Properties

Label 32.0.877...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $8.780\times 10^{48}$
Root discriminant \(33.84\)
Ramified primes $2,5,41$
Class number $32$ (GRH)
Class group [2, 2, 8] (GRH)
Galois group $C_2^4:C_4$ (as 32T262)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 13*x^28 + 78*x^24 + 356*x^20 + 1505*x^16 + 5696*x^12 + 19968*x^8 + 53248*x^4 + 65536)
 
gp: K = bnfinit(y^32 + 13*y^28 + 78*y^24 + 356*y^20 + 1505*y^16 + 5696*y^12 + 19968*y^8 + 53248*y^4 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 13*x^28 + 78*x^24 + 356*x^20 + 1505*x^16 + 5696*x^12 + 19968*x^8 + 53248*x^4 + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 13*x^28 + 78*x^24 + 356*x^20 + 1505*x^16 + 5696*x^12 + 19968*x^8 + 53248*x^4 + 65536)
 

\( x^{32} + 13x^{28} + 78x^{24} + 356x^{20} + 1505x^{16} + 5696x^{12} + 19968x^{8} + 53248x^{4} + 65536 \) 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:   \(8779518136340480467664896000000000000000000000000\) \(\medspace = 2^{64}\cdot 5^{24}\cdot 41^{8}\) 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:  \(33.84\)
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:  $2^{2}5^{3/4}41^{1/2}\approx 85.64054510685526$
Ramified primes:   \(2\), \(5\), \(41\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{15}a^{16}-\frac{1}{15}a^{12}+\frac{1}{15}a^{8}-\frac{1}{15}a^{4}+\frac{1}{15}$, $\frac{1}{30}a^{17}-\frac{1}{30}a^{13}-\frac{7}{15}a^{9}+\frac{7}{15}a^{5}+\frac{1}{30}a$, $\frac{1}{60}a^{18}+\frac{29}{60}a^{14}-\frac{7}{30}a^{10}-\frac{4}{15}a^{6}+\frac{1}{60}a^{2}$, $\frac{1}{120}a^{19}+\frac{29}{120}a^{15}+\frac{23}{60}a^{11}+\frac{11}{30}a^{7}+\frac{1}{120}a^{3}$, $\frac{1}{240}a^{20}-\frac{1}{80}a^{16}+\frac{13}{40}a^{12}-\frac{9}{20}a^{8}+\frac{11}{80}a^{4}-\frac{2}{15}$, $\frac{1}{480}a^{21}-\frac{1}{160}a^{17}-\frac{27}{80}a^{13}+\frac{11}{40}a^{9}+\frac{11}{160}a^{5}+\frac{13}{30}a$, $\frac{1}{960}a^{22}-\frac{1}{320}a^{18}+\frac{53}{160}a^{14}+\frac{11}{80}a^{10}+\frac{11}{320}a^{6}-\frac{17}{60}a^{2}$, $\frac{1}{1920}a^{23}-\frac{1}{640}a^{19}-\frac{107}{320}a^{15}+\frac{11}{160}a^{11}+\frac{11}{640}a^{7}+\frac{43}{120}a^{3}$, $\frac{1}{3840}a^{24}-\frac{1}{1280}a^{20}+\frac{21}{640}a^{16}-\frac{53}{320}a^{12}-\frac{373}{1280}a^{8}+\frac{23}{48}a^{4}+\frac{1}{5}$, $\frac{1}{7680}a^{25}-\frac{1}{2560}a^{21}+\frac{21}{1280}a^{17}-\frac{53}{640}a^{13}-\frac{373}{2560}a^{9}-\frac{25}{96}a^{5}+\frac{1}{10}a$, $\frac{1}{15360}a^{26}-\frac{1}{5120}a^{22}+\frac{21}{2560}a^{18}-\frac{53}{1280}a^{14}+\frac{2187}{5120}a^{10}-\frac{25}{192}a^{6}+\frac{1}{20}a^{2}$, $\frac{1}{30720}a^{27}-\frac{1}{10240}a^{23}+\frac{21}{5120}a^{19}+\frac{1227}{2560}a^{15}-\frac{2933}{10240}a^{11}+\frac{167}{384}a^{7}+\frac{1}{40}a^{3}$, $\frac{1}{5836800}a^{28}+\frac{397}{5836800}a^{24}+\frac{1101}{972800}a^{20}-\frac{6907}{291840}a^{16}-\frac{511187}{1167360}a^{12}-\frac{4801}{91200}a^{8}-\frac{829}{1900}a^{4}-\frac{479}{1425}$, $\frac{1}{11673600}a^{29}+\frac{397}{11673600}a^{25}+\frac{1101}{1945600}a^{21}-\frac{6907}{583680}a^{17}+\frac{656173}{2334720}a^{13}+\frac{86399}{182400}a^{9}+\frac{1071}{3800}a^{5}-\frac{479}{2850}a$, $\frac{1}{23347200}a^{30}+\frac{397}{23347200}a^{26}+\frac{1101}{3891200}a^{22}-\frac{6907}{1167360}a^{18}+\frac{656173}{4669440}a^{14}-\frac{96001}{364800}a^{10}+\frac{1071}{7600}a^{6}+\frac{2371}{5700}a^{2}$, $\frac{1}{46694400}a^{31}+\frac{397}{46694400}a^{27}+\frac{1101}{7782400}a^{23}-\frac{6907}{2334720}a^{19}+\frac{656173}{9338880}a^{15}+\frac{268799}{729600}a^{11}-\frac{6529}{15200}a^{7}-\frac{3329}{11400}a^{3}$ 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

$C_{2}\times C_{2}\times C_{8}$, which has order $32$ (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:   \( -\frac{469}{5836800} a^{31} - \frac{4553}{5836800} a^{27} - \frac{9727}{2918400} a^{23} - \frac{4829}{291840} a^{19} - \frac{78289}{1167360} a^{15} - \frac{170503}{729600} a^{11} - \frac{37391}{45600} a^{7} - \frac{5669}{3800} a^{3} \)  (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:   $\frac{71}{77824}a^{28}+\frac{10277}{1167360}a^{24}+\frac{4795}{116736}a^{20}+\frac{3575}{19456}a^{16}+\frac{59175}{77824}a^{12}+\frac{12971}{4864}a^{8}+\frac{21151}{2280}a^{4}+\frac{953}{57}$, $\frac{11557}{23347200}a^{30}+\frac{613}{389120}a^{28}+\frac{34203}{7782400}a^{26}+\frac{3319}{233472}a^{24}+\frac{237371}{11673600}a^{22}+\frac{2543}{38912}a^{20}+\frac{105761}{1167360}a^{18}+\frac{5745}{19456}a^{16}+\frac{1661761}{4669440}a^{14}+\frac{91905}{77824}a^{12}+\frac{1896947}{1459200}a^{10}+\frac{101713}{24320}a^{8}+\frac{201097}{45600}a^{6}+\frac{13039}{912}a^{4}+\frac{43547}{5700}a^{2}+\frac{471}{19}$, $\frac{2867}{5836800}a^{30}+\frac{8773}{1945600}a^{26}+\frac{62801}{2918400}a^{22}+\frac{27779}{291840}a^{18}+\frac{449239}{1167360}a^{14}+\frac{338453}{243200}a^{10}+\frac{435701}{91200}a^{6}+\frac{49793}{5700}a^{2}-1$, $\frac{191}{729600}a^{29}+\frac{721}{364800}a^{25}+\frac{5941}{729600}a^{21}+\frac{937}{24320}a^{17}+\frac{7381}{48640}a^{13}+\frac{393551}{729600}a^{9}+\frac{42301}{22800}a^{5}+\frac{6871}{2850}a+1$, $\frac{4099}{15564800}a^{31}+\frac{99989}{46694400}a^{27}+\frac{232111}{23347200}a^{23}+\frac{104669}{2334720}a^{19}+\frac{1623829}{9338880}a^{15}+\frac{930611}{1459200}a^{11}+\frac{380089}{182400}a^{7}+\frac{1513}{475}a^{3}+1$, $\frac{4099}{15564800}a^{31}+\frac{343}{1167360}a^{28}+\frac{99989}{46694400}a^{27}+\frac{3019}{1167360}a^{24}+\frac{232111}{23347200}a^{23}+\frac{7217}{583680}a^{20}+\frac{104669}{2334720}a^{19}+\frac{3011}{58368}a^{16}+\frac{1623829}{9338880}a^{15}+\frac{51403}{233472}a^{12}+\frac{930611}{1459200}a^{11}+\frac{29329}{36480}a^{8}+\frac{380089}{182400}a^{7}+\frac{4113}{1520}a^{4}+\frac{1513}{475}a^{3}+\frac{1364}{285}$, $\frac{2427}{3891200}a^{30}-\frac{191}{729600}a^{29}+\frac{67157}{11673600}a^{26}-\frac{721}{364800}a^{25}+\frac{52721}{1945600}a^{22}-\frac{5941}{729600}a^{21}+\frac{23667}{194560}a^{18}-\frac{937}{24320}a^{17}+\frac{378687}{778240}a^{14}-\frac{7381}{48640}a^{13}+\frac{838249}{486400}a^{10}-\frac{393551}{729600}a^{9}+\frac{546979}{91200}a^{6}-\frac{42301}{22800}a^{5}+\frac{20009}{1900}a^{2}-\frac{6871}{2850}a$, $\frac{6139}{5836800}a^{28}+\frac{59903}{5836800}a^{24}+\frac{143197}{2918400}a^{20}+\frac{21229}{97280}a^{16}+\frac{341429}{389120}a^{12}+\frac{70759}{22800}a^{8}+\frac{241373}{22800}a^{4}+\frac{28454}{1425}$, $\frac{9833}{46694400}a^{31}+\frac{3251}{2918400}a^{29}+\frac{109781}{46694400}a^{27}+\frac{33607}{2918400}a^{25}+\frac{278719}{23347200}a^{23}+\frac{77533}{1459200}a^{21}+\frac{116237}{2334720}a^{19}+\frac{34807}{145920}a^{17}+\frac{2057077}{9338880}a^{15}+\frac{572087}{583680}a^{13}+\frac{181869}{243200}a^{11}+\frac{616351}{182400}a^{9}+\frac{7549}{2850}a^{7}+\frac{556403}{45600}a^{5}+\frac{11653}{1900}a^{3}+\frac{65119}{2850}a$, $\frac{3567}{7782400}a^{30}-\frac{3301}{5836800}a^{28}+\frac{31379}{7782400}a^{26}-\frac{32177}{5836800}a^{24}+\frac{69441}{3891200}a^{22}-\frac{77003}{2918400}a^{20}+\frac{93497}{1167360}a^{18}-\frac{32977}{291840}a^{16}+\frac{1519177}{4669440}a^{14}-\frac{524417}{1167360}a^{12}+\frac{1721401}{1459200}a^{10}-\frac{198567}{121600}a^{8}+\frac{362837}{91200}a^{6}-\frac{44909}{7600}a^{4}+\frac{38041}{5700}a^{2}-\frac{5067}{475}$, $\frac{337}{15564800}a^{31}-\frac{9157}{23347200}a^{30}+\frac{11777}{11673600}a^{29}-\frac{5623}{5836800}a^{28}+\frac{3127}{46694400}a^{27}-\frac{77009}{23347200}a^{26}+\frac{36463}{3891200}a^{25}-\frac{52651}{5836800}a^{24}-\frac{1049}{7782400}a^{23}-\frac{173131}{11673600}a^{22}+\frac{85997}{1945600}a^{21}-\frac{124529}{2918400}a^{20}-\frac{445}{466944}a^{19}-\frac{77777}{1167360}a^{18}+\frac{38903}{194560}a^{17}-\frac{56003}{291840}a^{16}-\frac{20981}{1867776}a^{15}-\frac{1257697}{4669440}a^{14}+\frac{634383}{778240}a^{13}-\frac{868843}{1167360}a^{12}-\frac{110627}{1459200}a^{11}-\frac{1435217}{1459200}a^{10}+\frac{264319}{91200}a^{9}-\frac{468319}{182400}a^{8}-\frac{19079}{91200}a^{7}-\frac{322099}{91200}a^{6}+\frac{150263}{15200}a^{5}-\frac{16903}{1900}a^{4}-\frac{458}{1425}a^{3}-\frac{12079}{1900}a^{2}+\frac{8362}{475}a-\frac{23108}{1425}$, $\frac{4099}{15564800}a^{31}-\frac{871}{2918400}a^{30}-\frac{2561}{3891200}a^{29}+\frac{781}{364800}a^{28}+\frac{99989}{46694400}a^{27}-\frac{6827}{2918400}a^{26}-\frac{25677}{3891200}a^{25}+\frac{2399}{121600}a^{24}+\frac{232111}{23347200}a^{23}-\frac{5171}{486400}a^{22}-\frac{59463}{1945600}a^{21}+\frac{16733}{182400}a^{20}+\frac{104669}{2334720}a^{19}-\frac{477}{9728}a^{18}-\frac{26469}{194560}a^{17}+\frac{7447}{18240}a^{16}+\frac{1623829}{9338880}a^{15}-\frac{7093}{38912}a^{14}-\frac{428589}{778240}a^{13}+\frac{118937}{72960}a^{12}+\frac{930611}{1459200}a^{11}-\frac{120331}{182400}a^{10}-\frac{59097}{30400}a^{9}+\frac{176783}{30400}a^{8}+\frac{380089}{182400}a^{7}-\frac{208561}{91200}a^{6}-\frac{101637}{15200}a^{5}+\frac{230351}{11400}a^{4}+\frac{1513}{475}a^{3}-\frac{1604}{475}a^{2}-\frac{11511}{950}a+\frac{16842}{475}$, $\frac{2161}{7782400}a^{31}-\frac{469}{2918400}a^{30}-\frac{343}{1167360}a^{29}+\frac{71}{77824}a^{28}+\frac{64231}{23347200}a^{27}-\frac{4553}{2918400}a^{26}-\frac{3019}{1167360}a^{25}+\frac{10277}{1167360}a^{24}+\frac{146269}{11673600}a^{23}-\frac{9727}{1459200}a^{22}-\frac{7217}{583680}a^{21}+\frac{4795}{116736}a^{20}+\frac{4361}{77824}a^{19}-\frac{4829}{145920}a^{18}-\frac{3011}{58368}a^{17}+\frac{3575}{19456}a^{16}+\frac{72249}{311296}a^{15}-\frac{78289}{583680}a^{14}-\frac{51403}{233472}a^{13}+\frac{59175}{77824}a^{12}+\frac{395821}{486400}a^{11}-\frac{170503}{364800}a^{10}-\frac{29329}{36480}a^{9}+\frac{12971}{4864}a^{8}+\frac{529727}{182400}a^{7}-\frac{37391}{22800}a^{6}-\frac{4113}{1520}a^{5}+\frac{21151}{2280}a^{4}+\frac{15439}{2850}a^{3}-\frac{5669}{1900}a^{2}-\frac{1364}{285}a+\frac{953}{57}$, $\frac{8421}{15564800}a^{31}-\frac{2561}{7782400}a^{30}+\frac{2999}{11673600}a^{29}+\frac{1849}{972800}a^{28}+\frac{228451}{46694400}a^{27}-\frac{25677}{7782400}a^{26}+\frac{36923}{11673600}a^{25}+\frac{49079}{2918400}a^{24}+\frac{174883}{7782400}a^{23}-\frac{59463}{3891200}a^{22}+\frac{84217}{5836800}a^{21}+\frac{113281}{1459200}a^{20}+\frac{235499}{2334720}a^{19}-\frac{26469}{389120}a^{18}+\frac{2453}{38912}a^{17}+\frac{17249}{48640}a^{16}+\frac{3791299}{9338880}a^{15}-\frac{428589}{1556480}a^{14}+\frac{43877}{155648}a^{13}+\frac{272869}{194560}a^{12}+\frac{1059037}{729600}a^{11}-\frac{59097}{60800}a^{10}+\frac{706139}{729600}a^{9}+\frac{612563}{121600}a^{8}+\frac{37909}{7600}a^{7}-\frac{101637}{30400}a^{6}+\frac{160583}{45600}a^{5}+\frac{396913}{22800}a^{4}+\frac{24517}{2850}a^{3}-\frac{11511}{1900}a^{2}+\frac{10627}{1425}a+\frac{41599}{1425}$, $\frac{871}{2918400}a^{31}-\frac{11557}{23347200}a^{30}-\frac{2427}{1945600}a^{29}+\frac{613}{389120}a^{28}+\frac{6827}{2918400}a^{27}-\frac{34203}{7782400}a^{26}-\frac{67157}{5836800}a^{25}+\frac{3319}{233472}a^{24}+\frac{5171}{486400}a^{23}-\frac{237371}{11673600}a^{22}-\frac{52721}{972800}a^{21}+\frac{2543}{38912}a^{20}+\frac{477}{9728}a^{19}-\frac{105761}{1167360}a^{18}-\frac{23667}{97280}a^{17}+\frac{5745}{19456}a^{16}+\frac{7093}{38912}a^{15}-\frac{1661761}{4669440}a^{14}-\frac{378687}{389120}a^{13}+\frac{91905}{77824}a^{12}+\frac{120331}{182400}a^{11}-\frac{1896947}{1459200}a^{10}-\frac{838249}{243200}a^{9}+\frac{101713}{24320}a^{8}+\frac{208561}{91200}a^{7}-\frac{201097}{45600}a^{6}-\frac{546979}{45600}a^{5}+\frac{13039}{912}a^{4}+\frac{1604}{475}a^{3}-\frac{43547}{5700}a^{2}-\frac{20009}{950}a+\frac{452}{19}$ 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:  \( 150821321458.18454 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 150821321458.18454 \cdot 32}{40\cdot\sqrt{8779518136340480467664896000000000000000000000000}}\cr\approx \mathstrut & 0.240267639785288 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + 13*x^28 + 78*x^24 + 356*x^20 + 1505*x^16 + 5696*x^12 + 19968*x^8 + 53248*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 + 13*x^28 + 78*x^24 + 356*x^20 + 1505*x^16 + 5696*x^12 + 19968*x^8 + 53248*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 + 13*x^28 + 78*x^24 + 356*x^20 + 1505*x^16 + 5696*x^12 + 19968*x^8 + 53248*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 + 13*x^28 + 78*x^24 + 356*x^20 + 1505*x^16 + 5696*x^12 + 19968*x^8 + 53248*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

$C_2^4:C_4$ (as 32T262):

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 $C_2^4:C_4$
Character table for $C_2^4:C_4$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), 4.4.16400.1, 4.0.8000.2, 4.4.65600.1, \(\Q(\zeta_{5})\), 4.0.65600.4, \(\Q(\zeta_{20})^+\), 4.0.1025.1, 4.4.8000.1, \(\Q(\zeta_{8})\), 4.0.328000.2, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), 4.4.328000.1, \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{5})\), 4.4.5125.1, 4.0.82000.1, 8.0.1721344000000.31, 8.0.107584000000.6, 8.0.1721344000000.52, 8.0.40960000.1, 8.8.107584000000.4, 8.0.1721344000000.28, 8.0.6724000000.12, 8.0.1721344000000.37, 8.0.107584000000.17, 8.0.1721344000000.22, 8.0.107584000000.7, 8.8.68853760000.1, 8.0.64000000.2, 8.0.4303360000.4, \(\Q(\zeta_{40})^+\), 8.0.68853760000.27, 8.0.1024000000.1, 8.0.4303360000.2, 8.0.64000000.1, 8.0.107584000000.16, 8.8.1721344000000.4, 8.0.107584000000.24, 8.8.1721344000000.6, 8.0.68853760000.8, \(\Q(\zeta_{20})\), 8.0.268960000.3, 8.0.1024000000.2, 8.0.107584000000.22, 8.8.6724000000.1, 8.0.26265625.1, 8.8.107584000000.3, 8.0.1721344000000.1, 8.0.6724000000.7, 8.0.6724000000.3, 8.0.1721344000000.4, 16.0.2963025166336000000000000.12, 16.0.2963025166336000000000000.8, 16.0.2963025166336000000000000.7, 16.0.2963025166336000000000000.11, 16.0.11574317056000000000000.10, 16.0.2963025166336000000000000.13, 16.0.2963025166336000000000000.10, 16.0.4740840266137600000000.3, \(\Q(\zeta_{40})\), 16.0.11574317056000000000000.2, 16.16.2963025166336000000000000.1, 16.0.2963025166336000000000000.6, 16.0.2963025166336000000000000.9, 16.0.2963025166336000000000000.5, 16.0.45212176000000000000.9

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 ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.2.0.1}{2} }^{16}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{16}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ R ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(41\) Copy content Toggle raw display 41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 123$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 123$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 123$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 123$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 123$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 123$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 123$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 123$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$