{17}-\frac{962386951}{21165899776}a^{16}+\frac{30732789651}{423317995520}a^{15}+\frac{1809864907}{15118499840}a^{14}+\frac{233101475989}{105829498880}a^{13}-\frac{50475879731}{10582949888}a^{12}+\frac{2344389007}{472453120}a^{11}-\frac{23717030049}{3307171840}a^{10}+\frac{10244472593}{1653585920}a^{9}+\frac{110347851}{103349120}a^{8}+\frac{60381689}{41339648}a^{7}-\frac{428969259}{29528320}a^{6}+\frac{156116233}{7382080}a^{5}-\frac{2838429023}{51674560}a^{4}+\frac{9943877207}{103349120}a^{3}-\frac{450941343}{5167456}a^{2}+\frac{2648757341}{25837280}a-\frac{194903119}{2583728}$, $\frac{494267359}{84663599104}a^{31}-\frac{11903841}{755924992}a^{30}+\frac{78468475}{3023699968}a^{29}-\frac{3131919657}{52914749440}a^{28}+\frac{28898665011}{423317995520}a^{27}-\frac{1060489839}{21165899776}a^{26}+\frac{7908825737}{105829498880}a^{25}-\frac{419297413}{21165899776}a^{24}-\frac{173399327}{6614343680}a^{23}-\frac{25858158911}{105829498880}a^{22}+\frac{6001343955}{10582949888}a^{21}-\frac{19732764561}{26457374720}a^{20}+\frac{634060102099}{423317995520}a^{19}-\frac{153007418047}{105829498880}a^{18}-\frac{306416353}{1793720320}a^{17}-\frac{22514388439}{105829498880}a^{16}+\frac{24608684701}{60473999360}a^{15}+\frac{62508178129}{105829498880}a^{14}+\frac{808585823999}{105829498880}a^{13}-\frac{912211195877}{52914749440}a^{12}+\frac{56222763391}{3307171840}a^{11}-\frac{16847200127}{661434368}a^{10}+\frac{38891960887}{1653585920}a^{9}+\frac{116515063}{29528320}a^{8}+\frac{1086495133}{206698240}a^{7}-\frac{645494421}{12158720}a^{6}+\frac{541546651}{7382080}a^{5}-\frac{9805547121}{51674560}a^{4}+\frac{2631595}{7552}a^{3}-\frac{7937915959}{25837280}a^{2}+\frac{1339038641}{3691040}a-\frac{3748413089}{12918640}$, $\frac{2505886721}{211658997760}a^{31}-\frac{3201173643}{105829498880}a^{30}+\frac{95817523}{1889812480}a^{29}-\frac{6134159237}{52914749440}a^{28}+\frac{5452846277}{42331799552}a^{27}-\frac{1480618507}{15118499840}a^{26}+\frac{3824853969}{26457374720}a^{25}-\frac{120759117}{3779624960}a^{24}-\frac{1228339629}{26457374720}a^{23}-\frac{26076485273}{52914749440}a^{22}+\frac{2055751621}{1889812480}a^{21}-\frac{19077546491}{13228687360}a^{20}+\frac{36592599529}{12450529280}a^{19}-\frac{1776741253}{657326080}a^{18}-\frac{10123241913}{26457374720}a^{17}-\frac{710792687}{1322868736}a^{16}+\frac{28520696069}{42331799552}a^{15}+\frac{119114997067}{105829498880}a^{14}+\frac{51080476149}{3307171840}a^{13}-\frac{1747041440013}{52914749440}a^{12}+\frac{107883068779}{3307171840}a^{11}-\frac{41123342663}{826792960}a^{10}+\frac{14663472421}{330717184}a^{9}+\frac{1787535811}{206698240}a^{8}+\frac{1282364047}{103349120}a^{7}-\frac{672307637}{6459320}a^{6}+\frac{7225566981}{51674560}a^{5}-\frac{142869899}{379960}a^{4}+\frac{34611021371}{51674560}a^{3}-\frac{15170285829}{25837280}a^{2}+\frac{570005958}{807415}a-\frac{1005340939}{1845520}$, $\frac{117786357}{13228687360}a^{31}-\frac{984237669}{42331799552}a^{30}+\frac{4118442623}{105829498880}a^{29}-\frac{291746051}{3307171840}a^{28}+\frac{263356753}{2645737472}a^{27}-\frac{15738735713}{211658997760}a^{26}+\frac{2310541837}{21165899776}a^{25}-\frac{2337413}{89686016}a^{24}-\frac{1903870383}{52914749440}a^{23}-\frac{9832661007}{26457374720}a^{22}+\frac{6337113583}{7559249920}a^{21}-\frac{1837790509}{1653585920}a^{20}+\frac{7374886821}{3307171840}a^{19}-\frac{441027658017}{211658997760}a^{18}-\frac{30221241991}{105829498880}a^{17}-\frac{1845178637}{5291474944}a^{16}+\frac{28471685751}{52914749440}a^{15}+\frac{9699236639}{12450529280}a^{14}+\frac{1234154731689}{105829498880}a^{13}-\frac{167650304739}{6614343680}a^{12}+\frac{166459768201}{6614343680}a^{11}-\frac{124678701269}{3307171840}a^{10}+\frac{1999849409}{59056640}a^{9}+\frac{30691273}{4863488}a^{8}+\frac{365694519}{41339648}a^{7}-\frac{146048711}{1845520}a^{6}+\frac{2257931237}{20669824}a^{5}-\frac{14755757021}{51674560}a^{4}+\frac{287719989}{561680}a^{3}-\frac{666686531}{1476416}a^{2}+\frac{13844272017}{25837280}a-\frac{135294589}{322966}$, $\frac{483807719}{26457374720}a^{31}-\frac{1427619533}{30236999680}a^{30}+\frac{8402159009}{105829498880}a^{29}-\frac{9549515071}{52914749440}a^{28}+\frac{5344723457}{26457374720}a^{27}-\frac{6480849527}{42331799552}a^{26}+\frac{3372105217}{15118499840}a^{25}-\frac{2636904937}{52914749440}a^{24}-\frac{3800409549}{52914749440}a^{23}-\frac{20137146313}{26457374720}a^{22}+\frac{90057722599}{52914749440}a^{21}-\frac{15010958839}{6614343680}a^{20}+\frac{120933834527}{26457374720}a^{19}-\frac{893832135891}{211658997760}a^{18}-\frac{60861425713}{105829498880}a^{17}-\frac{39550160941}{52914749440}a^{16}+\frac{10690527355}{10582949888}a^{15}+\frac{344719669277}{211658997760}a^{14}+\frac{22090420733}{920256512}a^{13}-\frac{2729768542111}{52914749440}a^{12}+\frac{339357499107}{6614343680}a^{11}-\frac{255704443283}{3307171840}a^{10}+\frac{113688221931}{1653585920}a^{9}+\frac{1095696643}{82679296}a^{8}+\frac{3835107287}{206698240}a^{7}-\frac{16671725209}{103349120}a^{6}+\frac{22832359567}{103349120}a^{5}-\frac{30316076391}{51674560}a^{4}+\frac{26987527729}{25837280}a^{3}-\frac{2792630651}{3039680}a^{2}+\frac{47631381}{43424}a-\frac{312653581}{369104}$ 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:  \( 137041291644.51082 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 137041291644.51082 \cdot 18}{24\cdot\sqrt{1033818800357475102568684143000990752577071087616}}\cr\approx \mathstrut & 0.596442233123376 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 + 8*x^30 - 16*x^29 + 25*x^28 - 24*x^27 + 24*x^26 - 20*x^25 - 36*x^23 + 152*x^22 - 256*x^21 + 425*x^20 - 584*x^19 + 296*x^18 + 4*x^17 + 113*x^16 + 8*x^15 + 1184*x^14 - 4672*x^13 + 6800*x^12 - 8192*x^11 + 9728*x^10 - 4608*x^9 - 10240*x^7 + 24576*x^6 - 49152*x^5 + 102400*x^4 - 131072*x^3 + 131072*x^2 - 131072*x + 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^31 + 8*x^30 - 16*x^29 + 25*x^28 - 24*x^27 + 24*x^26 - 20*x^25 - 36*x^23 + 152*x^22 - 256*x^21 + 425*x^20 - 584*x^19 + 296*x^18 + 4*x^17 + 113*x^16 + 8*x^15 + 1184*x^14 - 4672*x^13 + 6800*x^12 - 8192*x^11 + 9728*x^10 - 4608*x^9 - 10240*x^7 + 24576*x^6 - 49152*x^5 + 102400*x^4 - 131072*x^3 + 131072*x^2 - 131072*x + 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^31 + 8*x^30 - 16*x^29 + 25*x^28 - 24*x^27 + 24*x^26 - 20*x^25 - 36*x^23 + 152*x^22 - 256*x^21 + 425*x^20 - 584*x^19 + 296*x^18 + 4*x^17 + 113*x^16 + 8*x^15 + 1184*x^14 - 4672*x^13 + 6800*x^12 - 8192*x^11 + 9728*x^10 - 4608*x^9 - 10240*x^7 + 24576*x^6 - 49152*x^5 + 102400*x^4 - 131072*x^3 + 131072*x^2 - 131072*x + 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^31 + 8*x^30 - 16*x^29 + 25*x^28 - 24*x^27 + 24*x^26 - 20*x^25 - 36*x^23 + 152*x^22 - 256*x^21 + 425*x^20 - 584*x^19 + 296*x^18 + 4*x^17 + 113*x^16 + 8*x^15 + 1184*x^14 - 4672*x^13 + 6800*x^12 - 8192*x^11 + 9728*x^10 - 4608*x^9 - 10240*x^7 + 24576*x^6 - 49152*x^5 + 102400*x^4 - 131072*x^3 + 131072*x^2 - 131072*x + 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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-1}) \), 4.4.94464.1, 4.0.10496.2, 4.4.2624.1, 4.0.23616.1, \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), 8.8.63021846528.1, 8.0.778047488.1, 8.8.12448759808.1, 8.0.1008349544448.1, \(\Q(\zeta_{24})\), 8.0.8923447296.2, 8.0.557715456.2, 8.8.8923447296.1, 8.0.8923447296.11, 8.0.8923447296.3, 8.0.110166016.2, 16.0.79627911644489711616.1, 16.0.3971753139798785654784.1, 16.0.1016768803788489127624704.1, 16.16.1016768803788489127624704.1, 16.0.1016768803788489127624704.3, 16.0.154971620757276196864.1, 16.0.1016768803788489127624704.2

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 R ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{8}$ ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{16}$ ${\href{/padicField/19.8.0.1}{8} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.8.0.1}{8} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{8}$ R ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
\(3\) 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}$
\(41\) Copy content Toggle raw display 41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(113\) Copy content Toggle raw display 113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
113.4.2.1$x^{4} + 18960 x^{3} + 90817911 x^{2} + 8982404280 x + 374946100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.4.2.1$x^{4} + 18960 x^{3} + 90817911 x^{2} + 8982404280 x + 374946100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$