Properties

Label 32.0.385...576.1
Degree $32$
Signature $[0, 16]$
Discriminant $3.850\times 10^{50}$
Root discriminant \(38.09\)
Ramified primes $2,17,997$
Class number $40$ (GRH)
Class group [2, 20] (GRH)
Galois group $C_2^6:S_4$ (as 32T96908)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 40*x^24 + 16*x^20 + 784*x^16 + 256*x^12 + 10240*x^8 + 65536)
 
gp: K = bnfinit(y^32 + 40*y^24 + 16*y^20 + 784*y^16 + 256*y^12 + 10240*y^8 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 40*x^24 + 16*x^20 + 784*x^16 + 256*x^12 + 10240*x^8 + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 40*x^24 + 16*x^20 + 784*x^16 + 256*x^12 + 10240*x^8 + 65536)
 

\( x^{32} + 40x^{24} + 16x^{20} + 784x^{16} + 256x^{12} + 10240x^{8} + 65536 \) 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:   \(385049567410499609358247677435067162906957579288576\) \(\medspace = 2^{72}\cdot 17^{4}\cdot 997^{8}\) 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:  \(38.09\)
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:  $2^{55/24}17^{1/2}997^{1/2}\approx 637.429922686193$
Ramified primes:   \(2\), \(17\), \(997\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{8}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{16}a^{12}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{13}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{16}a^{14}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{15}-\frac{1}{16}a^{11}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{32}a^{16}-\frac{1}{8}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{17}-\frac{1}{8}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{64}a^{18}-\frac{1}{32}a^{14}-\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{64}a^{19}+\frac{1}{4}a^{3}$, $\frac{1}{128}a^{20}-\frac{1}{8}a^{8}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{256}a^{21}-\frac{1}{32}a^{13}-\frac{1}{16}a^{9}-\frac{3}{16}a^{5}-\frac{1}{2}a$, $\frac{1}{512}a^{22}-\frac{1}{64}a^{14}-\frac{1}{16}a^{11}-\frac{1}{32}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{3}{32}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{512}a^{23}-\frac{1}{64}a^{15}-\frac{1}{32}a^{11}-\frac{1}{8}a^{9}-\frac{3}{32}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{2048}a^{24}-\frac{3}{256}a^{16}-\frac{1}{32}a^{14}+\frac{1}{128}a^{12}-\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{8}a^{9}+\frac{1}{128}a^{8}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2048}a^{25}-\frac{3}{256}a^{17}+\frac{1}{128}a^{13}-\frac{15}{128}a^{9}+\frac{1}{8}a^{5}-\frac{1}{2}a$, $\frac{1}{4096}a^{26}-\frac{3}{512}a^{18}+\frac{1}{256}a^{14}-\frac{15}{256}a^{10}+\frac{1}{16}a^{6}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8192}a^{27}-\frac{3}{1024}a^{19}+\frac{1}{512}a^{15}+\frac{17}{512}a^{11}-\frac{1}{8}a^{9}-\frac{3}{32}a^{7}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{638976}a^{28}+\frac{7}{79872}a^{24}-\frac{227}{79872}a^{20}-\frac{115}{39936}a^{16}+\frac{1097}{39936}a^{12}-\frac{1}{8}a^{9}+\frac{193}{4992}a^{8}+\frac{35}{624}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{11}{78}$, $\frac{1}{1277952}a^{29}+\frac{7}{159744}a^{25}-\frac{1}{1024}a^{23}-\frac{227}{159744}a^{21}-\frac{1}{128}a^{19}+\frac{1133}{79872}a^{17}-\frac{1}{128}a^{15}+\frac{1097}{79872}a^{13}-\frac{1}{64}a^{11}-\frac{431}{9984}a^{9}-\frac{1}{64}a^{7}-\frac{277}{1248}a^{5}-\frac{1}{8}a^{3}-\frac{67}{156}a$, $\frac{1}{2555904}a^{30}+\frac{7}{319488}a^{26}-\frac{1}{4096}a^{25}-\frac{1}{1024}a^{23}-\frac{227}{319488}a^{22}-\frac{1}{256}a^{20}+\frac{1133}{159744}a^{18}+\frac{3}{512}a^{17}-\frac{1}{128}a^{15}+\frac{1097}{159744}a^{14}-\frac{1}{256}a^{13}-\frac{1}{32}a^{12}+\frac{3}{64}a^{11}-\frac{431}{19968}a^{10}-\frac{17}{256}a^{9}+\frac{1}{16}a^{8}+\frac{7}{64}a^{7}+\frac{35}{2496}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}-\frac{1}{2}a^{3}+\frac{11}{312}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{2555904}a^{31}+\frac{7}{319488}a^{27}-\frac{227}{319488}a^{23}+\frac{1133}{159744}a^{19}+\frac{1097}{159744}a^{15}-\frac{431}{19968}a^{11}+\frac{35}{2496}a^{7}-\frac{145}{312}a^{3}$ 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:  $3$

Class group and class number

$C_{2}\times C_{20}$, which has order $40$ (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:   \( -\frac{53}{638976} a^{29} + \frac{19}{79872} a^{25} - \frac{137}{79872} a^{21} + \frac{167}{39936} a^{17} - \frac{1357}{39936} a^{13} + \frac{457}{4992} a^{9} - \frac{16}{39} a^{5} + \frac{40}{39} a \)  (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:   $\frac{9}{212992}a^{30}-\frac{53}{638976}a^{29}+\frac{9}{53248}a^{26}+\frac{19}{79872}a^{25}+\frac{37}{26624}a^{22}-\frac{137}{79872}a^{21}+\frac{83}{13312}a^{18}+\frac{167}{39936}a^{17}+\frac{253}{13312}a^{14}-\frac{1357}{39936}a^{13}+\frac{237}{3328}a^{10}+\frac{457}{4992}a^{9}+\frac{21}{104}a^{6}-\frac{16}{39}a^{5}+\frac{29}{52}a^{2}+\frac{40}{39}a+1$, $\frac{31}{1277952}a^{31}-\frac{11}{851968}a^{30}-\frac{49}{1277952}a^{29}-\frac{1}{16384}a^{28}-\frac{17}{159744}a^{27}+\frac{1}{106496}a^{26}+\frac{1}{19968}a^{25}+\frac{139}{159744}a^{23}+\frac{1}{106496}a^{22}-\frac{109}{159744}a^{21}-\frac{5}{2048}a^{20}-\frac{133}{79872}a^{19}-\frac{87}{53248}a^{18}+\frac{175}{79872}a^{17}-\frac{1}{1024}a^{16}+\frac{935}{79872}a^{15}+\frac{205}{53248}a^{14}-\frac{1025}{79872}a^{13}-\frac{49}{1024}a^{12}-\frac{179}{9984}a^{11}-\frac{173}{6656}a^{10}+\frac{439}{4992}a^{9}-\frac{1}{64}a^{8}+\frac{55}{624}a^{7}-\frac{21}{832}a^{6}-\frac{233}{1248}a^{5}-\frac{3}{8}a^{4}-\frac{59}{312}a^{3}-\frac{17}{104}a^{2}+\frac{31}{39}a-\frac{1}{2}$, $\frac{67}{1277952}a^{31}-\frac{85}{851968}a^{30}+\frac{5}{39936}a^{29}-\frac{1}{4096}a^{28}-\frac{19}{79872}a^{27}+\frac{29}{106496}a^{26}-\frac{11}{159744}a^{25}+\frac{235}{159744}a^{23}-\frac{257}{106496}a^{22}+\frac{31}{9984}a^{21}-\frac{3}{512}a^{20}-\frac{373}{79872}a^{19}+\frac{207}{53248}a^{18}+\frac{41}{19968}a^{17}-\frac{1}{256}a^{16}+\frac{1427}{79872}a^{15}-\frac{1725}{53248}a^{14}+\frac{529}{9984}a^{13}-\frac{25}{256}a^{12}-\frac{209}{2496}a^{11}+\frac{651}{6656}a^{10}-\frac{203}{9984}a^{9}+\frac{595}{2496}a^{7}-\frac{323}{832}a^{6}+\frac{191}{312}a^{5}-\frac{15}{16}a^{4}-\frac{121}{156}a^{3}+\frac{105}{104}a^{2}+\frac{5}{156}a-\frac{1}{2}$, $\frac{49}{1277952}a^{30}-\frac{11}{638976}a^{28}+\frac{31}{159744}a^{26}-\frac{19}{39936}a^{24}+\frac{109}{159744}a^{22}+\frac{1}{79872}a^{20}+\frac{605}{79872}a^{18}-\frac{451}{39936}a^{16}+\frac{1337}{79872}a^{14}+\frac{725}{39936}a^{12}+\frac{1033}{9984}a^{10}-\frac{209}{1248}a^{8}+\frac{311}{1248}a^{6}+\frac{161}{624}a^{4}+\frac{55}{78}a^{2}-\frac{41}{39}$, $\frac{5}{1277952}a^{30}-\frac{11}{638976}a^{28}-\frac{1}{39936}a^{26}-\frac{19}{39936}a^{24}+\frac{113}{159744}a^{22}+\frac{1}{79872}a^{20}-\frac{107}{79872}a^{18}-\frac{451}{39936}a^{16}+\frac{181}{79872}a^{14}+\frac{725}{39936}a^{12}-\frac{161}{4992}a^{10}-\frac{209}{1248}a^{8}-\frac{59}{1248}a^{6}+\frac{161}{624}a^{4}-\frac{23}{156}a^{2}-\frac{80}{39}$, $\frac{67}{1277952}a^{31}+\frac{193}{2555904}a^{30}-\frac{9}{212992}a^{29}+\frac{1}{4096}a^{28}+\frac{41}{319488}a^{27}+\frac{25}{319488}a^{26}-\frac{9}{53248}a^{25}+\frac{235}{159744}a^{23}+\frac{493}{319488}a^{22}-\frac{37}{26624}a^{21}+\frac{3}{512}a^{20}+\frac{173}{79872}a^{19}-\frac{43}{159744}a^{18}-\frac{83}{13312}a^{17}+\frac{1}{256}a^{16}+\frac{1895}{79872}a^{15}+\frac{3929}{159744}a^{14}-\frac{253}{13312}a^{13}+\frac{25}{256}a^{12}+\frac{317}{19968}a^{11}-\frac{269}{19968}a^{10}-\frac{237}{3328}a^{9}+\frac{517}{2496}a^{7}+\frac{593}{2496}a^{6}-\frac{21}{104}a^{5}+\frac{15}{16}a^{4}-\frac{47}{312}a^{3}-\frac{61}{312}a^{2}-\frac{55}{52}a-\frac{1}{2}$, $\frac{31}{851968}a^{31}+\frac{49}{851968}a^{30}-\frac{19}{159744}a^{29}+\frac{31}{638976}a^{28}+\frac{9}{106496}a^{27}-\frac{47}{106496}a^{26}-\frac{11}{159744}a^{25}+\frac{61}{79872}a^{24}+\frac{139}{106496}a^{23}+\frac{109}{106496}a^{22}-\frac{55}{19968}a^{21}+\frac{139}{79872}a^{20}+\frac{179}{53248}a^{19}-\frac{539}{53248}a^{18}-\frac{37}{19968}a^{17}+\frac{803}{39936}a^{16}+\frac{1143}{53248}a^{15}+\frac{713}{53248}a^{14}-\frac{223}{4992}a^{13}+\frac{1559}{39936}a^{12}+\frac{263}{6656}a^{11}-\frac{1125}{6656}a^{10}-\frac{203}{9984}a^{9}+\frac{1771}{4992}a^{8}+\frac{107}{416}a^{7}+\frac{155}{832}a^{6}-\frac{359}{624}a^{5}+\frac{133}{312}a^{4}+\frac{29}{104}a^{3}-\frac{163}{104}a^{2}+\frac{5}{156}a+\frac{151}{39}$, $\frac{31}{851968}a^{31}+\frac{5}{196608}a^{30}+\frac{9}{212992}a^{29}+\frac{25}{319488}a^{28}+\frac{9}{106496}a^{27}+\frac{5}{24576}a^{26}+\frac{9}{53248}a^{25}+\frac{19}{39936}a^{24}+\frac{139}{106496}a^{23}+\frac{17}{24576}a^{22}+\frac{37}{26624}a^{21}+\frac{97}{39936}a^{20}+\frac{179}{53248}a^{19}+\frac{73}{12288}a^{18}+\frac{83}{13312}a^{17}+\frac{245}{19968}a^{16}+\frac{1143}{53248}a^{15}+\frac{253}{12288}a^{14}+\frac{253}{13312}a^{13}+\frac{593}{19968}a^{12}+\frac{263}{6656}a^{11}+\frac{119}{1536}a^{10}+\frac{237}{3328}a^{9}+\frac{457}{2496}a^{8}+\frac{107}{416}a^{7}+\frac{43}{192}a^{6}+\frac{21}{104}a^{5}+\frac{73}{624}a^{4}+\frac{29}{104}a^{3}+\frac{13}{24}a^{2}+\frac{3}{52}a+\frac{121}{78}$, $\frac{31}{851968}a^{31}-\frac{5}{196608}a^{30}-\frac{9}{212992}a^{29}-\frac{25}{319488}a^{28}+\frac{9}{106496}a^{27}-\frac{5}{24576}a^{26}-\frac{9}{53248}a^{25}-\frac{19}{39936}a^{24}+\frac{139}{106496}a^{23}-\frac{17}{24576}a^{22}-\frac{37}{26624}a^{21}-\frac{97}{39936}a^{20}+\frac{179}{53248}a^{19}-\frac{73}{12288}a^{18}-\frac{83}{13312}a^{17}-\frac{245}{19968}a^{16}+\frac{1143}{53248}a^{15}-\frac{253}{12288}a^{14}-\frac{253}{13312}a^{13}-\frac{593}{19968}a^{12}+\frac{263}{6656}a^{11}-\frac{119}{1536}a^{10}-\frac{237}{3328}a^{9}-\frac{457}{2496}a^{8}+\frac{107}{416}a^{7}-\frac{43}{192}a^{6}-\frac{21}{104}a^{5}-\frac{73}{624}a^{4}+\frac{29}{104}a^{3}-\frac{13}{24}a^{2}-\frac{3}{52}a-\frac{121}{78}$, $\frac{7}{851968}a^{31}-\frac{11}{851968}a^{30}-\frac{49}{1277952}a^{29}+\frac{1}{16384}a^{28}+\frac{23}{106496}a^{27}+\frac{1}{106496}a^{26}+\frac{1}{19968}a^{25}+\frac{75}{106496}a^{23}+\frac{1}{106496}a^{22}-\frac{109}{159744}a^{21}+\frac{5}{2048}a^{20}+\frac{339}{53248}a^{19}-\frac{87}{53248}a^{18}+\frac{175}{79872}a^{17}+\frac{1}{1024}a^{16}+\frac{815}{53248}a^{15}+\frac{205}{53248}a^{14}-\frac{1025}{79872}a^{13}+\frac{49}{1024}a^{12}+\frac{701}{6656}a^{11}-\frac{173}{6656}a^{10}+\frac{439}{4992}a^{9}+\frac{1}{64}a^{8}+\frac{89}{832}a^{7}-\frac{21}{832}a^{6}-\frac{233}{1248}a^{5}+\frac{3}{8}a^{4}+\frac{45}{52}a^{3}-\frac{17}{104}a^{2}+\frac{31}{39}a-\frac{1}{2}$, $\frac{7}{851968}a^{31}+\frac{11}{851968}a^{30}-\frac{49}{1277952}a^{29}-\frac{1}{16384}a^{28}+\frac{23}{106496}a^{27}-\frac{1}{106496}a^{26}+\frac{1}{19968}a^{25}+\frac{75}{106496}a^{23}-\frac{1}{106496}a^{22}-\frac{109}{159744}a^{21}-\frac{5}{2048}a^{20}+\frac{339}{53248}a^{19}+\frac{87}{53248}a^{18}+\frac{175}{79872}a^{17}-\frac{1}{1024}a^{16}+\frac{815}{53248}a^{15}-\frac{205}{53248}a^{14}-\frac{1025}{79872}a^{13}-\frac{49}{1024}a^{12}+\frac{701}{6656}a^{11}+\frac{173}{6656}a^{10}+\frac{439}{4992}a^{9}-\frac{1}{64}a^{8}+\frac{89}{832}a^{7}+\frac{21}{832}a^{6}-\frac{233}{1248}a^{5}-\frac{3}{8}a^{4}+\frac{45}{52}a^{3}+\frac{17}{104}a^{2}+\frac{31}{39}a+\frac{1}{2}$, $\frac{21}{425984}a^{31}-\frac{379}{2555904}a^{30}+\frac{179}{638976}a^{29}-\frac{121}{638976}a^{28}+\frac{47}{106496}a^{27}-\frac{79}{319488}a^{26}+\frac{49}{159744}a^{25}+\frac{25}{39936}a^{24}+\frac{69}{53248}a^{23}-\frac{1327}{319488}a^{22}+\frac{551}{79872}a^{21}-\frac{301}{79872}a^{20}+\frac{315}{26624}a^{19}-\frac{1031}{159744}a^{18}+\frac{397}{39936}a^{17}+\frac{655}{39936}a^{16}+\frac{625}{26624}a^{15}-\frac{10787}{159744}a^{14}+\frac{4327}{39936}a^{13}-\frac{2009}{39936}a^{12}+\frac{1099}{6656}a^{11}-\frac{1309}{19968}a^{10}+\frac{1117}{9984}a^{9}+\frac{49}{156}a^{8}+\frac{209}{832}a^{7}-\frac{1877}{2496}a^{6}+\frac{43}{39}a^{5}-\frac{113}{156}a^{4}+\frac{163}{104}a^{3}-\frac{113}{312}a^{2}+\frac{77}{156}a+\frac{307}{78}$, $\frac{59}{2555904}a^{31}-\frac{25}{851968}a^{30}-\frac{5}{39936}a^{29}-\frac{23}{638976}a^{28}-\frac{1}{19968}a^{27}-\frac{19}{106496}a^{26}+\frac{11}{159744}a^{25}+\frac{17}{39936}a^{24}+\frac{23}{319488}a^{23}-\frac{149}{106496}a^{22}-\frac{31}{9984}a^{21}-\frac{83}{79872}a^{20}-\frac{389}{159744}a^{19}-\frac{245}{53248}a^{18}-\frac{41}{19968}a^{17}+\frac{305}{39936}a^{16}+\frac{139}{159744}a^{15}-\frac{1217}{53248}a^{14}-\frac{529}{9984}a^{13}+\frac{41}{39936}a^{12}-\frac{293}{9984}a^{11}-\frac{301}{6656}a^{10}+\frac{203}{9984}a^{9}+\frac{265}{1248}a^{8}+\frac{19}{624}a^{7}-\frac{147}{832}a^{6}-\frac{113}{312}a^{5}+\frac{7}{312}a^{4}-\frac{7}{156}a^{3}+\frac{11}{104}a^{2}+\frac{73}{156}a+\frac{137}{78}$, $\frac{1}{19968}a^{31}-\frac{11}{851968}a^{30}+\frac{37}{1277952}a^{29}-\frac{29}{319488}a^{28}-\frac{1}{319488}a^{27}+\frac{1}{106496}a^{26}-\frac{7}{79872}a^{25}+\frac{23}{79872}a^{24}+\frac{17}{19968}a^{23}+\frac{1}{106496}a^{22}+\frac{337}{159744}a^{21}-\frac{125}{39936}a^{20}-\frac{53}{39936}a^{19}-\frac{87}{53248}a^{18}+\frac{269}{79872}a^{17}+\frac{137}{19968}a^{16}+\frac{119}{19968}a^{15}+\frac{205}{53248}a^{14}+\frac{3461}{79872}a^{13}-\frac{769}{19968}a^{12}-\frac{1153}{19968}a^{11}-\frac{173}{6656}a^{10}+\frac{157}{2496}a^{9}+\frac{779}{4992}a^{8}+\frac{7}{156}a^{7}-\frac{21}{832}a^{6}+\frac{593}{1248}a^{5}-\frac{353}{624}a^{4}-\frac{37}{156}a^{3}-\frac{17}{104}a^{2}-\frac{11}{78}a+\frac{71}{39}$, $\frac{31}{638976}a^{31}-\frac{23}{1277952}a^{30}+\frac{1}{9984}a^{29}-\frac{1}{106496}a^{28}+\frac{49}{319488}a^{27}+\frac{17}{79872}a^{26}+\frac{19}{79872}a^{25}-\frac{7}{13312}a^{24}+\frac{139}{79872}a^{23}-\frac{83}{159744}a^{22}+\frac{17}{9984}a^{21}+\frac{19}{13312}a^{20}+\frac{35}{9984}a^{19}+\frac{305}{79872}a^{18}+\frac{71}{9984}a^{17}-\frac{93}{6656}a^{16}+\frac{1169}{39936}a^{15}+\frac{41}{79872}a^{14}+\frac{79}{4992}a^{13}+\frac{151}{6656}a^{12}+\frac{1273}{19968}a^{11}+\frac{109}{2496}a^{10}+\frac{379}{4992}a^{9}-\frac{89}{832}a^{8}+\frac{181}{1248}a^{7}+\frac{85}{624}a^{6}+\frac{95}{624}a^{5}+\frac{17}{104}a^{4}+\frac{233}{312}a^{3}+\frac{59}{156}a^{2}+\frac{1}{39}a-\frac{11}{13}$ 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:  \( 901078689200.9504 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 901078689200.9504 \cdot 40}{8\cdot\sqrt{385049567410499609358247677435067162906957579288576}}\cr\approx \mathstrut & 1.35472859179882 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + 40*x^24 + 16*x^20 + 784*x^16 + 256*x^12 + 10240*x^8 + 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 + 40*x^24 + 16*x^20 + 784*x^16 + 256*x^12 + 10240*x^8 + 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 + 40*x^24 + 16*x^20 + 784*x^16 + 256*x^12 + 10240*x^8 + 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 + 40*x^24 + 16*x^20 + 784*x^16 + 256*x^12 + 10240*x^8 + 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

$C_2^6:S_4$ (as 32T96908):

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 1536
The 80 conjugacy class representatives for $C_2^6:S_4$
Character table for $C_2^6:S_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.4.15952.1, \(\Q(\zeta_{8})\), 8.8.69214834688.1, 8.8.1107437355008.1, 8.0.1107437355008.1, 8.0.4325927168.1, 8.8.65143373824.1, 8.0.65143373824.1, 8.0.4071460864.1, 16.0.67898546450774534127616.1, 16.16.19622679924273840362881024.1, 16.0.1226417495267115022680064.1, 16.0.19622679924273840362881024.1, 16.0.1226417495267115022680064.2, 16.0.4790693340887168057344.1, 16.0.19622679924273840362881024.2

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 ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ ${\href{/padicField/5.8.0.1}{8} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{16}$ ${\href{/padicField/13.6.0.1}{6} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ R ${\href{/padicField/19.4.0.1}{4} }^{8}$ ${\href{/padicField/23.6.0.1}{6} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{12}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.6.0.1}{6} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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$$16$$1$$36$
Deg $16$$16$$1$$36$
\(17\) Copy content Toggle raw display 17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
\(997\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$