Properties

Label 32.0.766...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $7.670\times 10^{45}$
Root discriminant \(27.16\)
Ramified primes $2,5,17$
Class number $5$ (GRH)
Class group [5] (GRH)
Galois group $C_4^2:C_2^2$ (as 32T204)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 + 10*x^30 - 20*x^29 + 34*x^28 - 32*x^27 - 8*x^26 + 112*x^25 - 308*x^24 + 608*x^23 - 880*x^22 + 944*x^21 - 536*x^20 - 672*x^19 + 2960*x^18 - 6176*x^17 + 9744*x^16 - 12352*x^15 + 11840*x^14 - 5376*x^13 - 8576*x^12 + 30208*x^11 - 56320*x^10 + 77824*x^9 - 78848*x^8 + 57344*x^7 - 8192*x^6 - 65536*x^5 + 139264*x^4 - 163840*x^3 + 163840*x^2 - 131072*x + 65536)
 
gp: K = bnfinit(y^32 - 4*y^31 + 10*y^30 - 20*y^29 + 34*y^28 - 32*y^27 - 8*y^26 + 112*y^25 - 308*y^24 + 608*y^23 - 880*y^22 + 944*y^21 - 536*y^20 - 672*y^19 + 2960*y^18 - 6176*y^17 + 9744*y^16 - 12352*y^15 + 11840*y^14 - 5376*y^13 - 8576*y^12 + 30208*y^11 - 56320*y^10 + 77824*y^9 - 78848*y^8 + 57344*y^7 - 8192*y^6 - 65536*y^5 + 139264*y^4 - 163840*y^3 + 163840*y^2 - 131072*y + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 4*x^31 + 10*x^30 - 20*x^29 + 34*x^28 - 32*x^27 - 8*x^26 + 112*x^25 - 308*x^24 + 608*x^23 - 880*x^22 + 944*x^21 - 536*x^20 - 672*x^19 + 2960*x^18 - 6176*x^17 + 9744*x^16 - 12352*x^15 + 11840*x^14 - 5376*x^13 - 8576*x^12 + 30208*x^11 - 56320*x^10 + 77824*x^9 - 78848*x^8 + 57344*x^7 - 8192*x^6 - 65536*x^5 + 139264*x^4 - 163840*x^3 + 163840*x^2 - 131072*x + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 4*x^31 + 10*x^30 - 20*x^29 + 34*x^28 - 32*x^27 - 8*x^26 + 112*x^25 - 308*x^24 + 608*x^23 - 880*x^22 + 944*x^21 - 536*x^20 - 672*x^19 + 2960*x^18 - 6176*x^17 + 9744*x^16 - 12352*x^15 + 11840*x^14 - 5376*x^13 - 8576*x^12 + 30208*x^11 - 56320*x^10 + 77824*x^9 - 78848*x^8 + 57344*x^7 - 8192*x^6 - 65536*x^5 + 139264*x^4 - 163840*x^3 + 163840*x^2 - 131072*x + 65536)
 

\( x^{32} - 4 x^{31} + 10 x^{30} - 20 x^{29} + 34 x^{28} - 32 x^{27} - 8 x^{26} + 112 x^{25} - 308 x^{24} + 608 x^{23} - 880 x^{22} + 944 x^{21} - 536 x^{20} - 672 x^{19} + 2960 x^{18} + \cdots + 65536 \) 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:   \(7669926418924454281216000000000000000000000000\) \(\medspace = 2^{64}\cdot 5^{24}\cdot 17^{8}\) 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:  \(27.16\)
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:  $2^{2}5^{3/4}17^{1/2}\approx 55.14573827051111$
Ramified primes:   \(2\), \(5\), \(17\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$, $\frac{1}{8}a^{15}$, $\frac{1}{16}a^{16}$, $\frac{1}{16}a^{17}$, $\frac{1}{32}a^{18}-\frac{1}{16}a^{14}-\frac{1}{8}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{64}a^{19}-\frac{1}{32}a^{17}+\frac{1}{32}a^{15}-\frac{1}{16}a^{11}+\frac{1}{8}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{128}a^{20}-\frac{1}{64}a^{18}-\frac{1}{32}a^{17}+\frac{1}{64}a^{16}-\frac{1}{16}a^{15}-\frac{1}{32}a^{12}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}+\frac{1}{16}a^{8}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}$, $\frac{1}{128}a^{21}-\frac{1}{64}a^{17}+\frac{1}{32}a^{15}-\frac{1}{16}a^{14}-\frac{1}{32}a^{13}-\frac{1}{16}a^{11}+\frac{1}{16}a^{9}-\frac{1}{8}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{256}a^{22}-\frac{1}{128}a^{18}-\frac{1}{32}a^{17}+\frac{1}{64}a^{16}-\frac{1}{32}a^{15}-\frac{1}{64}a^{14}-\frac{1}{16}a^{13}-\frac{1}{32}a^{12}-\frac{1}{8}a^{11}+\frac{1}{32}a^{10}-\frac{1}{8}a^{9}-\frac{1}{16}a^{6}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}$, $\frac{1}{512}a^{23}-\frac{1}{256}a^{21}+\frac{1}{256}a^{19}+\frac{1}{64}a^{16}+\frac{7}{128}a^{15}+\frac{1}{32}a^{14}+\frac{1}{64}a^{11}-\frac{1}{8}a^{10}-\frac{1}{32}a^{9}+\frac{1}{32}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}$, $\frac{1}{1024}a^{24}-\frac{1}{512}a^{22}-\frac{1}{256}a^{21}+\frac{1}{512}a^{20}-\frac{1}{128}a^{19}-\frac{1}{64}a^{18}-\frac{1}{256}a^{16}+\frac{3}{64}a^{15}+\frac{1}{64}a^{13}-\frac{7}{128}a^{12}-\frac{5}{64}a^{10}-\frac{1}{32}a^{9}+\frac{1}{64}a^{8}+\frac{1}{8}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{1024}a^{25}-\frac{1}{512}a^{21}+\frac{1}{256}a^{19}+\frac{1}{128}a^{18}+\frac{7}{256}a^{17}-\frac{1}{32}a^{16}-\frac{1}{128}a^{15}-\frac{1}{32}a^{14}+\frac{1}{128}a^{13}-\frac{1}{16}a^{12}-\frac{1}{8}a^{11}-\frac{1}{64}a^{9}-\frac{1}{16}a^{8}-\frac{3}{32}a^{7}+\frac{1}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2048}a^{26}-\frac{1}{1024}a^{22}-\frac{1}{256}a^{21}+\frac{1}{512}a^{20}-\frac{1}{256}a^{19}-\frac{1}{512}a^{18}-\frac{3}{128}a^{17}+\frac{7}{256}a^{16}+\frac{1}{64}a^{15}+\frac{1}{256}a^{14}-\frac{1}{64}a^{13}+\frac{1}{16}a^{11}-\frac{9}{128}a^{10}-\frac{1}{16}a^{9}-\frac{3}{64}a^{8}+\frac{7}{32}a^{7}+\frac{1}{8}a^{5}$, $\frac{1}{4096}a^{27}-\frac{1}{2048}a^{25}+\frac{1}{2048}a^{23}+\frac{1}{512}a^{20}+\frac{7}{1024}a^{19}-\frac{3}{256}a^{18}+\frac{1}{64}a^{17}-\frac{1}{64}a^{16}-\frac{31}{512}a^{15}-\frac{1}{64}a^{14}+\frac{7}{256}a^{13}+\frac{1}{32}a^{12}-\frac{31}{256}a^{11}-\frac{1}{64}a^{10}+\frac{3}{32}a^{9}+\frac{3}{64}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8192}a^{28}-\frac{1}{4096}a^{26}-\frac{1}{2048}a^{25}+\frac{1}{4096}a^{24}-\frac{1}{1024}a^{23}-\frac{1}{512}a^{22}-\frac{1}{2048}a^{20}+\frac{3}{512}a^{19}-\frac{1}{64}a^{18}-\frac{7}{512}a^{17}-\frac{7}{1024}a^{16}+\frac{1}{32}a^{15}-\frac{5}{512}a^{14}-\frac{1}{256}a^{13}-\frac{31}{512}a^{12}-\frac{7}{64}a^{11}+\frac{3}{32}a^{10}-\frac{7}{64}a^{9}+\frac{1}{16}a^{8}-\frac{3}{32}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{8192}a^{29}-\frac{1}{4096}a^{25}+\frac{1}{2048}a^{23}+\frac{1}{1024}a^{22}+\frac{7}{2048}a^{21}-\frac{1}{256}a^{20}-\frac{1}{1024}a^{19}+\frac{3}{256}a^{18}+\frac{17}{1024}a^{17}-\frac{3}{128}a^{16}-\frac{1}{64}a^{15}-\frac{1}{16}a^{14}-\frac{17}{512}a^{13}+\frac{3}{128}a^{12}-\frac{3}{256}a^{11}+\frac{1}{128}a^{10}+\frac{1}{32}a^{9}+\frac{3}{64}a^{8}-\frac{1}{4}a^{7}-\frac{1}{16}a^{6}+\frac{1}{8}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{16384}a^{30}-\frac{1}{8192}a^{26}-\frac{1}{2048}a^{25}+\frac{1}{4096}a^{24}-\frac{1}{2048}a^{23}-\frac{1}{4096}a^{22}-\frac{3}{1024}a^{21}+\frac{7}{2048}a^{20}+\frac{1}{512}a^{19}+\frac{1}{2048}a^{18}-\frac{9}{512}a^{17}-\frac{1}{32}a^{16}+\frac{5}{128}a^{15}-\frac{41}{1024}a^{14}+\frac{7}{128}a^{13}-\frac{19}{512}a^{12}+\frac{23}{256}a^{11}+\frac{1}{16}a^{10}-\frac{7}{64}a^{9}+\frac{1}{16}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}$, $\frac{1}{32768}a^{31}-\frac{1}{16384}a^{29}+\frac{1}{16384}a^{27}+\frac{1}{4096}a^{24}+\frac{7}{8192}a^{23}-\frac{3}{2048}a^{22}+\frac{1}{512}a^{21}-\frac{1}{512}a^{20}-\frac{31}{4096}a^{19}-\frac{1}{512}a^{18}-\frac{25}{2048}a^{17}+\frac{1}{256}a^{16}-\frac{95}{2048}a^{15}-\frac{17}{512}a^{14}+\frac{3}{256}a^{13}-\frac{13}{512}a^{12}+\frac{3}{64}a^{11}-\frac{1}{8}a^{10}+\frac{3}{64}a^{8}+\frac{1}{16}a^{7}-\frac{1}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{5}$, which has order $5$ (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:   \( -\frac{1}{512} a^{31} - \frac{13}{4096} a^{30} + \frac{9}{2048} a^{29} - \frac{1}{64} a^{28} + \frac{17}{512} a^{27} - \frac{145}{2048} a^{26} + \frac{7}{1024} a^{25} + \frac{15}{512} a^{24} - \frac{1}{4} a^{23} + \frac{609}{1024} a^{22} - \frac{531}{512} a^{21} + \frac{281}{256} a^{20} - \frac{255}{256} a^{19} - \frac{143}{512} a^{18} + \frac{697}{256} a^{17} - \frac{1629}{256} a^{16} + \frac{685}{64} a^{15} - \frac{3681}{256} a^{14} + \frac{1927}{128} a^{13} - \frac{601}{64} a^{12} - \frac{35}{4} a^{11} + \frac{963}{32} a^{10} - \frac{545}{8} a^{9} + \frac{6241}{64} a^{8} - \frac{213}{2} a^{7} + \frac{229}{4} a^{6} - 40 a^{5} - 110 a^{4} + \frac{739}{4} a^{3} - 226 a^{2} + 168 a - 232 \)  (order $40$) 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:   $\frac{5}{1024}a^{31}-\frac{43}{4096}a^{30}+\frac{53}{2048}a^{29}-\frac{3}{64}a^{28}+\frac{33}{512}a^{27}-\frac{15}{2048}a^{26}-\frac{93}{1024}a^{25}+\frac{189}{512}a^{24}-\frac{201}{256}a^{23}+\frac{1271}{1024}a^{22}-\frac{711}{512}a^{21}+\frac{279}{256}a^{20}+\frac{109}{256}a^{19}-\frac{1713}{512}a^{18}+\frac{1957}{256}a^{17}-\frac{3251}{256}a^{16}+\frac{1081}{64}a^{15}-\frac{4463}{256}a^{14}+\frac{1339}{128}a^{13}+\frac{521}{64}a^{12}-\frac{139}{4}a^{11}+\frac{2349}{32}a^{10}-\frac{827}{8}a^{9}+\frac{7055}{64}a^{8}-\frac{147}{2}a^{7}+\frac{107}{4}a^{6}+100a^{5}-178a^{4}+216a^{3}-190a^{2}+184a-23$, $\frac{65}{16384}a^{31}-\frac{161}{16384}a^{30}+\frac{161}{8192}a^{29}-\frac{161}{4096}a^{28}+\frac{483}{8192}a^{27}-\frac{123}{8192}a^{26}-\frac{161}{2048}a^{25}+\frac{1127}{4096}a^{24}-\frac{2737}{4096}a^{23}+\frac{4669}{4096}a^{22}-\frac{349}{256}a^{21}+\frac{2415}{2048}a^{20}-\frac{161}{2048}a^{19}-\frac{5313}{2048}a^{18}+\frac{6923}{1024}a^{17}-\frac{6129}{512}a^{16}+\frac{17227}{1024}a^{15}-\frac{19159}{1024}a^{14}+\frac{1771}{128}a^{13}+\frac{1127}{512}a^{12}-\frac{443}{16}a^{11}+\frac{2093}{32}a^{10}-\frac{3381}{32}a^{9}+\frac{7889}{64}a^{8}-\frac{805}{8}a^{7}+\frac{1033}{16}a^{6}+\frac{161}{4}a^{5}-\frac{1449}{8}a^{4}+\frac{483}{2}a^{3}-\frac{483}{2}a^{2}+264a-162$, $\frac{15}{16384}a^{31}-\frac{11}{16384}a^{30}+\frac{51}{8192}a^{29}-\frac{31}{4096}a^{28}+\frac{45}{8192}a^{27}+\frac{63}{8192}a^{26}-\frac{25}{2048}a^{25}+\frac{385}{4096}a^{24}-\frac{479}{4096}a^{23}+\frac{415}{4096}a^{22}-\frac{13}{512}a^{21}-\frac{183}{2048}a^{20}+\frac{1033}{2048}a^{19}-\frac{1539}{2048}a^{18}+\frac{905}{1024}a^{17}-\frac{373}{512}a^{16}+\frac{69}{1024}a^{15}+\frac{1307}{1024}a^{14}-\frac{27}{8}a^{13}+\frac{3041}{512}a^{12}-\frac{113}{16}a^{11}+8a^{10}+\frac{73}{32}a^{9}-\frac{417}{32}a^{8}+\frac{217}{8}a^{7}-\frac{605}{16}a^{6}+\frac{239}{4}a^{5}+\frac{25}{8}a^{4}-\frac{51}{2}a^{3}+\frac{103}{2}a^{2}-80a+137$, $\frac{17}{16384}a^{31}-\frac{3}{4096}a^{30}+\frac{5}{4096}a^{29}+\frac{33}{8192}a^{28}-\frac{101}{8192}a^{27}+\frac{147}{4096}a^{26}-\frac{175}{4096}a^{25}+\frac{211}{4096}a^{24}+\frac{99}{4096}a^{23}-\frac{21}{128}a^{22}+\frac{837}{2048}a^{21}-\frac{1305}{2048}a^{20}+\frac{1709}{2048}a^{19}-\frac{81}{128}a^{18}-\frac{19}{256}a^{17}+\frac{1505}{1024}a^{16}-\frac{3661}{1024}a^{15}+\frac{3133}{512}a^{14}-\frac{4267}{512}a^{13}+\frac{4539}{512}a^{12}-\frac{143}{32}a^{11}-\frac{125}{32}a^{10}+\frac{1313}{64}a^{9}-\frac{293}{8}a^{8}+\frac{1649}{32}a^{7}-\frac{175}{4}a^{6}+\frac{245}{8}a^{5}+\frac{203}{8}a^{4}-\frac{113}{2}a^{3}+\frac{183}{2}a^{2}-83a+80$, $\frac{51}{8192}a^{31}-\frac{231}{16384}a^{30}+\frac{231}{8192}a^{29}-\frac{423}{8192}a^{28}+\frac{295}{4096}a^{27}+\frac{49}{8192}a^{26}-\frac{581}{4096}a^{25}+\frac{869}{2048}a^{24}-\frac{1869}{2048}a^{23}+\frac{5959}{4096}a^{22}-\frac{3231}{2048}a^{21}+\frac{1163}{1024}a^{20}+\frac{289}{512}a^{19}-\frac{8443}{2048}a^{18}+\frac{9579}{1024}a^{17}-\frac{15855}{1024}a^{16}+\frac{10519}{512}a^{15}-\frac{21639}{1024}a^{14}+\frac{6433}{512}a^{13}+\frac{2605}{256}a^{12}-\frac{1353}{32}a^{11}+\frac{1409}{16}a^{10}-\frac{8359}{64}a^{9}+\frac{9073}{64}a^{8}-\frac{3155}{32}a^{7}+\frac{837}{16}a^{6}+\frac{663}{8}a^{5}-\frac{1853}{8}a^{4}+289a^{3}-262a^{2}+293a-153$, $\frac{1}{512}a^{31}+\frac{13}{4096}a^{30}-\frac{9}{2048}a^{29}+\frac{1}{64}a^{28}-\frac{17}{512}a^{27}+\frac{145}{2048}a^{26}-\frac{7}{1024}a^{25}-\frac{15}{512}a^{24}+\frac{1}{4}a^{23}-\frac{609}{1024}a^{22}+\frac{531}{512}a^{21}-\frac{281}{256}a^{20}+\frac{255}{256}a^{19}+\frac{143}{512}a^{18}-\frac{697}{256}a^{17}+\frac{1629}{256}a^{16}-\frac{685}{64}a^{15}+\frac{3681}{256}a^{14}-\frac{1927}{128}a^{13}+\frac{601}{64}a^{12}+\frac{35}{4}a^{11}-\frac{963}{32}a^{10}+\frac{545}{8}a^{9}-\frac{6241}{64}a^{8}+\frac{213}{2}a^{7}-\frac{229}{4}a^{6}+40a^{5}+110a^{4}-\frac{739}{4}a^{3}+226a^{2}-168a+231$, $\frac{147}{16384}a^{31}-\frac{505}{16384}a^{30}+\frac{505}{8192}a^{29}-\frac{505}{4096}a^{28}+\frac{1515}{8192}a^{27}-\frac{763}{8192}a^{26}-\frac{505}{2048}a^{25}+\frac{3535}{4096}a^{24}-\frac{8585}{4096}a^{23}+\frac{14645}{4096}a^{22}-\frac{4533}{1024}a^{21}+\frac{7575}{2048}a^{20}-\frac{505}{2048}a^{19}-\frac{16665}{2048}a^{18}+\frac{21715}{1024}a^{17}-\frac{19219}{512}a^{16}+\frac{54035}{1024}a^{15}-\frac{60095}{1024}a^{14}+\frac{5555}{128}a^{13}+\frac{3535}{512}a^{12}-\frac{23411}{256}a^{11}+\frac{6565}{32}a^{10}-\frac{10605}{32}a^{9}+\frac{24745}{64}a^{8}-\frac{2525}{8}a^{7}+155a^{6}+\frac{505}{4}a^{5}-\frac{4545}{8}a^{4}+\frac{1515}{2}a^{3}-\frac{1515}{2}a^{2}+708a-506$, $\frac{1}{4096}a^{31}+\frac{51}{16384}a^{30}-\frac{49}{8192}a^{29}+\frac{123}{8192}a^{28}-\frac{53}{2048}a^{27}+\frac{295}{8192}a^{26}+\frac{41}{4096}a^{25}-\frac{141}{2048}a^{24}+\frac{31}{128}a^{23}-\frac{1951}{4096}a^{22}+\frac{1467}{2048}a^{21}-\frac{719}{1024}a^{20}+\frac{439}{1024}a^{19}+\frac{1291}{2048}a^{18}-\frac{2529}{1024}a^{17}+\frac{5131}{1024}a^{16}-\frac{1989}{256}a^{15}+\frac{9895}{1024}a^{14}-\frac{4589}{512}a^{13}+\frac{879}{256}a^{12}+\frac{151}{16}a^{11}-\frac{3285}{128}a^{10}+\frac{1563}{32}a^{9}-\frac{4083}{64}a^{8}+\frac{507}{8}a^{7}-\frac{515}{16}a^{6}+5a^{5}+\frac{161}{2}a^{4}-\frac{241}{2}a^{3}+\frac{279}{2}a^{2}-110a+118$, $\frac{41}{8192}a^{31}-\frac{259}{16384}a^{30}+\frac{261}{8192}a^{29}-\frac{281}{4096}a^{28}+\frac{439}{4096}a^{27}-\frac{533}{8192}a^{26}-\frac{385}{4096}a^{25}+\frac{1683}{4096}a^{24}-\frac{2289}{2048}a^{23}+\frac{8203}{4096}a^{22}-\frac{5357}{2048}a^{21}+\frac{5077}{2048}a^{20}-\frac{439}{512}a^{19}-\frac{7487}{2048}a^{18}+\frac{11303}{1024}a^{17}-\frac{10601}{512}a^{16}+\frac{15575}{512}a^{15}-\frac{36565}{1024}a^{14}+\frac{15229}{512}a^{13}-\frac{2165}{512}a^{12}-\frac{333}{8}a^{11}+\frac{1739}{16}a^{10}-\frac{12009}{64}a^{9}+\frac{14701}{64}a^{8}-\frac{413}{2}a^{7}+\frac{2101}{16}a^{6}+\frac{299}{8}a^{5}-310a^{4}+433a^{3}-465a^{2}+465a-325$, $\frac{29}{8192}a^{31}-\frac{101}{8192}a^{30}+\frac{41}{2048}a^{29}-\frac{41}{1024}a^{28}+\frac{123}{2048}a^{27}-\frac{81}{4096}a^{26}-\frac{117}{1024}a^{25}+\frac{593}{2048}a^{24}-\frac{179}{256}a^{23}+\frac{2435}{2048}a^{22}-\frac{1461}{1024}a^{21}+\frac{1125}{1024}a^{20}-\frac{3}{512}a^{19}-\frac{2915}{1024}a^{18}+\frac{929}{128}a^{17}-\frac{3259}{256}a^{16}+\frac{4545}{256}a^{15}-\frac{10005}{512}a^{14}+\frac{451}{32}a^{13}+\frac{821}{256}a^{12}-\frac{8027}{256}a^{11}+\frac{8691}{128}a^{10}-\frac{1817}{16}a^{9}+\frac{4227}{32}a^{8}-\frac{429}{4}a^{7}+\frac{487}{8}a^{6}+\frac{65}{4}a^{5}-\frac{795}{4}a^{4}+265a^{3}-265a^{2}+274a-251$, $\frac{15}{32768}a^{31}-\frac{53}{8192}a^{30}+\frac{161}{16384}a^{29}-\frac{11}{512}a^{28}+\frac{599}{16384}a^{27}-\frac{161}{4096}a^{26}-\frac{5}{128}a^{25}+\frac{453}{4096}a^{24}-\frac{2711}{8192}a^{23}+\frac{651}{1024}a^{22}-\frac{231}{256}a^{21}+\frac{845}{1024}a^{20}-\frac{1993}{4096}a^{19}-\frac{861}{1024}a^{18}+\frac{6401}{2048}a^{17}-\frac{799}{128}a^{16}+\frac{19623}{2048}a^{15}-\frac{189}{16}a^{14}+\frac{2763}{256}a^{13}-\frac{1941}{512}a^{12}-\frac{715}{64}a^{11}+\frac{3659}{128}a^{10}-\frac{1819}{32}a^{9}+\frac{587}{8}a^{8}-\frac{1095}{16}a^{7}+37a^{6}-\frac{43}{2}a^{5}-88a^{4}+132a^{3}-139a^{2}+118a-162$, $\frac{87}{32768}a^{31}-\frac{433}{16384}a^{30}+\frac{703}{16384}a^{29}-\frac{401}{4096}a^{28}+\frac{2751}{16384}a^{27}-\frac{1439}{8192}a^{26}-\frac{585}{4096}a^{25}+\frac{63}{128}a^{24}-\frac{12727}{8192}a^{23}+\frac{12351}{4096}a^{22}-\frac{8817}{2048}a^{21}+\frac{8423}{2048}a^{20}-\frac{10085}{4096}a^{19}-\frac{8257}{2048}a^{18}+\frac{31203}{2048}a^{17}-\frac{15683}{512}a^{16}+\frac{96687}{2048}a^{15}-\frac{59797}{1024}a^{14}+\frac{27507}{512}a^{13}-\frac{5045}{256}a^{12}-\frac{6809}{128}a^{11}+\frac{9391}{64}a^{10}-\frac{18351}{64}a^{9}+\frac{11955}{32}a^{8}-\frac{11495}{32}a^{7}+\frac{1731}{8}a^{6}-\frac{321}{4}a^{5}-\frac{3635}{8}a^{4}+\frac{2751}{4}a^{3}-758a^{2}+686a-778$, $\frac{99}{32768}a^{31}-\frac{207}{16384}a^{30}+\frac{445}{16384}a^{29}-\frac{59}{1024}a^{28}+\frac{1527}{16384}a^{27}-\frac{601}{8192}a^{26}-\frac{33}{512}a^{25}+\frac{173}{512}a^{24}-\frac{7627}{8192}a^{23}+\frac{6989}{4096}a^{22}-\frac{2343}{1024}a^{21}+\frac{4507}{2048}a^{20}-\frac{3561}{4096}a^{19}-\frac{5947}{2048}a^{18}+\frac{18729}{2048}a^{17}-\frac{557}{32}a^{16}+\frac{52935}{2048}a^{15}-\frac{31431}{1024}a^{14}+\frac{419}{16}a^{13}-\frac{1331}{256}a^{12}-\frac{2225}{64}a^{11}+\frac{5821}{64}a^{10}-\frac{5047}{32}a^{9}+\frac{12585}{64}a^{8}-\frac{709}{4}a^{7}+\frac{1649}{16}a^{6}+\frac{255}{8}a^{5}-\frac{2049}{8}a^{4}+\frac{737}{2}a^{3}-\frac{777}{2}a^{2}+364a-262$, $\frac{35}{16384}a^{31}-\frac{69}{16384}a^{30}+\frac{23}{4096}a^{29}-\frac{59}{4096}a^{28}+\frac{191}{8192}a^{27}+\frac{5}{8192}a^{26}-\frac{131}{4096}a^{25}+\frac{327}{4096}a^{24}-\frac{1019}{4096}a^{23}+\frac{1881}{4096}a^{22}-\frac{1143}{2048}a^{21}+\frac{1131}{2048}a^{20}-\frac{503}{2048}a^{19}-\frac{1705}{2048}a^{18}+\frac{335}{128}a^{17}-\frac{2555}{512}a^{16}+\frac{7599}{1024}a^{15}-\frac{9055}{1024}a^{14}+\frac{3883}{512}a^{13}-\frac{829}{512}a^{12}-\frac{65}{8}a^{11}+\frac{1611}{64}a^{10}-48a^{9}+\frac{3803}{64}a^{8}-\frac{451}{8}a^{7}+\frac{887}{16}a^{6}-\frac{41}{4}a^{5}-82a^{4}+\frac{455}{4}a^{3}-\frac{257}{2}a^{2}+176a-133$, $\frac{251}{32768}a^{31}-\frac{27}{1024}a^{30}+\frac{783}{16384}a^{29}-\frac{379}{4096}a^{28}+\frac{2223}{16384}a^{27}-\frac{23}{512}a^{26}-\frac{1035}{4096}a^{25}+\frac{2925}{4096}a^{24}-\frac{13499}{8192}a^{23}+\frac{5607}{2048}a^{22}-\frac{6617}{2048}a^{21}+\frac{2457}{1024}a^{20}+\frac{1827}{4096}a^{19}-\frac{913}{128}a^{18}+\frac{35343}{2048}a^{17}-\frac{15119}{512}a^{16}+\frac{82623}{2048}a^{15}-\frac{22131}{512}a^{14}+\frac{14863}{512}a^{13}+\frac{6399}{512}a^{12}-\frac{10093}{128}a^{11}+\frac{10501}{64}a^{10}-\frac{8325}{32}a^{9}+\frac{9463}{32}a^{8}-\frac{1827}{8}a^{7}+\frac{829}{8}a^{6}+\frac{393}{4}a^{5}-\frac{3699}{8}a^{4}+\frac{2415}{4}a^{3}-585a^{2}+562a-450$ 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:  \( 23735246685.523792 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 23735246685.523792 \cdot 5}{40\cdot\sqrt{7669926418924454281216000000000000000000000000}}\cr\approx \mathstrut & 0.199887693977467 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 + 10*x^30 - 20*x^29 + 34*x^28 - 32*x^27 - 8*x^26 + 112*x^25 - 308*x^24 + 608*x^23 - 880*x^22 + 944*x^21 - 536*x^20 - 672*x^19 + 2960*x^18 - 6176*x^17 + 9744*x^16 - 12352*x^15 + 11840*x^14 - 5376*x^13 - 8576*x^12 + 30208*x^11 - 56320*x^10 + 77824*x^9 - 78848*x^8 + 57344*x^7 - 8192*x^6 - 65536*x^5 + 139264*x^4 - 163840*x^3 + 163840*x^2 - 131072*x + 65536)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^32 - 4*x^31 + 10*x^30 - 20*x^29 + 34*x^28 - 32*x^27 - 8*x^26 + 112*x^25 - 308*x^24 + 608*x^23 - 880*x^22 + 944*x^21 - 536*x^20 - 672*x^19 + 2960*x^18 - 6176*x^17 + 9744*x^16 - 12352*x^15 + 11840*x^14 - 5376*x^13 - 8576*x^12 + 30208*x^11 - 56320*x^10 + 77824*x^9 - 78848*x^8 + 57344*x^7 - 8192*x^6 - 65536*x^5 + 139264*x^4 - 163840*x^3 + 163840*x^2 - 131072*x + 65536, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - 4*x^31 + 10*x^30 - 20*x^29 + 34*x^28 - 32*x^27 - 8*x^26 + 112*x^25 - 308*x^24 + 608*x^23 - 880*x^22 + 944*x^21 - 536*x^20 - 672*x^19 + 2960*x^18 - 6176*x^17 + 9744*x^16 - 12352*x^15 + 11840*x^14 - 5376*x^13 - 8576*x^12 + 30208*x^11 - 56320*x^10 + 77824*x^9 - 78848*x^8 + 57344*x^7 - 8192*x^6 - 65536*x^5 + 139264*x^4 - 163840*x^3 + 163840*x^2 - 131072*x + 65536);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 4*x^31 + 10*x^30 - 20*x^29 + 34*x^28 - 32*x^27 - 8*x^26 + 112*x^25 - 308*x^24 + 608*x^23 - 880*x^22 + 944*x^21 - 536*x^20 - 672*x^19 + 2960*x^18 - 6176*x^17 + 9744*x^16 - 12352*x^15 + 11840*x^14 - 5376*x^13 - 8576*x^12 + 30208*x^11 - 56320*x^10 + 77824*x^9 - 78848*x^8 + 57344*x^7 - 8192*x^6 - 65536*x^5 + 139264*x^4 - 163840*x^3 + 163840*x^2 - 131072*x + 65536);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_4^2:C_2^2$ (as 32T204):

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 64
The 40 conjugacy class representatives for $C_4^2:C_2^2$
Character table for $C_4^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), 4.0.8000.2, \(\Q(\zeta_{20})^+\), 4.4.8000.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{8})\), 4.4.108800.1, 4.0.27200.2, \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), 4.0.1088.2, 4.4.4352.1, \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.11837440000.20, 8.0.18939904.2, 8.0.40960000.1, 8.0.11837440000.5, 8.8.11837440000.1, 8.0.739840000.6, 8.0.11837440000.9, 8.0.1024000000.2, \(\Q(\zeta_{20})\), 8.0.1024000000.1, 8.0.64000000.1, \(\Q(\zeta_{40})^+\), 8.0.64000000.2, 8.8.18496000000.1, 8.0.295936000000.3, 16.0.140124985753600000000.3, \(\Q(\zeta_{40})\), 16.0.87578116096000000000000.3, 16.16.87578116096000000000000.1, 16.0.87578116096000000000000.1, 16.0.87578116096000000000000.2, 16.0.342102016000000000000.1

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 ${\href{/padicField/3.4.0.1}{4} }^{8}$ R ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ R ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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 Deg $16$$4$$4$$32$
Deg $16$$4$$4$$32$
\(5\) Copy content Toggle raw display 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
\(17\) Copy content Toggle raw display 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}$
17.8.4.1$x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$