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Properties

Label 32.0.596...336.1
Degree $32$
Signature $[0, 16]$
Discriminant $5.965\times 10^{49}$
Root discriminant \(35.93\)
Ramified primes $2,3,4093$
Class number $16$ (GRH)
Class group [2, 8] (GRH)
Galois group $C_2^5:S_4$ (as 32T34907)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^30 + x^28 - 8*x^26 + 20*x^24 - 16*x^22 + 32*x^20 - 128*x^18 + 64*x^16 - 512*x^14 + 512*x^12 - 1024*x^10 + 5120*x^8 - 8192*x^6 + 4096*x^4 - 16384*x^2 + 65536)
 
gp: K = bnfinit(y^32 - y^30 + y^28 - 8*y^26 + 20*y^24 - 16*y^22 + 32*y^20 - 128*y^18 + 64*y^16 - 512*y^14 + 512*y^12 - 1024*y^10 + 5120*y^8 - 8192*y^6 + 4096*y^4 - 16384*y^2 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - x^30 + x^28 - 8*x^26 + 20*x^24 - 16*x^22 + 32*x^20 - 128*x^18 + 64*x^16 - 512*x^14 + 512*x^12 - 1024*x^10 + 5120*x^8 - 8192*x^6 + 4096*x^4 - 16384*x^2 + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - x^30 + x^28 - 8*x^26 + 20*x^24 - 16*x^22 + 32*x^20 - 128*x^18 + 64*x^16 - 512*x^14 + 512*x^12 - 1024*x^10 + 5120*x^8 - 8192*x^6 + 4096*x^4 - 16384*x^2 + 65536)
 

\( x^{32} - x^{30} + x^{28} - 8 x^{26} + 20 x^{24} - 16 x^{22} + 32 x^{20} - 128 x^{18} + 64 x^{16} + \cdots + 65536 \) 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:   \(59647719207059933350876409342295283657151229198336\) \(\medspace = 2^{44}\cdot 3^{16}\cdot 4093^{8}\) 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:  \(35.93\)
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:  $2^{19/12}3^{1/2}4093^{1/2}\approx 332.05676010213426$
Ramified primes:   \(2\), \(3\), \(4093\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}+\frac{1}{8}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{9}+\frac{1}{8}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{10}+\frac{1}{16}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{11}+\frac{1}{16}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{32}a^{14}-\frac{1}{32}a^{12}+\frac{1}{32}a^{10}-\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{15}-\frac{1}{32}a^{13}+\frac{1}{32}a^{11}-\frac{1}{8}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{192}a^{16}+\frac{1}{192}a^{14}+\frac{1}{64}a^{12}-\frac{5}{96}a^{10}+\frac{1}{24}a^{8}-\frac{5}{24}a^{6}+\frac{1}{4}a^{4}-\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{384}a^{17}+\frac{1}{384}a^{15}+\frac{1}{128}a^{13}-\frac{5}{192}a^{11}-\frac{5}{48}a^{9}+\frac{1}{48}a^{7}+\frac{5}{12}a^{3}+\frac{1}{6}a$, $\frac{1}{768}a^{18}+\frac{1}{768}a^{16}+\frac{1}{256}a^{14}-\frac{5}{384}a^{12}-\frac{1}{16}a^{11}-\frac{5}{96}a^{10}-\frac{1}{16}a^{9}+\frac{1}{96}a^{8}-\frac{3}{16}a^{7}-\frac{3}{8}a^{5}-\frac{7}{24}a^{4}-\frac{1}{2}a^{3}-\frac{5}{12}a^{2}-\frac{1}{2}a$, $\frac{1}{768}a^{19}-\frac{1}{768}a^{17}+\frac{1}{768}a^{15}-\frac{1}{48}a^{13}-\frac{5}{192}a^{11}+\frac{11}{96}a^{9}-\frac{1}{48}a^{7}-\frac{7}{24}a^{5}+\frac{1}{6}a^{3}-\frac{1}{6}a$, $\frac{1}{1536}a^{20}-\frac{1}{1536}a^{18}+\frac{1}{1536}a^{16}-\frac{1}{96}a^{14}-\frac{1}{32}a^{13}-\frac{5}{384}a^{12}+\frac{1}{32}a^{11}+\frac{11}{192}a^{10}-\frac{1}{32}a^{9}-\frac{1}{96}a^{8}-\frac{7}{48}a^{6}+\frac{1}{8}a^{5}+\frac{1}{12}a^{4}-\frac{1}{4}a^{3}-\frac{1}{12}a^{2}-\frac{1}{2}a$, $\frac{1}{1536}a^{21}-\frac{1}{1536}a^{19}+\frac{1}{1536}a^{17}-\frac{1}{96}a^{15}-\frac{5}{384}a^{13}+\frac{11}{192}a^{11}-\frac{1}{96}a^{9}-\frac{7}{48}a^{7}+\frac{1}{12}a^{5}-\frac{1}{12}a^{3}$, $\frac{1}{3072}a^{22}-\frac{1}{3072}a^{20}+\frac{1}{3072}a^{18}-\frac{1}{64}a^{15}-\frac{1}{768}a^{14}+\frac{1}{64}a^{13}-\frac{7}{384}a^{12}-\frac{1}{64}a^{11}+\frac{1}{192}a^{10}-\frac{3}{32}a^{8}+\frac{1}{16}a^{7}-\frac{1}{6}a^{6}+\frac{3}{8}a^{5}+\frac{11}{24}a^{4}+\frac{1}{4}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3072}a^{23}-\frac{1}{3072}a^{21}+\frac{1}{3072}a^{19}-\frac{1}{768}a^{15}-\frac{7}{384}a^{13}+\frac{1}{192}a^{11}-\frac{3}{32}a^{9}-\frac{1}{6}a^{7}+\frac{11}{24}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{6144}a^{24}-\frac{1}{6144}a^{22}+\frac{1}{6144}a^{20}-\frac{1}{1536}a^{16}-\frac{1}{64}a^{15}-\frac{7}{768}a^{14}-\frac{1}{64}a^{13}+\frac{1}{384}a^{12}-\frac{3}{64}a^{11}-\frac{3}{64}a^{10}-\frac{3}{32}a^{9}-\frac{1}{12}a^{8}+\frac{1}{8}a^{7}+\frac{11}{48}a^{6}+\frac{3}{8}a^{5}-\frac{1}{3}a^{4}+\frac{1}{4}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}a$, $\frac{1}{6144}a^{25}-\frac{1}{6144}a^{23}+\frac{1}{6144}a^{21}-\frac{1}{1536}a^{17}-\frac{7}{768}a^{15}+\frac{1}{384}a^{13}-\frac{3}{64}a^{11}-\frac{1}{12}a^{9}+\frac{11}{48}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}$, $\frac{1}{12288}a^{26}-\frac{1}{12288}a^{24}+\frac{1}{12288}a^{22}-\frac{1}{3072}a^{18}+\frac{1}{1536}a^{16}+\frac{5}{768}a^{14}-\frac{1}{128}a^{12}+\frac{1}{32}a^{10}-\frac{1}{8}a^{9}+\frac{1}{32}a^{8}+\frac{1}{8}a^{7}-\frac{1}{4}a^{6}-\frac{1}{8}a^{5}-\frac{5}{12}a^{4}+\frac{1}{3}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{12288}a^{27}-\frac{1}{12288}a^{25}+\frac{1}{12288}a^{23}-\frac{1}{3072}a^{19}+\frac{1}{1536}a^{17}+\frac{5}{768}a^{15}-\frac{1}{128}a^{13}+\frac{1}{32}a^{11}+\frac{1}{32}a^{9}-\frac{1}{4}a^{7}-\frac{5}{12}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{24576}a^{28}-\frac{1}{24576}a^{26}+\frac{1}{24576}a^{24}-\frac{1}{6144}a^{20}+\frac{1}{3072}a^{18}-\frac{1}{512}a^{16}-\frac{7}{768}a^{14}-\frac{1}{16}a^{11}-\frac{11}{192}a^{10}+\frac{1}{16}a^{9}-\frac{1}{24}a^{8}-\frac{1}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{12}a^{4}-\frac{1}{4}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{24576}a^{29}-\frac{1}{24576}a^{27}+\frac{1}{24576}a^{25}-\frac{1}{6144}a^{21}+\frac{1}{3072}a^{19}+\frac{1}{1536}a^{17}-\frac{5}{768}a^{15}+\frac{1}{128}a^{13}+\frac{1}{24}a^{11}-\frac{1}{48}a^{9}-\frac{11}{48}a^{7}+\frac{1}{6}a^{5}+\frac{1}{4}a^{3}-\frac{1}{6}a$, $\frac{1}{344064}a^{30}-\frac{1}{114688}a^{28}+\frac{1}{49152}a^{26}-\frac{11}{172032}a^{24}-\frac{1}{7168}a^{22}-\frac{1}{10752}a^{20}+\frac{13}{21504}a^{18}+\frac{1}{2688}a^{16}+\frac{3}{896}a^{14}+\frac{55}{2688}a^{12}-\frac{1}{16}a^{11}-\frac{23}{672}a^{10}-\frac{1}{16}a^{9}+\frac{5}{112}a^{8}-\frac{3}{16}a^{7}-\frac{67}{336}a^{6}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{5}{28}a^{2}-\frac{5}{21}$, $\frac{1}{688128}a^{31}+\frac{11}{688128}a^{29}+\frac{1}{32768}a^{27}+\frac{5}{172032}a^{25}+\frac{3}{57344}a^{23}+\frac{5}{43008}a^{21}-\frac{11}{21504}a^{19}-\frac{1}{896}a^{17}+\frac{13}{3584}a^{15}+\frac{19}{1344}a^{13}+\frac{17}{336}a^{11}-\frac{19}{168}a^{9}+\frac{115}{672}a^{7}-\frac{1}{8}a^{5}+\frac{19}{56}a^{3}-\frac{19}{42}a$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:   \( -\frac{5}{229376} a^{31} + \frac{31}{688128} a^{29} - \frac{1}{98304} a^{27} + \frac{11}{43008} a^{25} - \frac{37}{172032} a^{23} - \frac{5}{43008} a^{21} - \frac{1}{896} a^{19} + \frac{5}{10752} a^{17} - \frac{11}{10752} a^{15} + \frac{3}{448} a^{13} - \frac{19}{1344} a^{11} - \frac{5}{224} a^{9} - \frac{115}{672} a^{7} + \frac{1}{8} a^{5} + \frac{9}{56} a^{3} + \frac{19}{42} a \)  (order $12$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{5}{229376}a^{31}-\frac{31}{688128}a^{29}+\frac{1}{98304}a^{27}-\frac{11}{43008}a^{25}+\frac{37}{172032}a^{23}+\frac{5}{43008}a^{21}+\frac{1}{896}a^{19}-\frac{5}{10752}a^{17}+\frac{11}{10752}a^{15}-\frac{3}{448}a^{13}+\frac{19}{1344}a^{11}+\frac{5}{224}a^{9}+\frac{115}{672}a^{7}-\frac{1}{8}a^{5}-\frac{9}{56}a^{3}-\frac{19}{42}a-1$, $\frac{5}{172032}a^{30}+\frac{13}{172032}a^{28}-\frac{1}{8192}a^{26}+\frac{1}{86016}a^{24}-\frac{25}{43008}a^{22}+\frac{1}{21504}a^{20}-\frac{1}{896}a^{18}-\frac{1}{672}a^{16}-\frac{1}{2688}a^{14}-\frac{1}{112}a^{12}-\frac{17}{336}a^{10}+\frac{5}{168}a^{8}-\frac{13}{168}a^{6}+\frac{1}{2}a^{4}-\frac{23}{42}a^{2}+\frac{13}{21}$, $\frac{1}{344064}a^{30}-\frac{17}{344064}a^{28}-\frac{1}{49152}a^{26}+\frac{1}{7168}a^{24}+\frac{17}{28672}a^{22}+\frac{13}{10752}a^{20}+\frac{5}{5376}a^{18}-\frac{5}{5376}a^{16}-\frac{15}{1792}a^{14}+\frac{1}{448}a^{12}+\frac{13}{336}a^{10}+\frac{1}{336}a^{8}-\frac{67}{336}a^{6}-\frac{1}{2}a^{4}-\frac{9}{28}a^{2}+\frac{3}{7}$, $\frac{1}{114688}a^{31}+\frac{5}{172032}a^{30}+\frac{5}{344064}a^{29}-\frac{1}{172032}a^{28}+\frac{5}{49152}a^{27}+\frac{1}{8192}a^{26}-\frac{1}{14336}a^{25}-\frac{5}{21504}a^{24}+\frac{41}{86016}a^{23}+\frac{19}{21504}a^{22}+\frac{1}{21504}a^{21}+\frac{1}{2688}a^{20}+\frac{1}{672}a^{19}+\frac{1}{672}a^{18}-\frac{23}{10752}a^{17}-\frac{1}{672}a^{16}+\frac{1}{448}a^{15}+\frac{11}{1792}a^{14}-\frac{1}{896}a^{13}-\frac{17}{2688}a^{12}+\frac{1}{84}a^{11}+\frac{5}{224}a^{10}-\frac{11}{336}a^{9}-\frac{19}{224}a^{8}+\frac{1}{168}a^{7}+\frac{11}{84}a^{6}-\frac{1}{6}a^{5}-\frac{7}{24}a^{4}+\frac{2}{7}a^{3}+\frac{17}{84}a^{2}-\frac{8}{21}a-\frac{29}{21}$, $\frac{1}{86016}a^{31}+\frac{1}{114688}a^{30}-\frac{1}{28672}a^{29}+\frac{5}{344064}a^{28}+\frac{1}{12288}a^{27}-\frac{11}{49152}a^{26}-\frac{1}{10752}a^{25}+\frac{11}{43008}a^{24}+\frac{11}{43008}a^{23}-\frac{43}{86016}a^{22}+\frac{5}{43008}a^{21}+\frac{29}{21504}a^{20}-\frac{11}{21504}a^{19}-\frac{1}{896}a^{18}-\frac{5}{10752}a^{17}+\frac{1}{896}a^{16}+\frac{1}{336}a^{15}-\frac{1}{2688}a^{14}-\frac{1}{672}a^{13}+\frac{25}{2688}a^{12}+\frac{13}{672}a^{11}-\frac{1}{112}a^{10}-\frac{9}{224}a^{9}+\frac{41}{672}a^{8}+\frac{13}{168}a^{7}-\frac{19}{336}a^{6}-\frac{1}{12}a^{5}+\frac{3}{8}a^{4}+\frac{1}{21}a^{3}-\frac{5}{7}a^{2}+\frac{23}{42}a-\frac{8}{21}$, $\frac{1}{229376}a^{31}-\frac{79}{688128}a^{29}-\frac{5}{32768}a^{27}-\frac{9}{57344}a^{25}+\frac{51}{57344}a^{23}+\frac{11}{21504}a^{21}+\frac{29}{10752}a^{19}-\frac{1}{10752}a^{17}+\frac{19}{10752}a^{15}-\frac{19}{2688}a^{13}+\frac{33}{448}a^{11}+\frac{15}{224}a^{9}+\frac{107}{672}a^{7}-\frac{1}{3}a^{5}-\frac{67}{168}a^{3}-\frac{32}{21}a$, $\frac{5}{229376}a^{31}-\frac{1}{172032}a^{30}-\frac{31}{688128}a^{29}+\frac{1}{57344}a^{28}-\frac{7}{98304}a^{27}+\frac{5}{24576}a^{26}-\frac{5}{28672}a^{25}+\frac{1}{21504}a^{24}+\frac{79}{172032}a^{23}-\frac{27}{28672}a^{22}+\frac{19}{43008}a^{21}-\frac{23}{14336}a^{20}+\frac{13}{5376}a^{19}-\frac{5}{21504}a^{18}-\frac{19}{10752}a^{17}-\frac{29}{10752}a^{16}-\frac{1}{3584}a^{15}-\frac{1}{672}a^{14}-\frac{11}{2688}a^{13}+\frac{37}{2688}a^{12}+\frac{47}{1344}a^{11}-\frac{1}{224}a^{10}+\frac{1}{672}a^{9}-\frac{17}{112}a^{8}+\frac{59}{672}a^{7}-\frac{17}{168}a^{6}-\frac{1}{12}a^{5}+\frac{5}{12}a^{4}-\frac{13}{168}a^{3}+\frac{25}{28}a^{2}-\frac{47}{42}a+\frac{1}{7}$, $\frac{11}{229376}a^{31}-\frac{1}{49152}a^{30}+\frac{9}{229376}a^{29}+\frac{1}{49152}a^{28}+\frac{11}{98304}a^{27}-\frac{1}{49152}a^{26}-\frac{73}{172032}a^{25}+\frac{1}{57344}a^{23}-\frac{7}{12288}a^{22}-\frac{17}{43008}a^{21}+\frac{1}{2048}a^{20}+\frac{3}{1792}a^{19}-\frac{1}{1024}a^{18}-\frac{13}{3584}a^{17}+\frac{5}{1536}a^{16}-\frac{5}{3584}a^{15}-\frac{1}{768}a^{14}-\frac{23}{896}a^{13}+\frac{1}{64}a^{12}-\frac{1}{56}a^{11}-\frac{11}{168}a^{9}+\frac{5}{96}a^{8}+\frac{47}{224}a^{7}-\frac{1}{24}a^{5}+\frac{5}{24}a^{4}+\frac{5}{168}a^{3}+\frac{5}{12}a^{2}-\frac{10}{7}a+\frac{1}{3}$, $\frac{5}{344064}a^{31}-\frac{1}{344064}a^{29}+\frac{7}{49152}a^{27}-\frac{17}{86016}a^{25}+\frac{73}{86016}a^{23}-\frac{31}{21504}a^{21}+\frac{5}{3584}a^{19}-\frac{25}{5376}a^{17}+\frac{1}{112}a^{15}-\frac{9}{448}a^{13}+\frac{1}{1344}a^{11}-\frac{11}{672}a^{9}-\frac{1}{56}a^{7}-\frac{1}{8}a^{5}+\frac{19}{84}a^{3}+\frac{13}{42}a$, $\frac{61}{688128}a^{31}+\frac{9}{229376}a^{29}-\frac{29}{98304}a^{27}-\frac{55}{43008}a^{25}-\frac{13}{57344}a^{23}+\frac{1}{1344}a^{21}+\frac{127}{21504}a^{19}+\frac{1}{3584}a^{17}+\frac{23}{3584}a^{15}-\frac{31}{1344}a^{13}+\frac{3}{224}a^{11}+\frac{17}{168}a^{9}+\frac{407}{672}a^{7}+\frac{7}{24}a^{5}-\frac{149}{168}a^{3}-\frac{193}{42}a$, $\frac{1}{172032}a^{31}+\frac{13}{344064}a^{30}+\frac{1}{43008}a^{29}+\frac{1}{114688}a^{28}-\frac{1}{6144}a^{27}-\frac{1}{49152}a^{26}-\frac{5}{57344}a^{25}-\frac{61}{86016}a^{24}-\frac{1}{3584}a^{23}+\frac{1}{7168}a^{22}+\frac{1}{7168}a^{21}-\frac{1}{1792}a^{20}-\frac{23}{21504}a^{19}+\frac{1}{336}a^{18}+\frac{5}{3584}a^{17}+\frac{17}{10752}a^{16}-\frac{5}{1344}a^{15}+\frac{1}{224}a^{14}+\frac{1}{224}a^{13}-\frac{41}{2688}a^{12}-\frac{11}{672}a^{11}+\frac{13}{448}a^{10}+\frac{53}{672}a^{9}-\frac{3}{224}a^{8}+\frac{1}{56}a^{7}+\frac{41}{112}a^{6}+\frac{1}{8}a^{5}-\frac{13}{42}a^{3}+\frac{4}{7}a^{2}-\frac{1}{7}a-\frac{17}{7}$, $\frac{11}{688128}a^{31}+\frac{13}{344064}a^{30}+\frac{3}{229376}a^{29}+\frac{31}{344064}a^{28}+\frac{1}{98304}a^{27}-\frac{13}{49152}a^{26}-\frac{85}{172032}a^{25}-\frac{13}{43008}a^{24}+\frac{43}{172032}a^{23}-\frac{1}{5376}a^{22}-\frac{1}{5376}a^{21}+\frac{53}{43008}a^{20}+\frac{9}{3584}a^{19}+\frac{11}{10752}a^{18}+\frac{1}{1344}a^{17}+\frac{17}{10752}a^{16}-\frac{5}{10752}a^{15}-\frac{3}{896}a^{14}-\frac{1}{168}a^{13}-\frac{5}{672}a^{12}+\frac{13}{1344}a^{11}-\frac{31}{1344}a^{10}-\frac{1}{224}a^{9}+\frac{47}{672}a^{8}+\frac{103}{672}a^{7}+\frac{67}{336}a^{6}-\frac{1}{24}a^{5}+\frac{1}{8}a^{4}+\frac{25}{168}a^{3}-\frac{16}{21}a^{2}-\frac{15}{7}a-\frac{10}{7}$, $\frac{17}{688128}a^{31}-\frac{1}{344064}a^{30}-\frac{37}{688128}a^{29}+\frac{17}{344064}a^{28}+\frac{1}{32768}a^{27}+\frac{5}{49152}a^{26}-\frac{83}{172032}a^{25}-\frac{5}{86016}a^{24}+\frac{97}{172032}a^{23}-\frac{29}{43008}a^{22}-\frac{1}{896}a^{21}+\frac{11}{43008}a^{20}+\frac{29}{10752}a^{19}-\frac{9}{7168}a^{18}-\frac{29}{10752}a^{17}+\frac{19}{5376}a^{16}+\frac{47}{10752}a^{15}-\frac{5}{448}a^{14}-\frac{47}{2688}a^{13}+\frac{43}{2688}a^{12}+\frac{11}{672}a^{11}-\frac{73}{1344}a^{10}+\frac{5}{336}a^{9}-\frac{1}{336}a^{8}+\frac{31}{224}a^{7}-\frac{19}{168}a^{6}-\frac{1}{24}a^{5}+\frac{3}{8}a^{4}+\frac{17}{168}a^{3}+\frac{5}{21}a^{2}-\frac{43}{42}a-\frac{2}{21}$, $\frac{17}{688128}a^{31}+\frac{1}{21504}a^{30}-\frac{3}{229376}a^{29}+\frac{11}{172032}a^{28}+\frac{7}{98304}a^{27}+\frac{1}{8192}a^{26}-\frac{31}{86016}a^{25}-\frac{29}{172032}a^{24}+\frac{139}{172032}a^{23}-\frac{13}{21504}a^{22}-\frac{1}{7168}a^{21}-\frac{19}{14336}a^{20}+\frac{65}{21504}a^{19}-\frac{11}{5376}a^{18}-\frac{11}{5376}a^{17}-\frac{13}{10752}a^{16}+\frac{61}{10752}a^{15}-\frac{1}{112}a^{14}-\frac{47}{2688}a^{13}-\frac{23}{2688}a^{12}+\frac{5}{448}a^{11}-\frac{1}{21}a^{10}-\frac{3}{112}a^{9}-\frac{1}{224}a^{8}+\frac{65}{672}a^{7}+\frac{17}{168}a^{6}-\frac{7}{24}a^{5}+\frac{11}{24}a^{4}-\frac{11}{168}a^{3}+\frac{6}{7}a^{2}-\frac{25}{21}a+\frac{6}{7}$, $\frac{41}{688128}a^{31}+\frac{1}{14336}a^{30}-\frac{27}{229376}a^{29}+\frac{1}{3584}a^{28}-\frac{25}{98304}a^{27}+\frac{5}{12288}a^{26}-\frac{13}{10752}a^{25}+\frac{1}{86016}a^{24}+\frac{145}{172032}a^{23}-\frac{155}{86016}a^{22}+\frac{11}{10752}a^{21}-\frac{89}{43008}a^{20}+\frac{41}{7168}a^{19}-\frac{101}{21504}a^{18}-\frac{1}{672}a^{17}-\frac{37}{10752}a^{16}+\frac{59}{10752}a^{15}-\frac{43}{2688}a^{14}-\frac{5}{1344}a^{13}-\frac{73}{2688}a^{12}+\frac{3}{56}a^{11}-\frac{55}{672}a^{10}+\frac{31}{336}a^{9}-\frac{5}{28}a^{8}+\frac{277}{672}a^{7}+\frac{5}{56}a^{6}-\frac{7}{24}a^{5}+\frac{11}{12}a^{4}-\frac{169}{168}a^{3}+\frac{199}{84}a^{2}-\frac{71}{21}a+\frac{34}{21}$ 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:  \( 658375568497.8936 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 658375568497.8936 \cdot 16}{12\cdot\sqrt{59647719207059933350876409342295283657151229198336}}\cr\approx \mathstrut & 0.670646119119364 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - x^30 + x^28 - 8*x^26 + 20*x^24 - 16*x^22 + 32*x^20 - 128*x^18 + 64*x^16 - 512*x^14 + 512*x^12 - 1024*x^10 + 5120*x^8 - 8192*x^6 + 4096*x^4 - 16384*x^2 + 65536)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^32 - x^30 + x^28 - 8*x^26 + 20*x^24 - 16*x^22 + 32*x^20 - 128*x^18 + 64*x^16 - 512*x^14 + 512*x^12 - 1024*x^10 + 5120*x^8 - 8192*x^6 + 4096*x^4 - 16384*x^2 + 65536, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - x^30 + x^28 - 8*x^26 + 20*x^24 - 16*x^22 + 32*x^20 - 128*x^18 + 64*x^16 - 512*x^14 + 512*x^12 - 1024*x^10 + 5120*x^8 - 8192*x^6 + 4096*x^4 - 16384*x^2 + 65536);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - x^30 + x^28 - 8*x^26 + 20*x^24 - 16*x^22 + 32*x^20 - 128*x^18 + 64*x^16 - 512*x^14 + 512*x^12 - 1024*x^10 + 5120*x^8 - 8192*x^6 + 4096*x^4 - 16384*x^2 + 65536);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^5:S_4$ (as 32T34907):

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 768
The 52 conjugacy class representatives for $C_2^5:S_4$ are not computed
Character table for $C_2^5:S_4$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.4.147348.1, \(\Q(\zeta_{12})\), 8.0.1389531718656.1, 8.8.1389531718656.2, 8.0.21711433104.1, 8.8.1389531718656.1, 8.0.86845732416.1, 8.8.86845732416.1, 8.0.1389531718656.2, 16.0.1930798397151097138446336.2, 16.0.7723193588604388553785344.3, 16.0.7723193588604388553785344.2, 16.16.7723193588604388553785344.1, 16.0.7723193588604388553785344.1, 16.0.1930798397151097138446336.1, 16.0.7542181238871473197056.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 32 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.4.0.1}{4} }^{8}$ ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.6.0.1}{6} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{8}$ ${\href{/padicField/17.6.0.1}{6} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.6.0.1}{6} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.6.0.1}{6} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{16}$ ${\href{/padicField/59.4.0.1}{4} }^{8}$

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 2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
Deg $24$$12$$2$$36$
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(4093\) 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$