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Properties

Label 32.0.492...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $4.924\times 10^{53}$
Root discriminant \(47.63\)
Ramified primes $2,5,29,769$
Class number $84$ (GRH)
Class group [84] (GRH)
Galois group $D_4^2:C_2^3$ (as 32T12882)

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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 + 14*x^24 + 60*x^20 - 95*x^16 + 960*x^12 + 3584*x^8 - 12288*x^4 + 65536)
 
gp: K = bnfinit(y^32 - 3*y^28 + 14*y^24 + 60*y^20 - 95*y^16 + 960*y^12 + 3584*y^8 - 12288*y^4 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 3*x^28 + 14*x^24 + 60*x^20 - 95*x^16 + 960*x^12 + 3584*x^8 - 12288*x^4 + 65536);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 3*x^28 + 14*x^24 + 60*x^20 - 95*x^16 + 960*x^12 + 3584*x^8 - 12288*x^4 + 65536)
 

\( x^{32} - 3x^{28} + 14x^{24} + 60x^{20} - 95x^{16} + 960x^{12} + 3584x^{8} - 12288x^{4} + 65536 \) 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:   \(492412573934221323439865515488952975360000000000000000\) \(\medspace = 2^{64}\cdot 5^{16}\cdot 29^{8}\cdot 769^{4}\) 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:  \(47.63\)
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:  $2^{2}5^{1/2}29^{1/2}769^{1/2}\approx 1335.6945758668035$
Ramified primes:   \(2\), \(5\), \(29\), \(769\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{4}a^{18}+\frac{1}{4}a^{14}-\frac{1}{2}a^{10}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{19}-\frac{3}{8}a^{15}-\frac{1}{4}a^{11}-\frac{1}{2}a^{7}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{20}-\frac{3}{16}a^{16}-\frac{1}{8}a^{12}-\frac{1}{4}a^{8}+\frac{1}{16}a^{4}$, $\frac{1}{32}a^{21}-\frac{3}{32}a^{17}+\frac{7}{16}a^{13}-\frac{1}{8}a^{9}+\frac{1}{32}a^{5}$, $\frac{1}{64}a^{22}-\frac{3}{64}a^{18}+\frac{7}{32}a^{14}-\frac{1}{16}a^{10}-\frac{31}{64}a^{6}$, $\frac{1}{128}a^{23}-\frac{3}{128}a^{19}+\frac{7}{64}a^{15}+\frac{15}{32}a^{11}+\frac{33}{128}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{256256}a^{24}+\frac{5069}{256256}a^{20}-\frac{13873}{128128}a^{16}-\frac{1003}{5824}a^{12}+\frac{29537}{256256}a^{8}+\frac{2025}{16016}a^{4}+\frac{16}{1001}$, $\frac{1}{512512}a^{25}+\frac{5069}{512512}a^{21}-\frac{13873}{256256}a^{17}+\frac{4821}{11648}a^{13}-\frac{226719}{512512}a^{9}-\frac{13991}{32032}a^{5}-\frac{985}{2002}a$, $\frac{1}{1025024}a^{26}+\frac{5069}{1025024}a^{22}-\frac{13873}{512512}a^{18}+\frac{4821}{23296}a^{14}-\frac{226719}{1025024}a^{10}-\frac{13991}{64064}a^{6}+\frac{1017}{4004}a^{2}$, $\frac{1}{2050048}a^{27}+\frac{5069}{2050048}a^{23}-\frac{13873}{1025024}a^{19}-\frac{18475}{46592}a^{15}+\frac{798305}{2050048}a^{11}-\frac{13991}{128128}a^{7}+\frac{1017}{8008}a^{3}$, $\frac{1}{127102976}a^{28}+\frac{125}{127102976}a^{24}-\frac{9723}{698368}a^{20}+\frac{200975}{31775744}a^{16}-\frac{51106911}{127102976}a^{12}-\frac{73531}{152768}a^{8}+\frac{31305}{70928}a^{4}+\frac{10071}{31031}$, $\frac{1}{254205952}a^{29}+\frac{125}{254205952}a^{25}-\frac{9723}{1396736}a^{21}+\frac{200975}{63551488}a^{17}-\frac{51106911}{254205952}a^{13}+\frac{79237}{305536}a^{9}-\frac{39623}{141856}a^{5}+\frac{10071}{62062}a$, $\frac{1}{508411904}a^{30}+\frac{125}{508411904}a^{26}-\frac{9723}{2793472}a^{22}+\frac{200975}{127102976}a^{18}-\frac{51106911}{508411904}a^{14}-\frac{226299}{611072}a^{10}-\frac{39623}{283712}a^{6}-\frac{51991}{124124}a^{2}$, $\frac{1}{1016823808}a^{31}+\frac{125}{1016823808}a^{27}-\frac{9723}{5586944}a^{23}+\frac{200975}{254205952}a^{19}+\frac{457304993}{1016823808}a^{15}-\frac{226299}{1222144}a^{11}-\frac{39623}{567424}a^{7}+\frac{72133}{248248}a^{3}$ 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

$C_{84}$, which has order $84$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Relative class number: $84$ (assuming GRH)

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:   \( -\frac{28471}{254205952} a^{29} + \frac{40597}{254205952} a^{25} - \frac{28265}{127102976} a^{21} - \frac{286001}{63551488} a^{17} + \frac{927657}{254205952} a^{13} + \frac{92107}{15887872} a^{9} - \frac{16767}{76384} a^{5} + \frac{43984}{31031} a \)  (order $8$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{3}{26624}a^{28}+\frac{89}{186368}a^{24}+\frac{111}{93184}a^{20}+\frac{43}{3584}a^{16}+\frac{489}{14336}a^{12}+\frac{1035}{23296}a^{8}+\frac{1011}{1456}a^{4}+\frac{38}{91}$, $\frac{1}{585728}a^{28}-\frac{2293}{4100096}a^{24}-\frac{639}{2050048}a^{20}-\frac{131}{78848}a^{16}-\frac{9277}{315392}a^{12}-\frac{2517}{128128}a^{8}-\frac{2571}{8008}a^{4}-\frac{1740}{1001}$, $\frac{80435}{508411904}a^{30}-\frac{40217}{508411904}a^{26}+\frac{9103}{23109632}a^{22}+\frac{813885}{127102976}a^{18}-\frac{5445357}{508411904}a^{14}+\frac{115779}{7943936}a^{10}+\frac{310175}{992992}a^{6}-\frac{8218}{4433}a^{2}$, $\frac{7175}{72630272}a^{30}+\frac{28471}{254205952}a^{29}-\frac{9813}{72630272}a^{26}-\frac{40597}{254205952}a^{25}+\frac{1491}{3301376}a^{22}+\frac{28265}{127102976}a^{21}+\frac{84937}{18157568}a^{18}+\frac{286001}{63551488}a^{17}-\frac{786329}{72630272}a^{14}-\frac{927657}{254205952}a^{13}+\frac{1015}{70928}a^{10}-\frac{92107}{15887872}a^{9}+\frac{68051}{283712}a^{6}+\frac{16767}{76384}a^{5}-\frac{18563}{17732}a^{2}-\frac{43984}{31031}a-1$, $\frac{7175}{72630272}a^{30}+\frac{8509}{127102976}a^{29}-\frac{9813}{72630272}a^{26}+\frac{13593}{127102976}a^{25}+\frac{1491}{3301376}a^{22}+\frac{74163}{63551488}a^{21}+\frac{84937}{18157568}a^{18}+\frac{8823}{2444288}a^{17}-\frac{786329}{72630272}a^{14}+\frac{97873}{9777152}a^{13}+\frac{1015}{70928}a^{10}+\frac{164057}{7943936}a^{9}+\frac{68051}{283712}a^{6}+\frac{235325}{992992}a^{5}-\frac{18563}{17732}a^{2}-\frac{8461}{31031}a-1$, $\frac{7175}{72630272}a^{30}+\frac{1173}{127102976}a^{29}-\frac{9813}{72630272}a^{26}+\frac{44449}{127102976}a^{25}+\frac{1491}{3301376}a^{22}+\frac{20291}{63551488}a^{21}+\frac{84937}{18157568}a^{18}+\frac{152107}{31775744}a^{17}-\frac{786329}{72630272}a^{14}+\frac{1553973}{127102976}a^{13}+\frac{1015}{70928}a^{10}+\frac{146593}{3971968}a^{9}+\frac{68051}{283712}a^{6}+\frac{7159}{90272}a^{5}-\frac{18563}{17732}a^{2}+\frac{55871}{62062}a-1$, $\frac{6091}{254205952}a^{31}-\frac{8509}{127102976}a^{29}-\frac{35}{292864}a^{28}-\frac{43261}{254205952}a^{27}-\frac{13593}{127102976}a^{25}-\frac{969}{2050048}a^{24}-\frac{7279}{18157568}a^{23}-\frac{74163}{63551488}a^{21}-\frac{2231}{1025024}a^{20}+\frac{25667}{63551488}a^{19}-\frac{8823}{2444288}a^{17}-\frac{387}{39424}a^{16}-\frac{4288741}{254205952}a^{15}-\frac{97873}{9777152}a^{13}-\frac{5129}{157696}a^{12}-\frac{1500535}{63551488}a^{11}-\frac{164057}{7943936}a^{9}-\frac{37581}{256256}a^{8}-\frac{9065}{141856}a^{7}-\frac{235325}{992992}a^{5}-\frac{839}{2002}a^{4}-\frac{231841}{248248}a^{3}+\frac{8461}{31031}a-\frac{608}{1001}$, $\frac{45055}{1016823808}a^{31}-\frac{15311}{127102976}a^{30}+\frac{2521}{19554304}a^{29}-\frac{16495}{63551488}a^{28}+\frac{11705}{92438528}a^{27}-\frac{1423}{127102976}a^{26}+\frac{46289}{254205952}a^{25}+\frac{161}{825344}a^{24}+\frac{30887}{72630272}a^{23}-\frac{781}{5777408}a^{22}+\frac{6099}{127102976}a^{21}+\frac{6855}{31775744}a^{20}+\frac{620985}{254205952}a^{19}-\frac{189187}{31775744}a^{18}+\frac{470747}{63551488}a^{17}-\frac{208977}{15887872}a^{16}+\frac{9094303}{1016823808}a^{15}+\frac{374849}{127102976}a^{14}-\frac{81677}{36315136}a^{13}+\frac{2155185}{63551488}a^{12}+\frac{548115}{63551488}a^{11}-\frac{346607}{31775744}a^{10}+\frac{219357}{7943936}a^{9}-\frac{10095}{283712}a^{8}+\frac{27575}{141856}a^{7}-\frac{777807}{1985984}a^{6}+\frac{139959}{248248}a^{5}-\frac{200217}{248248}a^{4}+\frac{1251}{4774}a^{3}+\frac{22167}{17732}a^{2}-\frac{67201}{62062}a+\frac{121141}{31031}$, $\frac{3959}{1016823808}a^{31}-\frac{80435}{508411904}a^{30}+\frac{7175}{36315136}a^{29}+\frac{43}{1444352}a^{28}-\frac{23827}{145260544}a^{27}+\frac{40217}{508411904}a^{26}-\frac{9813}{36315136}a^{25}-\frac{4363}{15887872}a^{24}-\frac{39693}{46219264}a^{23}-\frac{9103}{23109632}a^{22}+\frac{1491}{1650688}a^{21}-\frac{6593}{7943936}a^{20}-\frac{696247}{254205952}a^{19}-\frac{813885}{127102976}a^{18}+\frac{84937}{9078784}a^{17}-\frac{3607}{567424}a^{16}-\frac{17451689}{1016823808}a^{15}+\frac{5445357}{508411904}a^{14}-\frac{786329}{36315136}a^{13}-\frac{49333}{1444352}a^{12}-\frac{184913}{4539392}a^{11}-\frac{115779}{7943936}a^{10}+\frac{1015}{35464}a^{9}-\frac{27889}{248248}a^{8}-\frac{30909}{305536}a^{7}-\frac{310175}{992992}a^{6}+\frac{68051}{141856}a^{5}-\frac{206127}{496496}a^{4}-\frac{84457}{124124}a^{3}+\frac{8218}{4433}a^{2}-\frac{27429}{8866}a-\frac{39843}{31031}$, $\frac{77433}{1016823808}a^{31}-\frac{15311}{127102976}a^{30}-\frac{33207}{254205952}a^{29}+\frac{16495}{63551488}a^{28}-\frac{117371}{1016823808}a^{27}-\frac{1423}{127102976}a^{26}+\frac{95877}{254205952}a^{25}-\frac{161}{825344}a^{24}-\frac{147481}{508411904}a^{23}-\frac{781}{5777408}a^{22}+\frac{33519}{127102976}a^{21}-\frac{6855}{31775744}a^{20}+\frac{892511}{254205952}a^{19}-\frac{189187}{31775744}a^{18}-\frac{365161}{63551488}a^{17}+\frac{208977}{15887872}a^{16}-\frac{1727585}{145260544}a^{15}+\frac{374849}{127102976}a^{14}+\frac{8049001}{254205952}a^{13}-\frac{2155185}{63551488}a^{12}+\frac{145099}{63551488}a^{11}-\frac{346607}{31775744}a^{10}-\frac{63303}{7943936}a^{9}+\frac{10095}{283712}a^{8}+\frac{391757}{1985984}a^{7}-\frac{777807}{1985984}a^{6}-\frac{2739}{11284}a^{5}+\frac{200217}{248248}a^{4}-\frac{187695}{124124}a^{3}+\frac{22167}{17732}a^{2}+\frac{175081}{62062}a-\frac{121141}{31031}$, $\frac{3959}{1016823808}a^{31}+\frac{80435}{508411904}a^{30}+\frac{7175}{36315136}a^{29}-\frac{43}{1444352}a^{28}-\frac{23827}{145260544}a^{27}-\frac{40217}{508411904}a^{26}-\frac{9813}{36315136}a^{25}+\frac{4363}{15887872}a^{24}-\frac{39693}{46219264}a^{23}+\frac{9103}{23109632}a^{22}+\frac{1491}{1650688}a^{21}+\frac{6593}{7943936}a^{20}-\frac{696247}{254205952}a^{19}+\frac{813885}{127102976}a^{18}+\frac{84937}{9078784}a^{17}+\frac{3607}{567424}a^{16}-\frac{17451689}{1016823808}a^{15}-\frac{5445357}{508411904}a^{14}-\frac{786329}{36315136}a^{13}+\frac{49333}{1444352}a^{12}-\frac{184913}{4539392}a^{11}+\frac{115779}{7943936}a^{10}+\frac{1015}{35464}a^{9}+\frac{27889}{248248}a^{8}-\frac{30909}{305536}a^{7}+\frac{310175}{992992}a^{6}+\frac{68051}{141856}a^{5}+\frac{206127}{496496}a^{4}-\frac{84457}{124124}a^{3}-\frac{8218}{4433}a^{2}-\frac{27429}{8866}a+\frac{39843}{31031}$, $\frac{34899}{1016823808}a^{31}-\frac{16845}{254205952}a^{29}+\frac{135959}{1016823808}a^{27}+\frac{134807}{254205952}a^{25}+\frac{234669}{508411904}a^{23}-\frac{25187}{127102976}a^{21}+\frac{107971}{36315136}a^{19}+\frac{247077}{63551488}a^{17}+\frac{4885235}{1016823808}a^{15}+\frac{1007701}{36315136}a^{13}+\frac{2704789}{63551488}a^{11}+\frac{469863}{15887872}a^{9}+\frac{102111}{3971968}a^{7}+\frac{118103}{496496}a^{5}+\frac{12202}{31031}a^{3}+\frac{5031}{2387}a$, $\frac{110401}{1016823808}a^{31}-\frac{111469}{508411904}a^{30}+\frac{3021}{36315136}a^{29}+\frac{65389}{1016823808}a^{27}+\frac{62999}{508411904}a^{26}-\frac{47921}{254205952}a^{25}+\frac{216191}{508411904}a^{23}-\frac{13561}{23109632}a^{22}+\frac{3021}{127102976}a^{21}+\frac{1785527}{254205952}a^{19}-\frac{1351307}{127102976}a^{18}-\frac{62331}{63551488}a^{17}+\frac{9066529}{1016823808}a^{15}+\frac{7003699}{508411904}a^{14}-\frac{6697989}{254205952}a^{13}+\frac{191213}{5777408}a^{11}-\frac{801327}{31775744}a^{10}+\frac{30479}{7943936}a^{9}+\frac{1288909}{3971968}a^{7}-\frac{313541}{496496}a^{6}+\frac{105659}{992992}a^{5}-\frac{52193}{124124}a^{3}+\frac{20365}{8866}a^{2}-\frac{110039}{62062}a$, $\frac{1571}{11554816}a^{31}-\frac{1}{16384}a^{30}-\frac{40511}{254205952}a^{29}+\frac{101}{292864}a^{28}-\frac{41929}{127102976}a^{27}+\frac{3}{16384}a^{26}-\frac{68163}{254205952}a^{25}+\frac{2927}{2050048}a^{24}-\frac{709}{3971968}a^{23}-\frac{7}{8192}a^{22}-\frac{191929}{127102976}a^{21}+\frac{4673}{1025024}a^{20}+\frac{11131}{2269696}a^{19}-\frac{15}{4096}a^{18}-\frac{471281}{63551488}a^{17}+\frac{1333}{39424}a^{16}-\frac{237109}{11554816}a^{15}+\frac{95}{16384}a^{14}+\frac{149335}{36315136}a^{13}+\frac{15887}{157696}a^{12}-\frac{1868963}{63551488}a^{11}-\frac{15}{256}a^{10}-\frac{243861}{3971968}a^{9}+\frac{60351}{256256}a^{8}+\frac{38629}{248248}a^{7}-\frac{7}{32}a^{6}-\frac{419733}{992992}a^{5}+\frac{14477}{8008}a^{4}-\frac{44901}{19096}a^{3}+\frac{7}{4}a^{2}+\frac{35545}{31031}a+\frac{3446}{1001}$, $\frac{11019}{508411904}a^{30}-\frac{10877}{127102976}a^{29}-\frac{893}{2269696}a^{28}+\frac{74383}{508411904}a^{26}+\frac{1277}{11554816}a^{25}-\frac{23}{1222144}a^{24}-\frac{7313}{23109632}a^{22}-\frac{43271}{63551488}a^{21}-\frac{16827}{7943936}a^{20}+\frac{162189}{127102976}a^{18}-\frac{154279}{31775744}a^{17}-\frac{80919}{3971968}a^{16}+\frac{4004907}{508411904}a^{14}+\frac{2288323}{127102976}a^{13}+\frac{220709}{15887872}a^{12}-\frac{108113}{31775744}a^{10}-\frac{546827}{15887872}a^{9}-\frac{305387}{3971968}a^{8}+\frac{150725}{992992}a^{6}-\frac{129193}{496496}a^{5}-\frac{635759}{496496}a^{4}+\frac{3532}{4433}a^{2}+\frac{104035}{62062}a+\frac{107032}{31031}$ 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:  \( 2768304842151.426 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 2768304842151.426 \cdot 84}{8\cdot\sqrt{492412573934221323439865515488952975360000000000000000}}\cr\approx \mathstrut & 0.244408630772899 \end{aligned}\] (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 + 14*x^24 + 60*x^20 - 95*x^16 + 960*x^12 + 3584*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 + 14*x^24 + 60*x^20 - 95*x^16 + 960*x^12 + 3584*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 + 14*x^24 + 60*x^20 - 95*x^16 + 960*x^12 + 3584*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 + 14*x^24 + 60*x^20 - 95*x^16 + 960*x^12 + 3584*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

$D_4^2:C_2^3$ (as 32T12882):

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 512
The 80 conjugacy class representatives for $D_4^2:C_2^3$
Character table for $D_4^2:C_2^3$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-1}) \), 4.0.11600.1, 4.0.46400.1, 4.4.46400.1, 4.4.725.1, \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{5})\), 8.0.1655626240000.1, 8.0.404205625.1, 8.8.103476640000.1, 8.8.1655626240000.1, 8.0.40960000.1, 8.0.34447360000.26, 8.8.2152960000.1, 8.0.34447360000.23, 8.0.2152960000.5, 8.0.34447360000.3, 8.0.134560000.4, 16.0.1186620610969600000000.3, 16.0.2741098246576537600000000.1, 16.16.701721151123593625600000000.1, 16.0.701721151123593625600000000.1, 16.0.2741098246576537600000000.2, 16.0.10707415025689600000000.1, 16.0.701721151123593625600000000.2

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

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

Sibling fields

Degree 32 siblings: data not computed
Minimal sibling: not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{8}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.2.0.1}{2} }^{16}$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ R ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.8.0.1}{8} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$

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 Deg $16$$4$$4$$32$
Deg $16$$4$$4$$32$
\(5\) Copy content Toggle raw display 5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(29\) Copy content Toggle raw display 29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
\(769\) 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$$2$$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 $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$