Properties

Label 32.0.334...416.1
Degree $32$
Signature $[0, 16]$
Discriminant $3.343\times 10^{50}$
Root discriminant \(37.92\)
Ramified primes $2,3,5077$
Class number $16$ (GRH)
Class group [4, 4] (GRH)
Galois group $C_2^5:S_4$ (as 32T34907)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^30 - 3*x^28 + 20*x^26 - 12*x^24 - 64*x^22 + 160*x^20 + 64*x^18 - 704*x^16 + 256*x^14 + 2560*x^12 - 4096*x^10 - 3072*x^8 + 20480*x^6 - 12288*x^4 - 16384*x^2 + 65536)
 
gp: K = bnfinit(y^32 - y^30 - 3*y^28 + 20*y^26 - 12*y^24 - 64*y^22 + 160*y^20 + 64*y^18 - 704*y^16 + 256*y^14 + 2560*y^12 - 4096*y^10 - 3072*y^8 + 20480*y^6 - 12288*y^4 - 16384*y^2 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - x^30 - 3*x^28 + 20*x^26 - 12*x^24 - 64*x^22 + 160*x^20 + 64*x^18 - 704*x^16 + 256*x^14 + 2560*x^12 - 4096*x^10 - 3072*x^8 + 20480*x^6 - 12288*x^4 - 16384*x^2 + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - x^30 - 3*x^28 + 20*x^26 - 12*x^24 - 64*x^22 + 160*x^20 + 64*x^18 - 704*x^16 + 256*x^14 + 2560*x^12 - 4096*x^10 - 3072*x^8 + 20480*x^6 - 12288*x^4 - 16384*x^2 + 65536)
 

\( x^{32} - x^{30} - 3 x^{28} + 20 x^{26} - 12 x^{24} - 64 x^{22} + 160 x^{20} + 64 x^{18} - 704 x^{16} + \cdots + 65536 \) 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:   \(334285232879825411498534254159851729760457815228416\) \(\medspace = 2^{44}\cdot 3^{16}\cdot 5077^{8}\) 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:  \(37.92\)
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:  $2^{19/12}3^{1/2}5077^{1/2}\approx 369.8239522562059$
Ramified primes:   \(2\), \(3\), \(5077\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}+\frac{1}{8}a^{6}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{9}+\frac{1}{8}a^{7}$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{10}+\frac{1}{16}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{11}+\frac{1}{16}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{32}a^{14}-\frac{1}{32}a^{12}+\frac{1}{32}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{15}-\frac{1}{32}a^{13}+\frac{1}{32}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{64}a^{16}-\frac{1}{64}a^{14}+\frac{1}{64}a^{12}-\frac{1}{8}a^{8}-\frac{1}{4}a^{4}$, $\frac{1}{128}a^{17}+\frac{1}{128}a^{15}-\frac{1}{128}a^{13}-\frac{3}{64}a^{11}-\frac{1}{8}a^{9}-\frac{1}{16}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{256}a^{18}+\frac{1}{256}a^{16}-\frac{1}{256}a^{14}-\frac{1}{32}a^{13}-\frac{3}{128}a^{12}-\frac{1}{32}a^{11}-\frac{1}{16}a^{10}+\frac{1}{32}a^{9}-\frac{1}{32}a^{8}-\frac{1}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{3}{8}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{256}a^{19}-\frac{1}{256}a^{17}-\frac{3}{256}a^{15}-\frac{1}{64}a^{13}-\frac{1}{64}a^{11}+\frac{3}{32}a^{9}-\frac{1}{16}a^{7}-\frac{3}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{512}a^{20}-\frac{1}{512}a^{18}-\frac{3}{512}a^{16}-\frac{1}{64}a^{15}-\frac{1}{128}a^{14}+\frac{1}{64}a^{13}-\frac{1}{128}a^{12}+\frac{3}{64}a^{11}+\frac{3}{64}a^{10}-\frac{1}{16}a^{9}-\frac{1}{32}a^{8}-\frac{1}{16}a^{7}-\frac{3}{16}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{512}a^{21}-\frac{1}{512}a^{19}+\frac{1}{512}a^{17}-\frac{1}{64}a^{13}+\frac{3}{32}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{1024}a^{22}-\frac{1}{1024}a^{20}+\frac{1}{1024}a^{18}-\frac{1}{64}a^{15}-\frac{1}{128}a^{14}+\frac{1}{64}a^{13}-\frac{1}{64}a^{11}+\frac{3}{64}a^{10}+\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{1024}a^{23}-\frac{1}{1024}a^{21}+\frac{1}{1024}a^{19}-\frac{1}{128}a^{15}+\frac{3}{64}a^{11}-\frac{1}{8}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{2048}a^{24}-\frac{1}{2048}a^{22}+\frac{1}{2048}a^{20}-\frac{1}{256}a^{16}-\frac{1}{64}a^{15}+\frac{1}{64}a^{13}+\frac{3}{128}a^{12}+\frac{3}{64}a^{11}+\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{3}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2048}a^{25}-\frac{1}{2048}a^{23}+\frac{1}{2048}a^{21}-\frac{1}{256}a^{17}+\frac{3}{128}a^{13}-\frac{1}{16}a^{9}-\frac{1}{8}a^{7}+\frac{1}{8}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4096}a^{26}-\frac{1}{4096}a^{24}+\frac{1}{4096}a^{22}-\frac{1}{512}a^{18}-\frac{1}{64}a^{15}+\frac{3}{256}a^{14}-\frac{1}{64}a^{13}+\frac{1}{64}a^{11}-\frac{1}{32}a^{10}-\frac{1}{32}a^{9}-\frac{1}{16}a^{8}-\frac{1}{8}a^{7}+\frac{1}{16}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a$, $\frac{1}{4096}a^{27}-\frac{1}{4096}a^{25}+\frac{1}{4096}a^{23}-\frac{1}{512}a^{19}+\frac{3}{256}a^{15}-\frac{1}{32}a^{11}-\frac{1}{16}a^{9}+\frac{1}{16}a^{7}-\frac{1}{4}a^{5}$, $\frac{1}{24576}a^{28}-\frac{1}{8192}a^{26}-\frac{1}{24576}a^{24}+\frac{5}{12288}a^{22}-\frac{1}{6144}a^{20}-\frac{1}{1024}a^{18}+\frac{1}{192}a^{16}+\frac{1}{128}a^{14}-\frac{1}{384}a^{12}-\frac{1}{32}a^{10}-\frac{1}{8}a^{9}+\frac{11}{96}a^{8}-\frac{1}{8}a^{7}+\frac{1}{6}a^{6}+\frac{1}{8}a^{5}-\frac{1}{6}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{3}$, $\frac{1}{24576}a^{29}-\frac{1}{8192}a^{27}-\frac{1}{24576}a^{25}+\frac{5}{12288}a^{23}-\frac{1}{6144}a^{21}-\frac{1}{1024}a^{19}-\frac{1}{384}a^{17}+\frac{1}{192}a^{13}+\frac{1}{64}a^{11}-\frac{1}{96}a^{9}-\frac{1}{48}a^{7}+\frac{1}{3}a^{5}-\frac{1}{2}a^{3}+\frac{1}{6}a$, $\frac{1}{147456}a^{30}+\frac{1}{147456}a^{28}-\frac{13}{147456}a^{26}+\frac{5}{24576}a^{24}+\frac{5}{12288}a^{22}-\frac{1}{2304}a^{20}+\frac{5}{9216}a^{18}+\frac{1}{288}a^{16}-\frac{19}{2304}a^{14}-\frac{35}{1152}a^{12}-\frac{11}{288}a^{10}-\frac{1}{24}a^{8}-\frac{1}{4}a^{7}+\frac{1}{6}a^{6}+\frac{1}{4}a^{5}-\frac{13}{36}a^{4}-\frac{1}{4}a^{3}+\frac{1}{36}a^{2}+\frac{4}{9}$, $\frac{1}{294912}a^{31}-\frac{5}{294912}a^{29}-\frac{31}{294912}a^{27}+\frac{1}{8192}a^{23}-\frac{7}{18432}a^{21}-\frac{1}{4608}a^{19}-\frac{1}{1152}a^{17}+\frac{17}{4608}a^{15}+\frac{1}{576}a^{13}-\frac{5}{144}a^{11}+\frac{3}{32}a^{9}+\frac{5}{32}a^{7}+\frac{5}{18}a^{5}-\frac{17}{72}a^{3}+\frac{7}{18}a$ 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

$C_{4}\times C_{4}$, 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:   \( \frac{7}{147456} a^{31} - \frac{35}{147456} a^{29} + \frac{35}{147456} a^{27} + \frac{5}{4096} a^{25} - \frac{7}{2048} a^{23} + \frac{7}{4608} a^{21} + \frac{49}{4608} a^{19} - \frac{7}{576} a^{17} - \frac{35}{1152} a^{15} + \frac{91}{1152} a^{13} + \frac{35}{576} a^{11} - \frac{7}{16} a^{9} + \frac{7}{16} a^{7} + \frac{7}{18} a^{5} - \frac{14}{9} a^{3} + \frac{35}{18} a \)  (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:   $\frac{37}{294912}a^{31}-\frac{1}{49152}a^{30}-\frac{137}{294912}a^{29}-\frac{7}{49152}a^{28}+\frac{149}{294912}a^{27}+\frac{43}{49152}a^{26}+\frac{7}{3072}a^{25}-\frac{5}{4096}a^{24}-\frac{173}{24576}a^{23}-\frac{7}{4096}a^{22}+\frac{49}{9216}a^{21}+\frac{7}{768}a^{20}+\frac{169}{9216}a^{19}-\frac{7}{1536}a^{18}-\frac{151}{4608}a^{17}-\frac{5}{192}a^{16}-\frac{217}{4608}a^{15}+\frac{19}{768}a^{14}+\frac{47}{288}a^{13}+\frac{19}{192}a^{12}+\frac{43}{576}a^{11}-\frac{25}{96}a^{10}-\frac{85}{96}a^{9}-\frac{1}{8}a^{8}+\frac{127}{96}a^{7}+\frac{17}{16}a^{6}+\frac{11}{18}a^{5}-\frac{17}{12}a^{4}-\frac{269}{72}a^{3}-\frac{1}{12}a^{2}+\frac{50}{9}a+\frac{5}{3}$, $\frac{127}{294912}a^{31}-\frac{7}{147456}a^{30}-\frac{335}{294912}a^{29}-\frac{31}{147456}a^{28}+\frac{347}{294912}a^{27}+\frac{127}{147456}a^{26}+\frac{143}{24576}a^{25}-\frac{85}{24576}a^{24}-\frac{401}{24576}a^{23}+\frac{11}{3072}a^{22}+\frac{85}{9216}a^{21}+\frac{59}{18432}a^{20}+\frac{367}{9216}a^{19}-\frac{179}{9216}a^{18}-\frac{277}{4608}a^{17}+\frac{29}{1152}a^{16}-\frac{631}{4608}a^{15}+\frac{43}{2304}a^{14}+\frac{101}{288}a^{13}+\frac{5}{1152}a^{12}+\frac{71}{288}a^{11}-\frac{19}{72}a^{10}-\frac{49}{24}a^{9}+\frac{59}{96}a^{8}+\frac{313}{96}a^{7}-\frac{7}{48}a^{6}+\frac{49}{36}a^{5}-\frac{55}{18}a^{4}-\frac{521}{72}a^{3}+\frac{50}{9}a^{2}+\frac{104}{9}a-\frac{70}{9}$, $\frac{11}{98304}a^{31}-\frac{7}{98304}a^{29}-\frac{53}{98304}a^{27}+\frac{3}{2048}a^{25}-\frac{7}{8192}a^{23}-\frac{19}{3072}a^{21}+\frac{5}{768}a^{19}+\frac{25}{1536}a^{17}-\frac{53}{1536}a^{15}-\frac{7}{192}a^{13}+\frac{47}{192}a^{11}-\frac{1}{8}a^{9}-\frac{13}{32}a^{7}+\frac{25}{24}a^{5}-\frac{13}{24}a^{3}-\frac{7}{6}a$, $\frac{49}{98304}a^{31}+\frac{11}{49152}a^{30}-\frac{149}{98304}a^{29}-\frac{31}{49152}a^{28}+\frac{137}{98304}a^{27}+\frac{7}{49152}a^{26}+\frac{33}{4096}a^{25}+\frac{21}{4096}a^{24}-\frac{191}{8192}a^{23}-\frac{27}{2048}a^{22}+\frac{5}{384}a^{21}+\frac{35}{6144}a^{20}+\frac{193}{3072}a^{19}+\frac{121}{3072}a^{18}-\frac{151}{1536}a^{17}-\frac{91}{1536}a^{16}-\frac{277}{1536}a^{15}-\frac{71}{768}a^{14}+\frac{197}{384}a^{13}+\frac{107}{384}a^{12}+\frac{67}{192}a^{11}+\frac{23}{96}a^{10}-3a^{9}-\frac{27}{16}a^{8}+\frac{139}{32}a^{7}+\frac{33}{16}a^{6}+\frac{31}{12}a^{5}+\frac{31}{12}a^{4}-\frac{305}{24}a^{3}-\frac{28}{3}a^{2}+\frac{103}{6}a+\frac{32}{3}$, $\frac{1}{294912}a^{31}-\frac{29}{147456}a^{30}+\frac{7}{294912}a^{29}+\frac{73}{147456}a^{28}+\frac{5}{294912}a^{27}-\frac{73}{147456}a^{26}-\frac{7}{24576}a^{25}-\frac{11}{4096}a^{24}+\frac{19}{24576}a^{23}+\frac{1}{128}a^{22}-\frac{5}{9216}a^{21}-\frac{71}{18432}a^{20}-\frac{29}{9216}a^{19}-\frac{181}{9216}a^{18}+\frac{5}{1152}a^{17}+\frac{71}{2304}a^{16}+\frac{35}{4608}a^{15}+\frac{73}{1152}a^{14}-\frac{19}{1152}a^{13}-\frac{215}{1152}a^{12}-\frac{11}{576}a^{11}-\frac{73}{576}a^{10}+\frac{17}{96}a^{9}+a^{8}-\frac{5}{96}a^{7}-\frac{13}{8}a^{6}-\frac{19}{72}a^{5}-\frac{71}{72}a^{4}+\frac{55}{72}a^{3}+\frac{71}{18}a^{2}-\frac{4}{9}a-\frac{50}{9}$, $\frac{23}{294912}a^{31}-\frac{7}{36864}a^{30}-\frac{79}{294912}a^{29}+\frac{5}{36864}a^{28}+\frac{43}{294912}a^{27}+\frac{19}{36864}a^{26}+\frac{11}{8192}a^{25}-\frac{19}{6144}a^{24}-\frac{27}{8192}a^{23}+\frac{5}{3072}a^{22}-\frac{53}{18432}a^{21}+\frac{55}{9216}a^{20}+\frac{125}{9216}a^{19}-\frac{77}{9216}a^{18}+\frac{1}{288}a^{17}-\frac{49}{4608}a^{16}-\frac{293}{4608}a^{15}+\frac{25}{576}a^{14}+\frac{1}{18}a^{13}+\frac{47}{1152}a^{12}+\frac{29}{144}a^{11}-\frac{35}{144}a^{10}-\frac{7}{16}a^{9}+\frac{29}{96}a^{8}+\frac{1}{32}a^{7}-\frac{7}{48}a^{6}+\frac{73}{72}a^{5}-\frac{35}{36}a^{4}-\frac{31}{72}a^{3}+\frac{11}{9}a^{2}-\frac{14}{9}a-\frac{46}{9}$, $\frac{25}{294912}a^{31}+\frac{1}{49152}a^{30}-\frac{53}{294912}a^{29}-\frac{1}{16384}a^{28}+\frac{89}{294912}a^{27}-\frac{1}{49152}a^{26}+\frac{1}{2048}a^{25}-\frac{7}{24576}a^{24}-\frac{9}{8192}a^{23}+\frac{17}{12288}a^{22}+\frac{1}{576}a^{21}-\frac{7}{4608}a^{19}-\frac{19}{3072}a^{18}-\frac{1}{4608}a^{17}+\frac{5}{512}a^{16}-\frac{25}{4608}a^{15}+\frac{5}{768}a^{14}+\frac{7}{576}a^{13}-\frac{1}{64}a^{12}-\frac{7}{288}a^{11}-\frac{13}{192}a^{10}-\frac{1}{16}a^{9}+\frac{17}{96}a^{8}+\frac{15}{32}a^{7}+\frac{5}{48}a^{6}-\frac{49}{72}a^{5}-a^{4}+\frac{43}{72}a^{3}+\frac{5}{6}a^{2}+\frac{31}{18}a$, $\frac{127}{294912}a^{31}+\frac{7}{147456}a^{30}-\frac{335}{294912}a^{29}+\frac{31}{147456}a^{28}+\frac{347}{294912}a^{27}-\frac{127}{147456}a^{26}+\frac{143}{24576}a^{25}+\frac{85}{24576}a^{24}-\frac{401}{24576}a^{23}-\frac{11}{3072}a^{22}+\frac{85}{9216}a^{21}-\frac{59}{18432}a^{20}+\frac{367}{9216}a^{19}+\frac{179}{9216}a^{18}-\frac{277}{4608}a^{17}-\frac{29}{1152}a^{16}-\frac{631}{4608}a^{15}-\frac{43}{2304}a^{14}+\frac{101}{288}a^{13}-\frac{5}{1152}a^{12}+\frac{71}{288}a^{11}+\frac{19}{72}a^{10}-\frac{49}{24}a^{9}-\frac{59}{96}a^{8}+\frac{313}{96}a^{7}+\frac{7}{48}a^{6}+\frac{49}{36}a^{5}+\frac{55}{18}a^{4}-\frac{521}{72}a^{3}-\frac{50}{9}a^{2}+\frac{104}{9}a+\frac{70}{9}$, $\frac{1}{9216}a^{31}+\frac{1}{49152}a^{30}-\frac{1}{4608}a^{29}-\frac{5}{49152}a^{28}+\frac{1}{18432}a^{27}+\frac{5}{49152}a^{26}+\frac{13}{6144}a^{25}-\frac{1}{4096}a^{24}-\frac{19}{6144}a^{23}-\frac{1}{512}a^{22}-\frac{25}{9216}a^{21}+\frac{7}{6144}a^{20}+\frac{29}{2304}a^{19}+\frac{17}{3072}a^{18}-\frac{1}{576}a^{17}-\frac{7}{768}a^{16}-\frac{133}{2304}a^{15}-\frac{5}{384}a^{14}+\frac{5}{144}a^{13}+\frac{31}{384}a^{12}+\frac{29}{144}a^{11}+\frac{5}{192}a^{10}-\frac{19}{48}a^{9}-\frac{1}{4}a^{8}+\frac{5}{24}a^{7}+\frac{1}{4}a^{6}+\frac{19}{18}a^{5}+\frac{7}{24}a^{4}-\frac{29}{36}a^{3}-\frac{7}{6}a^{2}+\frac{16}{9}a-\frac{5}{3}$, $\frac{7}{147456}a^{31}-\frac{5}{49152}a^{30}-\frac{23}{147456}a^{29}+\frac{13}{49152}a^{28}-\frac{1}{147456}a^{27}-\frac{13}{49152}a^{26}+\frac{7}{6144}a^{25}-\frac{1}{2048}a^{24}-\frac{1}{384}a^{23}+\frac{3}{2048}a^{22}+\frac{11}{9216}a^{21}+\frac{1}{1536}a^{20}+\frac{5}{576}a^{19}-\frac{7}{3072}a^{18}-\frac{11}{1152}a^{17}-\frac{1}{192}a^{16}-\frac{13}{576}a^{15}+\frac{13}{384}a^{14}+\frac{29}{576}a^{13}-\frac{1}{48}a^{12}+\frac{11}{144}a^{11}-\frac{13}{192}a^{10}-\frac{35}{96}a^{9}+\frac{3}{16}a^{8}+\frac{1}{3}a^{7}-\frac{1}{4}a^{6}+\frac{5}{9}a^{5}+\frac{1}{6}a^{4}-\frac{37}{18}a^{3}-\frac{2}{3}a^{2}+\frac{16}{9}a+\frac{1}{3}$, $\frac{5}{16384}a^{31}-\frac{23}{73728}a^{30}-\frac{9}{16384}a^{29}+\frac{47}{36864}a^{28}+\frac{9}{16384}a^{27}-\frac{35}{36864}a^{26}+\frac{17}{4096}a^{25}-\frac{107}{24576}a^{24}-\frac{5}{512}a^{23}+\frac{43}{3072}a^{22}+\frac{11}{2048}a^{21}-\frac{25}{4608}a^{20}+\frac{3}{128}a^{19}-\frac{365}{9216}a^{18}-\frac{11}{256}a^{17}+\frac{191}{4608}a^{16}-\frac{9}{128}a^{15}+\frac{79}{576}a^{14}+\frac{25}{128}a^{13}-\frac{185}{576}a^{12}+\frac{9}{64}a^{11}-\frac{167}{576}a^{10}-\frac{5}{4}a^{9}+\frac{169}{96}a^{8}+\frac{35}{16}a^{7}-\frac{55}{24}a^{6}+\frac{11}{8}a^{5}-\frac{23}{36}a^{4}-\frac{11}{2}a^{3}+\frac{233}{36}a^{2}+11a-\frac{49}{9}$, $\frac{43}{294912}a^{31}+\frac{41}{147456}a^{30}-\frac{143}{294912}a^{29}-\frac{79}{147456}a^{28}-\frac{37}{294912}a^{27}-\frac{137}{147456}a^{26}+\frac{9}{2048}a^{25}+\frac{61}{8192}a^{24}-\frac{79}{8192}a^{23}-\frac{25}{2048}a^{22}-\frac{13}{18432}a^{21}-\frac{97}{18432}a^{20}+\frac{319}{9216}a^{19}+\frac{7}{144}a^{18}-\frac{145}{4608}a^{17}-\frac{83}{2304}a^{16}-\frac{457}{4608}a^{15}-\frac{311}{2304}a^{14}+\frac{97}{576}a^{13}+\frac{173}{1152}a^{12}+\frac{193}{576}a^{11}+\frac{349}{576}a^{10}-\frac{41}{32}a^{9}-\frac{55}{32}a^{8}+\frac{27}{32}a^{7}+\frac{15}{16}a^{6}+\frac{97}{36}a^{5}+\frac{163}{36}a^{4}-\frac{461}{72}a^{3}-\frac{373}{36}a^{2}+\frac{47}{9}a+\frac{98}{9}$, $\frac{13}{73728}a^{31}-\frac{7}{147456}a^{30}-\frac{23}{73728}a^{29}+\frac{35}{147456}a^{28}+\frac{29}{73728}a^{27}-\frac{71}{147456}a^{26}+\frac{5}{3072}a^{25}-\frac{1}{512}a^{24}-\frac{47}{12288}a^{23}+\frac{29}{4096}a^{22}+\frac{31}{9216}a^{21}-\frac{17}{2304}a^{20}+\frac{5}{1152}a^{19}-\frac{179}{9216}a^{18}-\frac{61}{4608}a^{17}+\frac{227}{4608}a^{16}-\frac{31}{1152}a^{15}+\frac{61}{2304}a^{14}+\frac{5}{72}a^{13}-\frac{181}{1152}a^{12}+\frac{31}{576}a^{11}+\frac{7}{144}a^{10}-\frac{19}{48}a^{9}+\frac{29}{32}a^{8}+\frac{55}{48}a^{7}-\frac{11}{8}a^{6}-\frac{1}{72}a^{5}-\frac{8}{9}a^{4}-\frac{37}{36}a^{3}+\frac{173}{36}a^{2}+\frac{91}{18}a-\frac{58}{9}$, $\frac{7}{49152}a^{31}+\frac{11}{147456}a^{30}-\frac{17}{49152}a^{29}-\frac{19}{147456}a^{28}+\frac{5}{49152}a^{27}-\frac{17}{147456}a^{26}+\frac{15}{8192}a^{25}+\frac{3}{4096}a^{24}-\frac{17}{4096}a^{23}-\frac{3}{4096}a^{22}-\frac{5}{6144}a^{21}-\frac{41}{9216}a^{20}+\frac{1}{96}a^{19}+\frac{37}{9216}a^{18}-\frac{5}{1536}a^{17}+\frac{37}{2304}a^{16}-\frac{37}{768}a^{15}-\frac{101}{2304}a^{14}+\frac{11}{192}a^{13}+\frac{1}{144}a^{12}+\frac{19}{96}a^{11}+\frac{109}{576}a^{10}-\frac{17}{32}a^{9}-\frac{3}{16}a^{8}+\frac{1}{2}a^{7}-\frac{1}{16}a^{6}+\frac{19}{24}a^{5}+\frac{11}{18}a^{4}-\frac{23}{12}a^{3}-\frac{4}{9}a^{2}+\frac{4}{3}a-\frac{13}{9}$, $\frac{5}{32768}a^{31}+\frac{1}{16384}a^{30}-\frac{23}{98304}a^{29}-\frac{31}{49152}a^{28}+\frac{9}{32768}a^{27}+\frac{17}{16384}a^{26}+\frac{65}{24576}a^{25}+\frac{1}{3072}a^{24}-\frac{125}{24576}a^{23}-\frac{97}{12288}a^{22}+\frac{5}{1536}a^{21}+\frac{17}{1536}a^{20}+\frac{7}{512}a^{19}+\frac{13}{1024}a^{18}-\frac{5}{192}a^{17}-\frac{1}{24}a^{16}-\frac{21}{512}a^{15}-\frac{7}{256}a^{14}+\frac{35}{384}a^{13}+\frac{23}{96}a^{12}+\frac{1}{16}a^{11}-\frac{7}{64}a^{10}-\frac{35}{48}a^{9}-\frac{11}{12}a^{8}+\frac{121}{96}a^{7}+\frac{83}{48}a^{6}+\frac{5}{6}a^{5}-\frac{11}{12}a^{4}-\frac{23}{8}a^{3}-4a^{2}+\frac{23}{3}a+\frac{8}{3}$ 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:  \( 1000974895371.5886 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 1000974895371.5886 \cdot 16}{12\cdot\sqrt{334285232879825411498534254159851729760457815228416}}\cr\approx \mathstrut & 0.430706295531231 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - x^30 - 3*x^28 + 20*x^26 - 12*x^24 - 64*x^22 + 160*x^20 + 64*x^18 - 704*x^16 + 256*x^14 + 2560*x^12 - 4096*x^10 - 3072*x^8 + 20480*x^6 - 12288*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 - 3*x^28 + 20*x^26 - 12*x^24 - 64*x^22 + 160*x^20 + 64*x^18 - 704*x^16 + 256*x^14 + 2560*x^12 - 4096*x^10 - 3072*x^8 + 20480*x^6 - 12288*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 - 3*x^28 + 20*x^26 - 12*x^24 - 64*x^22 + 160*x^20 + 64*x^18 - 704*x^16 + 256*x^14 + 2560*x^12 - 4096*x^10 - 3072*x^8 + 20480*x^6 - 12288*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 - 3*x^28 + 20*x^26 - 12*x^24 - 64*x^22 + 160*x^20 + 64*x^18 - 704*x^16 + 256*x^14 + 2560*x^12 - 4096*x^10 - 3072*x^8 + 20480*x^6 - 12288*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 $C_2^5:S_4$
Character table for $C_2^5:S_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.4.20308.1, \(\Q(\zeta_{12})\), 8.8.2137958654976.1, 8.0.26394551296.1, 8.0.33405603984.1, 8.8.26394551296.1, 8.8.133622415936.1, 8.0.1649659456.5, 8.0.2137958654976.1, 16.0.4570867210386787009560576.1, 16.16.18283468841547148038242304.1, 16.0.18283468841547148038242304.2, 16.0.2786689352468701118464.1, 16.0.18283468841547148038242304.1, 16.0.4570867210386787009560576.2, 16.0.17854950040573386756096.1

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 ${\href{/padicField/5.4.0.1}{4} }^{8}$ ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.6.0.1}{6} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.6.0.1}{6} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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 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 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.12.6.2$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(5077\) Copy content Toggle raw display $\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5077}$$x$$1$$1$$0$Trivial$[\ ]$
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}$
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}$