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Properties

Label 32.0.480...776.2
Degree $32$
Signature $[0, 16]$
Discriminant $4.810\times 10^{50}$
Root discriminant \(38.35\)
Ramified primes $2,3,13,17$
Class number $64$ (GRH)
Class group [2, 4, 8] (GRH)
Galois group $C_2^4:D_4$ (as 32T1369)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 3*x^28 + 13*x^24 - 108*x^20 - 384*x^16 - 1728*x^12 + 3328*x^8 + 12288*x^4 + 65536)
 
gp: K = bnfinit(y^32 + 3*y^28 + 13*y^24 - 108*y^20 - 384*y^16 - 1728*y^12 + 3328*y^8 + 12288*y^4 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 3*x^28 + 13*x^24 - 108*x^20 - 384*x^16 - 1728*x^12 + 3328*x^8 + 12288*x^4 + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 3*x^28 + 13*x^24 - 108*x^20 - 384*x^16 - 1728*x^12 + 3328*x^8 + 12288*x^4 + 65536)
 

\( x^{32} + 3x^{28} + 13x^{24} - 108x^{20} - 384x^{16} - 1728x^{12} + 3328x^{8} + 12288x^{4} + 65536 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(480960519379403029833827263813614000556122650443776\) \(\medspace = 2^{48}\cdot 3^{16}\cdot 13^{8}\cdot 17^{16}\) 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:  \(38.35\)
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:  $2^{2}3^{1/2}13^{1/2}17^{1/2}\approx 102.99514551666987$
Ramified primes:   \(2\), \(3\), \(13\), \(17\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $16$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}+\frac{1}{4}a^{5}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{10}+\frac{1}{8}a^{6}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{11}-\frac{1}{2}a^{9}+\frac{1}{16}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{96}a^{16}-\frac{1}{8}a^{13}-\frac{3}{32}a^{12}-\frac{1}{4}a^{10}+\frac{1}{8}a^{9}-\frac{7}{96}a^{8}+\frac{1}{4}a^{6}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{3}$, $\frac{1}{192}a^{17}-\frac{1}{16}a^{14}+\frac{5}{64}a^{13}-\frac{1}{8}a^{11}+\frac{1}{16}a^{10}+\frac{65}{192}a^{9}+\frac{1}{8}a^{7}-\frac{1}{16}a^{6}-\frac{1}{2}a^{5}-\frac{1}{8}a^{3}+\frac{1}{3}a$, $\frac{1}{192}a^{18}-\frac{3}{64}a^{14}-\frac{1}{8}a^{12}-\frac{7}{192}a^{10}-\frac{1}{2}a^{9}+\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{8}a^{4}-\frac{1}{6}a^{2}-\frac{1}{2}a$, $\frac{1}{384}a^{19}-\frac{3}{128}a^{15}+\frac{1}{16}a^{13}+\frac{89}{384}a^{11}-\frac{1}{4}a^{10}+\frac{7}{16}a^{9}+\frac{3}{16}a^{7}+\frac{1}{4}a^{6}-\frac{7}{16}a^{5}+\frac{1}{6}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{384}a^{20}-\frac{1}{384}a^{16}-\frac{1}{16}a^{14}+\frac{17}{384}a^{12}-\frac{1}{4}a^{11}+\frac{1}{16}a^{10}+\frac{1}{24}a^{8}+\frac{1}{4}a^{7}-\frac{1}{16}a^{6}-\frac{1}{3}a^{4}-\frac{1}{4}a^{3}+\frac{1}{3}$, $\frac{1}{768}a^{21}-\frac{1}{768}a^{17}-\frac{1}{32}a^{15}+\frac{17}{768}a^{13}-\frac{1}{8}a^{12}+\frac{1}{32}a^{11}-\frac{23}{48}a^{9}+\frac{1}{8}a^{8}+\frac{15}{32}a^{7}+\frac{1}{3}a^{5}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{6}a$, $\frac{1}{768}a^{22}-\frac{1}{768}a^{18}+\frac{17}{768}a^{14}-\frac{11}{48}a^{10}+\frac{1}{12}a^{6}+\frac{5}{12}a^{2}$, $\frac{1}{1536}a^{23}-\frac{1}{1536}a^{19}+\frac{17}{1536}a^{15}-\frac{11}{96}a^{11}-\frac{11}{24}a^{7}-\frac{7}{24}a^{3}$, $\frac{1}{9216}a^{24}-\frac{1}{1024}a^{20}-\frac{13}{3072}a^{16}-\frac{1}{16}a^{14}-\frac{7}{128}a^{12}-\frac{1}{4}a^{11}+\frac{1}{16}a^{10}-\frac{7}{24}a^{8}+\frac{1}{4}a^{7}-\frac{1}{16}a^{6}+\frac{5}{16}a^{4}-\frac{1}{4}a^{3}+\frac{1}{9}$, $\frac{1}{18432}a^{25}-\frac{1}{2048}a^{21}-\frac{13}{6144}a^{17}-\frac{1}{32}a^{15}+\frac{25}{256}a^{13}-\frac{1}{8}a^{12}-\frac{7}{32}a^{11}+\frac{11}{48}a^{9}+\frac{1}{8}a^{8}-\frac{9}{32}a^{7}-\frac{7}{32}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{4}{9}a$, $\frac{1}{36864}a^{26}+\frac{5}{12288}a^{22}-\frac{7}{4096}a^{18}+\frac{23}{384}a^{14}-\frac{1}{8}a^{12}+\frac{1}{8}a^{10}-\frac{1}{2}a^{9}+\frac{1}{8}a^{8}-\frac{37}{192}a^{6}-\frac{1}{2}a^{5}-\frac{1}{8}a^{4}-\frac{7}{18}a^{2}-\frac{1}{2}a$, $\frac{1}{73728}a^{27}+\frac{5}{24576}a^{23}-\frac{7}{8192}a^{19}+\frac{23}{768}a^{15}-\frac{1}{16}a^{13}+\frac{1}{16}a^{11}-\frac{1}{4}a^{10}-\frac{7}{16}a^{9}-\frac{37}{384}a^{7}+\frac{1}{4}a^{6}+\frac{7}{16}a^{5}-\frac{7}{36}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{147456}a^{28}-\frac{1}{147456}a^{24}-\frac{37}{49152}a^{20}-\frac{1}{3072}a^{16}-\frac{1}{8}a^{13}+\frac{1}{768}a^{12}-\frac{1}{4}a^{10}+\frac{1}{8}a^{9}-\frac{293}{768}a^{8}+\frac{1}{4}a^{6}-\frac{1}{8}a^{5}+\frac{1}{144}a^{4}-\frac{1}{4}a^{2}-\frac{1}{9}$, $\frac{1}{147456}a^{29}-\frac{1}{147456}a^{25}+\frac{9}{16384}a^{21}-\frac{5}{3072}a^{17}-\frac{1}{32}a^{15}+\frac{3}{128}a^{13}-\frac{1}{8}a^{12}-\frac{7}{32}a^{11}+\frac{107}{768}a^{9}+\frac{1}{8}a^{8}-\frac{9}{32}a^{7}+\frac{49}{144}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}+\frac{1}{18}a$, $\frac{1}{589824}a^{30}-\frac{1}{294912}a^{29}-\frac{1}{294912}a^{28}-\frac{1}{589824}a^{26}+\frac{1}{294912}a^{25}+\frac{1}{294912}a^{24}-\frac{37}{196608}a^{22}+\frac{37}{98304}a^{21}+\frac{37}{98304}a^{20}+\frac{31}{12288}a^{18}+\frac{1}{6144}a^{17}-\frac{31}{6144}a^{16}+\frac{121}{3072}a^{14}-\frac{1}{1536}a^{13}-\frac{121}{1536}a^{12}+\frac{1}{8}a^{11}+\frac{227}{3072}a^{10}+\frac{293}{1536}a^{9}+\frac{541}{1536}a^{8}-\frac{1}{8}a^{7}-\frac{287}{576}a^{6}+\frac{143}{288}a^{5}+\frac{143}{288}a^{4}+\frac{1}{8}a^{3}-\frac{1}{9}a^{2}-\frac{4}{9}a-\frac{5}{18}$, $\frac{1}{1179648}a^{31}-\frac{1}{589824}a^{29}-\frac{1}{294912}a^{28}-\frac{1}{1179648}a^{27}+\frac{1}{589824}a^{25}+\frac{1}{294912}a^{24}-\frac{37}{393216}a^{23}+\frac{37}{196608}a^{21}+\frac{37}{98304}a^{20}+\frac{31}{24576}a^{19}-\frac{31}{12288}a^{17}-\frac{31}{6144}a^{16}-\frac{71}{6144}a^{15}-\frac{1}{16}a^{14}-\frac{121}{3072}a^{13}+\frac{71}{1536}a^{12}-\frac{1117}{6144}a^{11}+\frac{1}{16}a^{10}-\frac{995}{3072}a^{9}+\frac{349}{1536}a^{8}-\frac{35}{1152}a^{7}-\frac{1}{16}a^{6}-\frac{145}{576}a^{5}-\frac{109}{288}a^{4}+\frac{7}{36}a^{3}-\frac{5}{36}a-\frac{5}{18}$ 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

$C_{2}\times C_{4}\times C_{8}$, which has order $64$ (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:   \( \frac{5}{294912} a^{30} - \frac{101}{294912} a^{26} - \frac{25}{98304} a^{22} - \frac{5}{2048} a^{18} + \frac{55}{1536} a^{14} + \frac{5}{512} a^{10} + \frac{35}{288} a^{6} - \frac{55}{36} a^{2} \)  (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:   $\frac{7}{294912}a^{30}-\frac{31}{294912}a^{26}+\frac{5}{98304}a^{22}+\frac{1}{12288}a^{18}+\frac{25}{1536}a^{14}-\frac{35}{1536}a^{10}-\frac{13}{576}a^{6}-\frac{35}{36}a^{2}-1$, $\frac{5}{1179648}a^{31}+\frac{43}{589824}a^{30}+\frac{29}{589824}a^{29}+\frac{11}{1179648}a^{27}+\frac{181}{589824}a^{26}+\frac{35}{589824}a^{25}+\frac{151}{393216}a^{23}-\frac{215}{196608}a^{22}+\frac{5}{65536}a^{21}-\frac{5}{12288}a^{19}-\frac{43}{4096}a^{18}-\frac{65}{12288}a^{17}-\frac{7}{6144}a^{15}-\frac{121}{3072}a^{14}-\frac{5}{1024}a^{13}-\frac{113}{6144}a^{11}+\frac{43}{1024}a^{10}-\frac{25}{3072}a^{9}+\frac{7}{576}a^{7}+\frac{301}{576}a^{6}+\frac{53}{576}a^{5}+\frac{1}{36}a^{3}+\frac{13}{9}a^{2}+\frac{5}{36}a$, $\frac{65}{589824}a^{30}-\frac{3}{32768}a^{29}+\frac{35}{294912}a^{28}+\frac{127}{589824}a^{26}+\frac{59}{294912}a^{25}+\frac{21}{32768}a^{24}+\frac{91}{196608}a^{22}+\frac{45}{32768}a^{21}+\frac{27}{32768}a^{20}-\frac{125}{12288}a^{18}+\frac{27}{2048}a^{17}-\frac{19}{2048}a^{16}-\frac{79}{3072}a^{14}-\frac{1}{512}a^{13}-\frac{27}{512}a^{12}-\frac{157}{3072}a^{10}-\frac{27}{512}a^{9}-\frac{45}{512}a^{8}+\frac{173}{576}a^{6}-\frac{21}{32}a^{5}+\frac{113}{288}a^{4}+\frac{19}{36}a^{2}-\frac{5}{9}a+\frac{3}{2}$, $\frac{1}{32768}a^{31}-\frac{1}{49152}a^{29}+\frac{1}{18432}a^{28}+\frac{5}{98304}a^{27}+\frac{3}{16384}a^{25}+\frac{5}{6144}a^{24}+\frac{43}{98304}a^{23}-\frac{25}{49152}a^{21}-\frac{5}{6144}a^{20}-\frac{49}{12288}a^{19}+\frac{7}{6144}a^{17}-\frac{1}{128}a^{16}-\frac{7}{1536}a^{15}-\frac{23}{768}a^{13}-\frac{31}{384}a^{12}-\frac{37}{1536}a^{11}+\frac{19}{768}a^{9}+\frac{1}{32}a^{8}+\frac{35}{192}a^{7}+\frac{11}{96}a^{5}+\frac{7}{18}a^{4}+\frac{1}{8}a^{3}+\frac{7}{6}a+\frac{7}{3}$, $\frac{1}{32768}a^{31}-\frac{1}{589824}a^{30}-\frac{29}{294912}a^{29}+\frac{35}{294912}a^{28}+\frac{5}{98304}a^{27}+\frac{97}{589824}a^{26}-\frac{163}{294912}a^{25}+\frac{21}{32768}a^{24}+\frac{43}{98304}a^{23}+\frac{5}{196608}a^{22}+\frac{113}{98304}a^{21}+\frac{27}{32768}a^{20}-\frac{49}{12288}a^{19}+\frac{1}{4096}a^{18}+\frac{89}{6144}a^{17}-\frac{19}{2048}a^{16}-\frac{7}{1536}a^{15}-\frac{17}{3072}a^{14}+\frac{115}{1536}a^{13}-\frac{27}{512}a^{12}-\frac{37}{1536}a^{11}-\frac{1}{1024}a^{10}+\frac{1}{1536}a^{9}-\frac{45}{512}a^{8}+\frac{35}{192}a^{7}-\frac{7}{576}a^{6}-\frac{137}{288}a^{5}+\frac{113}{288}a^{4}+\frac{1}{8}a^{3}+\frac{13}{36}a^{2}-\frac{55}{18}a+\frac{3}{2}$, $\frac{5}{589824}a^{31}+\frac{7}{196608}a^{30}+\frac{5}{24576}a^{29}-\frac{71}{294912}a^{28}-\frac{245}{589824}a^{27}+\frac{283}{589824}a^{26}+\frac{61}{73728}a^{25}-\frac{185}{294912}a^{24}+\frac{23}{196608}a^{23}-\frac{25}{196608}a^{22}-\frac{5}{8192}a^{21}+\frac{67}{98304}a^{20}-\frac{1}{6144}a^{19}-\frac{5}{3072}a^{18}-\frac{49}{2048}a^{17}+\frac{45}{2048}a^{16}+\frac{139}{3072}a^{15}-\frac{167}{3072}a^{14}-\frac{23}{256}a^{13}+\frac{125}{1536}a^{12}-\frac{49}{3072}a^{11}-\frac{49}{3072}a^{10}+\frac{1}{128}a^{9}-\frac{7}{512}a^{8}+\frac{11}{144}a^{7}+\frac{1}{16}a^{6}+\frac{65}{96}a^{5}-\frac{227}{288}a^{4}-\frac{14}{9}a^{3}+\frac{67}{36}a^{2}+\frac{61}{18}a-\frac{35}{18}$, $\frac{1}{49152}a^{31}-\frac{5}{98304}a^{30}+\frac{1}{24576}a^{29}-\frac{5}{147456}a^{28}-\frac{35}{147456}a^{27}-\frac{89}{294912}a^{26}+\frac{5}{73728}a^{25}-\frac{91}{147456}a^{24}-\frac{5}{16384}a^{23}+\frac{35}{98304}a^{22}-\frac{13}{8192}a^{21}-\frac{13}{16384}a^{20}-\frac{17}{3072}a^{19}+\frac{47}{12288}a^{18}-\frac{31}{3072}a^{17}+\frac{19}{3072}a^{16}+\frac{3}{256}a^{15}+\frac{37}{1536}a^{14}-\frac{11}{256}a^{13}+\frac{13}{256}a^{12}-\frac{1}{768}a^{11}+\frac{11}{1536}a^{10}+\frac{17}{384}a^{9}+\frac{65}{768}a^{8}+\frac{23}{96}a^{7}-\frac{11}{64}a^{6}+\frac{7}{24}a^{5}-\frac{29}{144}a^{4}-\frac{7}{18}a^{3}-\frac{25}{36}a^{2}+\frac{35}{18}a-\frac{22}{9}$, $\frac{1}{12288}a^{31}-\frac{3}{65536}a^{30}-\frac{77}{294912}a^{29}-\frac{17}{294912}a^{28}+\frac{47}{73728}a^{27}+\frac{59}{589824}a^{26}-\frac{11}{32768}a^{25}-\frac{143}{294912}a^{24}-\frac{5}{24576}a^{23}+\frac{45}{65536}a^{22}+\frac{49}{98304}a^{21}+\frac{85}{98304}a^{20}-\frac{67}{8192}a^{19}+\frac{27}{4096}a^{18}+\frac{43}{1536}a^{17}+\frac{17}{2048}a^{16}-\frac{97}{1536}a^{15}-\frac{1}{1024}a^{14}+\frac{77}{1536}a^{13}+\frac{107}{1536}a^{12}-\frac{5}{128}a^{11}-\frac{27}{1024}a^{10}+\frac{65}{1536}a^{9}-\frac{17}{512}a^{8}+\frac{33}{128}a^{7}-\frac{21}{64}a^{6}-\frac{83}{72}a^{5}-\frac{119}{288}a^{4}+\frac{85}{36}a^{3}+\frac{2}{9}a^{2}-\frac{11}{6}a-\frac{65}{18}$, $\frac{7}{393216}a^{31}-\frac{29}{589824}a^{30}+\frac{5}{65536}a^{29}-\frac{23}{147456}a^{28}+\frac{539}{1179648}a^{27}-\frac{35}{589824}a^{26}+\frac{275}{589824}a^{25}-\frac{89}{147456}a^{24}-\frac{25}{393216}a^{23}-\frac{5}{65536}a^{22}-\frac{65}{196608}a^{21}+\frac{35}{49152}a^{20}-\frac{7}{2048}a^{19}+\frac{65}{12288}a^{18}-\frac{113}{12288}a^{17}+\frac{29}{1536}a^{16}-\frac{299}{6144}a^{15}+\frac{5}{1024}a^{14}-\frac{139}{3072}a^{13}+\frac{61}{768}a^{12}+\frac{21}{2048}a^{11}+\frac{25}{3072}a^{10}+\frac{23}{3072}a^{9}-\frac{13}{768}a^{8}+\frac{3}{32}a^{7}-\frac{53}{576}a^{6}+\frac{55}{192}a^{5}-\frac{43}{72}a^{4}+\frac{119}{72}a^{3}-\frac{5}{36}a^{2}+\frac{65}{36}a-\frac{26}{9}$, $\frac{17}{196608}a^{31}-\frac{1}{65536}a^{30}-\frac{5}{24576}a^{29}-\frac{19}{98304}a^{28}+\frac{21}{65536}a^{27}+\frac{35}{196608}a^{26}-\frac{7}{24576}a^{25}-\frac{263}{294912}a^{24}+\frac{27}{65536}a^{23}+\frac{15}{65536}a^{22}-\frac{3}{8192}a^{21}-\frac{1}{32768}a^{20}-\frac{21}{2048}a^{19}+\frac{9}{4096}a^{18}+\frac{51}{2048}a^{17}+\frac{157}{6144}a^{16}-\frac{43}{1024}a^{15}-\frac{31}{1024}a^{14}+\frac{3}{128}a^{13}+\frac{61}{512}a^{12}-\frac{29}{1024}a^{11}-\frac{9}{1024}a^{10}+\frac{5}{128}a^{9}+\frac{101}{1536}a^{8}+\frac{31}{96}a^{7}-\frac{7}{64}a^{6}-\frac{83}{96}a^{5}-\frac{79}{96}a^{4}+\frac{3}{2}a^{3}+\frac{7}{6}a^{2}-\frac{2}{3}a-\frac{83}{18}$, $\frac{35}{589824}a^{31}-\frac{79}{589824}a^{30}+\frac{29}{147456}a^{29}-\frac{73}{294912}a^{28}+\frac{61}{589824}a^{27}-\frac{65}{589824}a^{26}-\frac{53}{147456}a^{25}+\frac{1}{32768}a^{24}-\frac{175}{196608}a^{23}-\frac{101}{196608}a^{22}-\frac{41}{49152}a^{21}-\frac{179}{98304}a^{20}-\frac{35}{4096}a^{19}+\frac{31}{3072}a^{18}-\frac{107}{6144}a^{17}+\frac{31}{2048}a^{16}-\frac{59}{3072}a^{15}+\frac{29}{3072}a^{14}+\frac{23}{768}a^{13}-\frac{25}{1536}a^{12}+\frac{35}{1024}a^{11}+\frac{227}{3072}a^{10}+\frac{19}{768}a^{9}+\frac{71}{512}a^{8}+\frac{245}{576}a^{7}-\frac{5}{18}a^{6}+\frac{211}{288}a^{5}-\frac{109}{288}a^{4}+\frac{10}{9}a^{3}+\frac{4}{9}a^{2}-\frac{14}{9}a+\frac{5}{2}$, $\frac{113}{1179648}a^{31}+\frac{5}{589824}a^{30}-\frac{25}{196608}a^{29}-\frac{5}{147456}a^{28}+\frac{607}{1179648}a^{27}+\frac{11}{589824}a^{26}-\frac{71}{196608}a^{25}-\frac{91}{147456}a^{24}-\frac{23}{131072}a^{23}+\frac{151}{196608}a^{22}+\frac{55}{196608}a^{21}-\frac{13}{16384}a^{20}-\frac{145}{12288}a^{19}-\frac{5}{6144}a^{18}+\frac{181}{12288}a^{17}+\frac{19}{3072}a^{16}-\frac{125}{2048}a^{15}-\frac{7}{3072}a^{14}+\frac{137}{3072}a^{13}+\frac{13}{256}a^{12}-\frac{13}{6144}a^{11}-\frac{113}{3072}a^{10}-\frac{1}{3072}a^{9}+\frac{65}{768}a^{8}+\frac{235}{576}a^{7}+\frac{7}{288}a^{6}-\frac{25}{64}a^{5}-\frac{29}{144}a^{4}+\frac{65}{36}a^{3}+\frac{1}{18}a^{2}-\frac{19}{12}a-\frac{13}{9}$, $\frac{17}{589824}a^{31}+\frac{1}{32768}a^{29}+\frac{1}{24576}a^{28}+\frac{13}{196608}a^{27}+\frac{23}{294912}a^{25}-\frac{1}{24576}a^{24}+\frac{35}{196608}a^{23}+\frac{83}{98304}a^{21}+\frac{27}{8192}a^{20}-\frac{37}{24576}a^{19}-\frac{1}{2048}a^{17}+\frac{1}{1536}a^{16}-\frac{29}{3072}a^{15}+\frac{13}{1536}a^{13}+\frac{3}{64}a^{12}-\frac{61}{3072}a^{11}-\frac{31}{512}a^{9}-\frac{91}{384}a^{8}-\frac{23}{1152}a^{7}-\frac{1}{96}a^{5}-\frac{5}{24}a^{4}+\frac{1}{6}a^{3}-\frac{5}{9}a-3$, $\frac{1}{294912}a^{31}-\frac{13}{147456}a^{30}+\frac{1}{12288}a^{29}+\frac{3}{32768}a^{27}-\frac{23}{147456}a^{26}-\frac{1}{36864}a^{25}+\frac{13}{32768}a^{23}-\frac{83}{49152}a^{22}+\frac{11}{12288}a^{21}-\frac{7}{24576}a^{19}+\frac{27}{4096}a^{18}-\frac{37}{6144}a^{17}-\frac{1}{256}a^{15}-\frac{1}{768}a^{14}+\frac{7}{768}a^{13}-\frac{29}{1536}a^{11}+\frac{27}{256}a^{10}-\frac{13}{192}a^{9}+\frac{7}{1152}a^{7}-\frac{61}{576}a^{6}+\frac{3}{32}a^{5}+\frac{5}{24}a^{3}+\frac{13}{36}a^{2}-\frac{7}{9}a$, $\frac{7}{1179648}a^{31}+\frac{77}{589824}a^{30}-\frac{23}{196608}a^{29}+\frac{41}{147456}a^{28}-\frac{47}{131072}a^{27}+\frac{227}{589824}a^{26}+\frac{37}{589824}a^{25}+\frac{29}{49152}a^{24}-\frac{35}{393216}a^{23}-\frac{49}{196608}a^{22}+\frac{89}{196608}a^{21}+\frac{19}{49152}a^{20}+\frac{43}{24576}a^{19}-\frac{17}{1024}a^{18}+\frac{53}{4096}a^{17}-\frac{73}{3072}a^{16}+\frac{275}{6144}a^{15}-\frac{143}{3072}a^{14}-\frac{65}{3072}a^{13}-\frac{25}{768}a^{12}+\frac{101}{6144}a^{11}-\frac{3}{1024}a^{10}-\frac{53}{1024}a^{9}-\frac{29}{768}a^{8}-\frac{59}{1152}a^{7}+\frac{23}{36}a^{6}-\frac{35}{64}a^{5}+\frac{77}{144}a^{4}-\frac{19}{12}a^{3}+\frac{59}{36}a^{2}+\frac{43}{36}a-\frac{2}{3}$ 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:  \( 950005676337.5021 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 950005676337.5021 \cdot 64}{12\cdot\sqrt{480960519379403029833827263813614000556122650443776}}\cr\approx \mathstrut & 1.36316399913236 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + 3*x^28 + 13*x^24 - 108*x^20 - 384*x^16 - 1728*x^12 + 3328*x^8 + 12288*x^4 + 65536)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^32 + 3*x^28 + 13*x^24 - 108*x^20 - 384*x^16 - 1728*x^12 + 3328*x^8 + 12288*x^4 + 65536, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 + 3*x^28 + 13*x^24 - 108*x^20 - 384*x^16 - 1728*x^12 + 3328*x^8 + 12288*x^4 + 65536);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 3*x^28 + 13*x^24 - 108*x^20 - 384*x^16 - 1728*x^12 + 3328*x^8 + 12288*x^4 + 65536);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^4:D_4$ (as 32T1369):

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 128
The 56 conjugacy class representatives for $C_2^4:D_4$
Character table for $C_2^4:D_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{51}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-51}) \), 4.0.2312.1, 4.0.83232.1, 4.4.1082016.2, 4.4.30056.2, 4.0.120224.2, 4.0.270504.2, 4.4.20808.1, 4.4.9248.1, \(\Q(i, \sqrt{51})\), \(\Q(\sqrt{-3}, \sqrt{-17})\), \(\Q(\sqrt{3}, \sqrt{17})\), 4.0.541008.2, 4.4.60112.1, \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{17})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{3}, \sqrt{-17})\), 4.4.33813.1, 4.0.3757.1, 8.0.292689656064.39, 8.8.292689656064.1, 8.0.292689656064.9, 8.0.1731891456.1, 8.0.292689656064.35, 8.0.3613452544.3, 8.0.1143318969.1, 8.0.27710263296.13, 8.8.4683034497024.3, 8.0.4683034497024.26, 8.8.27710263296.2, 8.0.4683034497024.78, 8.0.4683034497024.33, 8.0.4683034497024.4, 8.0.4683034497024.35, 8.0.57815240704.13, 8.0.4683034497024.123, 8.8.4683034497024.7, 8.8.57815240704.2, 8.0.57815240704.11, 8.0.4683034497024.73, 8.0.27710263296.14, 8.0.342102016.5, 8.0.1170758624256.15, 8.0.432972864.2, 8.0.73172414016.8, 8.0.6927565824.3, 8.0.1170758624256.25, 8.0.73172414016.11, 8.8.73172414016.2, 8.8.1170758624256.1, 8.0.903363136.2, 8.0.1170758624256.22, 8.0.14453810176.7, 8.0.73172414016.10, 16.0.85667234766862611972096.1, 16.0.21930812100316828664856576.10, 16.0.21930812100316828664856576.9, 16.0.21930812100316828664856576.12, 16.16.21930812100316828664856576.2, 16.0.21930812100316828664856576.5, 16.0.21930812100316828664856576.3, 16.0.21930812100316828664856576.6, 16.0.767858691933644783616.8, 16.0.21930812100316828664856576.11, 16.0.21930812100316828664856576.14, 16.0.3342602057661458415616.3, 16.0.21930812100316828664856576.7, 16.0.5354202172928913248256.1, 16.0.1370675756269801791553536.2

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: 32.0.480960519379403029833827263813614000556122650443776.1

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 ${\href{/padicField/5.4.0.1}{4} }^{8}$ ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ R R ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.2.0.1}{2} }^{16}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{16}$ ${\href{/padicField/47.2.0.1}{2} }^{16}$ ${\href{/padicField/53.2.0.1}{2} }^{16}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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}$
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}$
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 3.8.4.1$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}$
\(13\) Copy content Toggle raw display 13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
\(17\) Copy content Toggle raw display 17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$