Properties

Label 32.0.170...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.704\times 10^{50}$
Root discriminant \(37.13\)
Ramified primes $2,3,5,11$
Class number $64$ (GRH)
Class group [2, 4, 8] (GRH)
Galois group $D_4\times C_2^3$ (as 32T273)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 24*x^30 + 407*x^28 - 3048*x^26 + 15745*x^24 - 51120*x^22 + 120578*x^20 - 196608*x^18 + 236974*x^16 - 196608*x^14 + 120578*x^12 - 51120*x^10 + 15745*x^8 - 3048*x^6 + 407*x^4 - 24*x^2 + 1)
 
gp: K = bnfinit(y^32 - 24*y^30 + 407*y^28 - 3048*y^26 + 15745*y^24 - 51120*y^22 + 120578*y^20 - 196608*y^18 + 236974*y^16 - 196608*y^14 + 120578*y^12 - 51120*y^10 + 15745*y^8 - 3048*y^6 + 407*y^4 - 24*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 24*x^30 + 407*x^28 - 3048*x^26 + 15745*x^24 - 51120*x^22 + 120578*x^20 - 196608*x^18 + 236974*x^16 - 196608*x^14 + 120578*x^12 - 51120*x^10 + 15745*x^8 - 3048*x^6 + 407*x^4 - 24*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 24*x^30 + 407*x^28 - 3048*x^26 + 15745*x^24 - 51120*x^22 + 120578*x^20 - 196608*x^18 + 236974*x^16 - 196608*x^14 + 120578*x^12 - 51120*x^10 + 15745*x^8 - 3048*x^6 + 407*x^4 - 24*x^2 + 1)
 

\( x^{32} - 24 x^{30} + 407 x^{28} - 3048 x^{26} + 15745 x^{24} - 51120 x^{22} + 120578 x^{20} - 196608 x^{18} + \cdots + 1 \) 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:   \(170408552095468407540899423358812160000000000000000\) \(\medspace = 2^{64}\cdot 3^{24}\cdot 5^{16}\cdot 11^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(37.13\)
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:  $2^{2}3^{3/4}5^{1/2}11^{1/2}\approx 67.62110714844763$
Ramified primes:   \(2\), \(3\), \(5\), \(11\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{16}a^{12}-\frac{1}{2}a^{6}-\frac{7}{16}$, $\frac{1}{16}a^{13}-\frac{1}{2}a^{7}-\frac{7}{16}a$, $\frac{1}{16}a^{14}-\frac{1}{2}a^{8}-\frac{7}{16}a^{2}$, $\frac{1}{16}a^{15}-\frac{1}{2}a^{9}-\frac{7}{16}a^{3}$, $\frac{1}{48}a^{16}+\frac{1}{48}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{23}{48}a^{4}-\frac{23}{48}$, $\frac{1}{48}a^{17}+\frac{1}{48}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{23}{48}a^{5}-\frac{23}{48}a$, $\frac{1}{48}a^{18}+\frac{1}{48}a^{14}-\frac{1}{2}a^{8}-\frac{23}{48}a^{6}-\frac{23}{48}a^{2}-\frac{1}{2}$, $\frac{1}{48}a^{19}+\frac{1}{48}a^{15}-\frac{1}{2}a^{9}-\frac{23}{48}a^{7}-\frac{23}{48}a^{3}-\frac{1}{2}a$, $\frac{1}{48}a^{20}-\frac{1}{48}a^{12}-\frac{23}{48}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}+\frac{23}{48}$, $\frac{1}{48}a^{21}-\frac{1}{48}a^{13}-\frac{23}{48}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}+\frac{23}{48}a$, $\frac{1}{48}a^{22}-\frac{1}{48}a^{14}-\frac{23}{48}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}+\frac{23}{48}a^{2}$, $\frac{1}{48}a^{23}-\frac{1}{48}a^{15}-\frac{23}{48}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}+\frac{23}{48}a^{3}$, $\frac{1}{1536}a^{24}-\frac{1}{96}a^{18}+\frac{1}{48}a^{14}+\frac{1}{768}a^{12}-\frac{1}{2}a^{8}-\frac{1}{96}a^{6}-\frac{23}{48}a^{2}-\frac{383}{1536}$, $\frac{1}{1536}a^{25}-\frac{1}{96}a^{19}+\frac{1}{48}a^{15}+\frac{1}{768}a^{13}-\frac{1}{2}a^{9}-\frac{1}{96}a^{7}-\frac{23}{48}a^{3}-\frac{383}{1536}a$, $\frac{1}{1536}a^{26}-\frac{1}{96}a^{20}+\frac{1}{768}a^{14}-\frac{1}{48}a^{12}-\frac{1}{96}a^{8}-\frac{1}{2}a^{6}-\frac{383}{1536}a^{2}+\frac{23}{48}$, $\frac{1}{1536}a^{27}-\frac{1}{96}a^{21}+\frac{1}{768}a^{15}-\frac{1}{48}a^{13}-\frac{1}{96}a^{9}-\frac{1}{2}a^{7}-\frac{383}{1536}a^{3}+\frac{23}{48}a$, $\frac{1}{614389248}a^{28}-\frac{4337}{68265472}a^{26}-\frac{16831}{76798656}a^{24}+\frac{87743}{12799776}a^{22}+\frac{4415}{556512}a^{20}-\frac{63517}{6399888}a^{18}+\frac{2967793}{307194624}a^{16}-\frac{1104179}{102398208}a^{14}-\frac{479011}{38399328}a^{12}+\frac{4539551}{12799776}a^{10}-\frac{273841}{556512}a^{8}+\frac{910523}{6399888}a^{6}+\frac{294660193}{614389248}a^{4}+\frac{8520173}{204796416}a^{2}+\frac{15949721}{76798656}$, $\frac{1}{614389248}a^{29}-\frac{4337}{68265472}a^{27}-\frac{16831}{76798656}a^{25}+\frac{87743}{12799776}a^{23}+\frac{4415}{556512}a^{21}-\frac{63517}{6399888}a^{19}+\frac{2967793}{307194624}a^{17}-\frac{1104179}{102398208}a^{15}-\frac{479011}{38399328}a^{13}+\frac{4539551}{12799776}a^{11}-\frac{273841}{556512}a^{9}+\frac{910523}{6399888}a^{7}+\frac{294660193}{614389248}a^{5}+\frac{8520173}{204796416}a^{3}+\frac{15949721}{76798656}a$, $\frac{1}{14524776211968}a^{30}+\frac{1115}{1613864023552}a^{28}-\frac{4529360147}{14524776211968}a^{26}-\frac{344884181}{1613864023552}a^{24}+\frac{2245511119}{302599504416}a^{22}+\frac{273696395}{100866501472}a^{20}+\frac{68553446617}{7262388105984}a^{18}+\frac{55504709}{17168766208}a^{16}+\frac{245335235}{154518895872}a^{14}+\frac{6937519315}{806932011776}a^{12}+\frac{88696067}{887388576}a^{10}-\frac{19183447061}{100866501472}a^{8}+\frac{1870878670129}{14524776211968}a^{6}+\frac{280119113083}{1613864023552}a^{4}+\frac{6090288513325}{14524776211968}a^{2}-\frac{750810721797}{1613864023552}$, $\frac{1}{14524776211968}a^{31}+\frac{1115}{1613864023552}a^{29}-\frac{4529360147}{14524776211968}a^{27}-\frac{344884181}{1613864023552}a^{25}+\frac{2245511119}{302599504416}a^{23}+\frac{273696395}{100866501472}a^{21}+\frac{68553446617}{7262388105984}a^{19}+\frac{55504709}{17168766208}a^{17}+\frac{245335235}{154518895872}a^{15}+\frac{6937519315}{806932011776}a^{13}+\frac{88696067}{887388576}a^{11}-\frac{19183447061}{100866501472}a^{9}+\frac{1870878670129}{14524776211968}a^{7}+\frac{280119113083}{1613864023552}a^{5}+\frac{6090288513325}{14524776211968}a^{3}-\frac{750810721797}{1613864023552}a$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

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:   \( \frac{36033961741}{660217100544} a^{31} - \frac{19171119113099}{14524776211968} a^{29} + \frac{163070398171481}{7262388105984} a^{27} - \frac{2475592237058459}{14524776211968} a^{25} + \frac{269332003570057}{302599504416} a^{23} - \frac{111564248757713}{37824938052} a^{21} + \frac{25793310111599507}{3631194052992} a^{19} - \frac{1850690782190389}{154518895872} a^{17} + \frac{1159510078533847}{77259447936} a^{15} - \frac{3117392626573781}{234270584064} a^{13} + \frac{2673977643717241}{302599504416} a^{11} - \frac{162069059908859}{37824938052} a^{9} + \frac{11405563394861975}{7262388105984} a^{7} - \frac{5941360886911595}{14524776211968} a^{5} + \frac{524269443581849}{7262388105984} a^{3} - \frac{82634753254523}{14524776211968} a \)  (order $24$) 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:   $\frac{287624135567}{1210398017664}a^{31}-\frac{41513076978877}{7262388105984}a^{29}+\frac{5106166357943}{52626000768}a^{27}-\frac{26\!\cdots\!91}{3631194052992}a^{25}+\frac{4333436759277}{1146210244}a^{23}-\frac{155684551752509}{12608312684}a^{21}+\frac{571788262651217}{19522548672}a^{19}-\frac{37\!\cdots\!67}{77259447936}a^{17}+\frac{754072760056643}{12876574656}a^{15}-\frac{89\!\cdots\!55}{1815597026496}a^{13}+\frac{383426801674863}{12608312684}a^{11}-\frac{14839834850329}{1146210244}a^{9}+\frac{47\!\cdots\!71}{1210398017664}a^{7}-\frac{52\!\cdots\!17}{7262388105984}a^{5}+\frac{101762136142825}{1210398017664}a^{3}-\frac{14854071783059}{3631194052992}a$, $\frac{175620231}{9169681952}a^{31}-\frac{17170297263}{36678727808}a^{29}+\frac{4579148469}{573105122}a^{27}-\frac{4520908677033}{73357455616}a^{25}+\frac{742902560043}{2292420488}a^{23}-\frac{73367798553}{67424132}a^{21}+\frac{750477445501}{286552561}a^{19}-\frac{1716499853649}{390199232}a^{17}+\frac{32452690620}{6096863}a^{15}-\frac{162272977941249}{36678727808}a^{13}+\frac{5684429625819}{2292420488}a^{11}-\frac{1072387185345}{1146210244}a^{9}+\frac{1864223879049}{9169681952}a^{7}-\frac{1096565649039}{36678727808}a^{5}+\frac{1035711999}{573105122}a^{3}-\frac{4015002503}{2366369536}a$, $\frac{8484018335}{52626000768}a^{30}-\frac{1226311319719}{315756004608}a^{28}+\frac{433922146061}{6578250096}a^{26}-\frac{314044496142137}{631512009216}a^{24}+\frac{514744276399}{199340912}a^{22}-\frac{1092004219919}{128985296}a^{20}+\frac{530385854757245}{26313000384}a^{18}-\frac{112024910655809}{3359106432}a^{16}+\frac{1427397309029}{34990692}a^{14}-\frac{10\!\cdots\!01}{315756004608}a^{12}+\frac{47373853825085}{2192750032}a^{10}-\frac{1860251023509}{199340912}a^{8}+\frac{150995252539547}{52626000768}a^{6}-\frac{171247741771063}{315756004608}a^{4}+\frac{434640683207}{6578250096}a^{2}-\frac{45625833479}{20371355136}$, $\frac{43240761731}{660217100544}a^{31}-\frac{2756090059889}{1815597026496}a^{29}+\frac{368597562358307}{14524776211968}a^{27}-\frac{12\!\cdots\!01}{7262388105984}a^{25}+\frac{66003399489629}{75649876104}a^{23}-\frac{768633067277525}{302599504416}a^{21}+\frac{626431519786015}{117135292032}a^{19}-\frac{135723799404781}{19314861984}a^{17}+\frac{990327286447645}{154518895872}a^{15}-\frac{87\!\cdots\!33}{3631194052992}a^{13}-\frac{4905298191199}{75649876104}a^{11}+\frac{351398660458363}{302599504416}a^{9}-\frac{44\!\cdots\!83}{7262388105984}a^{7}+\frac{447079883509003}{1815597026496}a^{5}-\frac{481155256444669}{14524776211968}a^{3}+\frac{37888890062503}{7262388105984}a-1$, $\frac{36033961741}{660217100544}a^{31}+\frac{7670143}{99859584}a^{30}-\frac{19171119113099}{14524776211968}a^{29}-\frac{8731015}{4680918}a^{28}+\frac{163070398171481}{7262388105984}a^{27}+\frac{396719035}{12482448}a^{26}-\frac{24\!\cdots\!59}{14524776211968}a^{25}-\frac{18193744075}{74894688}a^{24}+\frac{269332003570057}{302599504416}a^{23}+\frac{241076835}{189128}a^{22}-\frac{111564248757713}{37824938052}a^{21}-\frac{3323927899}{780153}a^{20}+\frac{25\!\cdots\!07}{3631194052992}a^{19}+\frac{515596409407}{49929792}a^{18}-\frac{18\!\cdots\!89}{154518895872}a^{17}-\frac{874250042}{49797}a^{16}+\frac{11\!\cdots\!47}{77259447936}a^{15}+\frac{366062332}{16599}a^{14}-\frac{31\!\cdots\!81}{234270584064}a^{13}-\frac{730120712281}{37447344}a^{12}+\frac{26\!\cdots\!41}{302599504416}a^{11}+\frac{26137682841}{2080408}a^{10}-\frac{162069059908859}{37824938052}a^{9}-\frac{402188048}{70923}a^{8}+\frac{11\!\cdots\!75}{7262388105984}a^{7}+\frac{178173807103}{99859584}a^{6}-\frac{59\!\cdots\!95}{14524776211968}a^{5}-\frac{1648457347}{4680918}a^{4}+\frac{524269443581849}{7262388105984}a^{3}+\frac{492368917}{12482448}a^{2}-\frac{82634753254523}{14524776211968}a-\frac{125862247}{74894688}$, $\frac{4552039199}{4854537504}a^{31}+\frac{740308829}{8771000128}a^{30}-\frac{327072103800889}{14524776211968}a^{29}-\frac{212450657293}{105252001536}a^{28}+\frac{13\!\cdots\!99}{3631194052992}a^{27}+\frac{9368800609}{274093754}a^{26}-\frac{41\!\cdots\!57}{14524776211968}a^{25}-\frac{53544948700579}{210504003072}a^{24}+\frac{14\!\cdots\!69}{100866501472}a^{23}+\frac{260649292309}{199340912}a^{22}-\frac{72\!\cdots\!67}{151299752208}a^{21}-\frac{1627344421853}{386955888}a^{20}+\frac{10\!\cdots\!27}{907798513248}a^{19}+\frac{21555545583313}{2192750032}a^{18}-\frac{28\!\cdots\!11}{154518895872}a^{17}-\frac{17683833274219}{1119702144}a^{16}+\frac{86\!\cdots\!77}{38629723968}a^{15}+\frac{218579304391}{11663564}a^{14}-\frac{13\!\cdots\!57}{7262388105984}a^{13}-\frac{15\!\cdots\!03}{105252001536}a^{12}+\frac{11\!\cdots\!85}{100866501472}a^{11}+\frac{19824736110671}{2192750032}a^{10}-\frac{72\!\cdots\!31}{151299752208}a^{9}-\frac{2189503449791}{598022736}a^{8}+\frac{66\!\cdots\!95}{453899256624}a^{7}+\frac{9516276032711}{8771000128}a^{6}-\frac{40\!\cdots\!81}{14524776211968}a^{5}-\frac{6672155885759}{35084000512}a^{4}+\frac{11\!\cdots\!83}{3631194052992}a^{3}+\frac{14596855329}{548187508}a^{2}-\frac{198332896797757}{14524776211968}a-\frac{3529184159}{2263483904}$, $\frac{352522500619}{660217100544}a^{31}+\frac{1084511831771}{7262388105984}a^{30}-\frac{92613188903773}{7262388105984}a^{29}-\frac{51711012330445}{14524776211968}a^{28}+\frac{31\!\cdots\!39}{14524776211968}a^{27}+\frac{27329743140527}{453899256624}a^{26}-\frac{29\!\cdots\!75}{1815597026496}a^{25}-\frac{32\!\cdots\!57}{7262388105984}a^{24}+\frac{207270273335799}{25216625368}a^{23}+\frac{230003864422041}{100866501472}a^{22}-\frac{79\!\cdots\!25}{302599504416}a^{21}-\frac{91683414969297}{12608312684}a^{20}+\frac{71\!\cdots\!87}{117135292032}a^{19}+\frac{61\!\cdots\!95}{3631194052992}a^{18}-\frac{75\!\cdots\!39}{77259447936}a^{17}-\frac{41\!\cdots\!59}{154518895872}a^{16}+\frac{17\!\cdots\!45}{154518895872}a^{15}+\frac{18882770153513}{603589437}a^{14}-\frac{83\!\cdots\!75}{907798513248}a^{13}-\frac{89\!\cdots\!85}{3631194052992}a^{12}+\frac{13\!\cdots\!95}{25216625368}a^{11}+\frac{14\!\cdots\!49}{100866501472}a^{10}-\frac{63\!\cdots\!01}{302599504416}a^{9}-\frac{73856336236813}{12608312684}a^{8}+\frac{43\!\cdots\!21}{7262388105984}a^{7}+\frac{12\!\cdots\!47}{7262388105984}a^{6}-\frac{68\!\cdots\!09}{7262388105984}a^{5}-\frac{205279391482523}{631512009216}a^{4}+\frac{15\!\cdots\!91}{14524776211968}a^{3}+\frac{18217662382481}{453899256624}a^{2}-\frac{2625784837915}{1815597026496}a-\frac{886043978683}{660217100544}$, $\frac{229504873173}{1613864023552}a^{31}-\frac{15739201439}{210504003072}a^{30}-\frac{356388225701}{105252001536}a^{29}+\frac{140216118055}{78939001152}a^{28}+\frac{2716929778687}{47466588928}a^{27}-\frac{3156475858091}{105252001536}a^{26}-\frac{511567000219787}{1210398017664}a^{25}+\frac{34793607275297}{157878002304}a^{24}+\frac{2475419324219}{1146210244}a^{23}-\frac{223259503089}{199340912}a^{22}-\frac{10\!\cdots\!97}{151299752208}a^{21}+\frac{23174976945311}{6578250096}a^{20}+\frac{753316413894453}{47466588928}a^{19}-\frac{844370407955375}{105252001536}a^{18}-\frac{214820854220631}{8584383104}a^{17}+\frac{10344691024313}{839776608}a^{16}+\frac{14701740146073}{504963712}a^{15}-\frac{15381297735181}{1119702144}a^{14}-\frac{13\!\cdots\!53}{605199008832}a^{13}+\frac{775963901734799}{78939001152}a^{12}+\frac{164790953975293}{12608312684}a^{11}-\frac{10733462382627}{2192750032}a^{10}-\frac{67664361083923}{13754522928}a^{9}+\frac{806957288101}{598022736}a^{8}+\frac{129384455468901}{94933177856}a^{7}-\frac{35773621535807}{210504003072}a^{6}-\frac{483984280014467}{2420796035328}a^{5}-\frac{3413700486041}{78939001152}a^{4}+\frac{24345982841887}{806932011776}a^{3}+\frac{1231256340709}{105252001536}a^{2}-\frac{702004213525}{403466005888}a-\frac{13078073747}{9286941312}$, $\frac{27255869981}{110036183424}a^{31}+\frac{1270037449}{14352545664}a^{30}-\frac{24768802497}{4127529472}a^{29}-\frac{7606047949565}{3631194052992}a^{28}+\frac{494554116375383}{4841592070656}a^{27}+\frac{512964419657131}{14524776211968}a^{26}-\frac{37\!\cdots\!91}{4841592070656}a^{25}-\frac{936373537664375}{3631194052992}a^{24}+\frac{12\!\cdots\!91}{302599504416}a^{23}+\frac{98685819767729}{75649876104}a^{22}-\frac{40\!\cdots\!53}{302599504416}a^{21}-\frac{53483273248499}{13156500192}a^{20}+\frac{19\!\cdots\!29}{605199008832}a^{19}+\frac{16\!\cdots\!65}{1815597026496}a^{18}-\frac{28\!\cdots\!19}{51506298624}a^{17}-\frac{536725220768083}{38629723968}a^{16}+\frac{35\!\cdots\!53}{51506298624}a^{15}+\frac{23\!\cdots\!53}{154518895872}a^{14}-\frac{14\!\cdots\!47}{2420796035328}a^{13}-\frac{19\!\cdots\!79}{1815597026496}a^{12}+\frac{11\!\cdots\!11}{302599504416}a^{11}+\frac{386646398581217}{75649876104}a^{10}-\frac{53\!\cdots\!53}{302599504416}a^{9}-\frac{16097874884867}{13156500192}a^{8}+\frac{23\!\cdots\!33}{403466005888}a^{7}+\frac{272859020116645}{3631194052992}a^{6}-\frac{60\!\cdots\!49}{4841592070656}a^{5}+\frac{352785519902467}{3631194052992}a^{4}+\frac{15145493375469}{94933177856}a^{3}-\frac{298453309021877}{14524776211968}a^{2}-\frac{15317096508741}{1613864023552}a+\frac{501085420937}{213599650176}$, $\frac{63981841037}{73357455616}a^{31}+\frac{1141496051}{440144733696}a^{30}-\frac{33891487035753}{1613864023552}a^{29}-\frac{4373574125}{330108550272}a^{28}+\frac{17\!\cdots\!63}{4841592070656}a^{27}-\frac{415028515}{3438630732}a^{26}-\frac{13\!\cdots\!85}{4841592070656}a^{25}+\frac{1987277682677}{165054275136}a^{24}+\frac{183255458592917}{13156500192}a^{23}-\frac{497653590847}{4584840976}a^{22}-\frac{270434983390859}{5933323616}a^{21}+\frac{8777146957975}{13754522928}a^{20}+\frac{13\!\cdots\!81}{1210398017664}a^{19}-\frac{15505005076867}{7099108608}a^{18}-\frac{91\!\cdots\!21}{51506298624}a^{17}+\frac{18777457658921}{3511793088}a^{16}+\frac{11\!\cdots\!21}{51506298624}a^{15}-\frac{1293684766961}{146324712}a^{14}-\frac{44\!\cdots\!97}{2420796035328}a^{13}+\frac{440071111157791}{41263568784}a^{12}+\frac{14\!\cdots\!37}{13156500192}a^{11}-\frac{39508356061471}{4584840976}a^{10}-\frac{49\!\cdots\!75}{100866501472}a^{9}+\frac{68219590155295}{13754522928}a^{8}+\frac{36\!\cdots\!01}{2420796035328}a^{7}-\frac{819516794658349}{440144733696}a^{6}-\frac{14\!\cdots\!35}{4841592070656}a^{5}+\frac{152288143002619}{330108550272}a^{4}+\frac{17\!\cdots\!67}{4841592070656}a^{3}-\frac{394307712101}{6877261464}a^{2}-\frac{1324206168709}{70168001024}a+\frac{791406729191}{165054275136}$, $\frac{6593166341}{9709075008}a^{31}-\frac{24214803835}{4841592070656}a^{30}-\frac{237033086447933}{14524776211968}a^{29}+\frac{1016811722147}{7262388105984}a^{28}+\frac{20\!\cdots\!51}{7262388105984}a^{27}-\frac{22423986329}{8899985424}a^{26}-\frac{30\!\cdots\!41}{14524776211968}a^{25}+\frac{341059945486327}{14524776211968}a^{24}+\frac{32\!\cdots\!57}{302599504416}a^{23}-\frac{162100669971}{1146210244}a^{22}-\frac{26\!\cdots\!65}{75649876104}a^{21}+\frac{3675458519241}{6304156342}a^{20}+\frac{37\!\cdots\!97}{453899256624}a^{19}-\frac{240373690676483}{142399766784}a^{18}-\frac{20\!\cdots\!95}{154518895872}a^{17}+\frac{276900115254877}{77259447936}a^{16}+\frac{12\!\cdots\!49}{77259447936}a^{15}-\frac{523420249009}{94680696}a^{14}-\frac{99\!\cdots\!61}{7262388105984}a^{13}+\frac{45\!\cdots\!07}{7262388105984}a^{12}+\frac{25\!\cdots\!81}{302599504416}a^{11}-\frac{64232784947523}{12608312684}a^{10}-\frac{27\!\cdots\!61}{75649876104}a^{9}+\frac{1690809509315}{573105122}a^{8}+\frac{20\!\cdots\!21}{1815597026496}a^{7}-\frac{331122955748027}{284799533568}a^{6}-\frac{31\!\cdots\!53}{14524776211968}a^{5}+\frac{21\!\cdots\!95}{7262388105984}a^{4}+\frac{19\!\cdots\!71}{7262388105984}a^{3}-\frac{6735498094069}{151299752208}a^{2}-\frac{218891561598617}{14524776211968}a+\frac{53356569970951}{14524776211968}$, $\frac{896685237253}{3631194052992}a^{31}-\frac{1484397227}{12608312684}a^{30}-\frac{10828323904097}{1815597026496}a^{29}+\frac{41267145064963}{14524776211968}a^{28}+\frac{92039418516145}{907798513248}a^{27}-\frac{116887900458413}{2420796035328}a^{26}-\frac{11\!\cdots\!07}{14524776211968}a^{25}+\frac{53\!\cdots\!53}{14524776211968}a^{24}+\frac{302258756058971}{75649876104}a^{23}-\frac{574550326283167}{302599504416}a^{22}-\frac{58575677819401}{4449992712}a^{21}+\frac{157485745060595}{25216625368}a^{20}+\frac{51\!\cdots\!29}{165054275136}a^{19}-\frac{195990589710689}{13156500192}a^{18}-\frac{10\!\cdots\!49}{19314861984}a^{17}+\frac{38\!\cdots\!77}{154518895872}a^{16}+\frac{617758723994801}{9657430992}a^{15}-\frac{70789816556009}{2341195392}a^{14}-\frac{39\!\cdots\!19}{7262388105984}a^{13}+\frac{18\!\cdots\!81}{7262388105984}a^{12}+\frac{25\!\cdots\!59}{75649876104}a^{11}-\frac{47\!\cdots\!39}{302599504416}a^{10}-\frac{10\!\cdots\!17}{75649876104}a^{9}+\frac{170479745880095}{25216625368}a^{8}+\frac{15\!\cdots\!37}{3631194052992}a^{7}-\frac{597422098465087}{302599504416}a^{6}-\frac{14\!\cdots\!37}{1815597026496}a^{5}+\frac{49\!\cdots\!47}{14524776211968}a^{4}+\frac{60756863780365}{907798513248}a^{3}-\frac{65239277939309}{2420796035328}a^{2}-\frac{53269978081591}{14524776211968}a+\frac{15386131495693}{14524776211968}$, $\frac{2015489550007}{7262388105984}a^{31}-\frac{168021983417}{3631194052992}a^{30}-\frac{48654243790139}{7262388105984}a^{29}+\frac{5451385825091}{4841592070656}a^{28}+\frac{16\!\cdots\!15}{14524776211968}a^{27}-\frac{139451284631983}{7262388105984}a^{26}-\frac{12\!\cdots\!65}{14524776211968}a^{25}+\frac{7667868260149}{52060129792}a^{24}+\frac{113054727636383}{25216625368}a^{23}-\frac{13796972953139}{17799970848}a^{22}-\frac{44\!\cdots\!95}{302599504416}a^{21}+\frac{131348481950819}{50433250736}a^{20}+\frac{12\!\cdots\!95}{3631194052992}a^{19}-\frac{11\!\cdots\!95}{1815597026496}a^{18}-\frac{45\!\cdots\!45}{77259447936}a^{17}+\frac{50918174156359}{4682390784}a^{16}+\frac{11\!\cdots\!33}{154518895872}a^{15}-\frac{10\!\cdots\!37}{77259447936}a^{14}-\frac{40\!\cdots\!67}{660217100544}a^{13}+\frac{29\!\cdots\!37}{2420796035328}a^{12}+\frac{976185851113039}{25216625368}a^{11}-\frac{24\!\cdots\!27}{302599504416}a^{10}-\frac{51\!\cdots\!27}{302599504416}a^{9}+\frac{184490750425907}{50433250736}a^{8}+\frac{38\!\cdots\!27}{7262388105984}a^{7}-\frac{41\!\cdots\!25}{3631194052992}a^{6}-\frac{74\!\cdots\!39}{7262388105984}a^{5}+\frac{357909342782721}{1613864023552}a^{4}+\frac{16\!\cdots\!43}{14524776211968}a^{3}-\frac{142389486622831}{7262388105984}a^{2}-\frac{63341113860481}{14524776211968}a+\frac{904580914657}{4841592070656}$, $\frac{13507494317}{94933177856}a^{31}+\frac{1121666966771}{7262388105984}a^{30}-\frac{48403072918177}{14524776211968}a^{29}-\frac{5925237747791}{1613864023552}a^{28}+\frac{90262522031965}{1613864023552}a^{27}+\frac{225200951404525}{3631194052992}a^{26}-\frac{58\!\cdots\!47}{14524776211968}a^{25}-\frac{184189522615279}{403466005888}a^{24}+\frac{601540603862525}{302599504416}a^{23}+\frac{234654986199409}{100866501472}a^{22}-\frac{18\!\cdots\!75}{302599504416}a^{21}-\frac{11\!\cdots\!29}{151299752208}a^{20}+\frac{31\!\cdots\!23}{2420796035328}a^{19}+\frac{61\!\cdots\!79}{3631194052992}a^{18}-\frac{27\!\cdots\!87}{154518895872}a^{17}-\frac{453764202880881}{17168766208}a^{16}+\frac{296445329212467}{17168766208}a^{15}+\frac{106918491032533}{3511793088}a^{14}-\frac{59\!\cdots\!03}{7262388105984}a^{13}-\frac{14\!\cdots\!49}{605199008832}a^{12}+\frac{242455926691613}{302599504416}a^{11}+\frac{13\!\cdots\!57}{100866501472}a^{10}+\frac{759820740168053}{302599504416}a^{9}-\frac{767804679035381}{151299752208}a^{8}-\frac{80\!\cdots\!85}{4841592070656}a^{7}+\frac{10\!\cdots\!83}{7262388105984}a^{6}+\frac{86\!\cdots\!35}{14524776211968}a^{5}-\frac{16762129588521}{70168001024}a^{4}-\frac{116711402012195}{1613864023552}a^{3}+\frac{100055560573285}{3631194052992}a^{2}+\frac{1923463036747}{1320434201088}a-\frac{1099782526349}{1210398017664}$, $\frac{903685075813}{4841592070656}a^{31}-\frac{260605535963}{58567646016}a^{29}+\frac{3915928898129}{52060129792}a^{27}-\frac{80\!\cdots\!79}{14524776211968}a^{25}+\frac{429151424066989}{151299752208}a^{23}-\frac{27\!\cdots\!65}{302599504416}a^{21}+\frac{16\!\cdots\!95}{806932011776}a^{19}-\frac{19897326776413}{623060064}a^{17}+\frac{18\!\cdots\!47}{51506298624}a^{15}-\frac{17\!\cdots\!01}{660217100544}a^{13}+\frac{20\!\cdots\!77}{151299752208}a^{11}-\frac{39802884596995}{9761274336}a^{9}+\frac{33\!\cdots\!81}{4841592070656}a^{7}-\frac{60958720849973}{1815597026496}a^{5}-\frac{65547975629299}{4841592070656}a^{3}+\frac{20940249034325}{14524776211968}a$ 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:  \( 547885182394.1923 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 547885182394.1923 \cdot 64}{24\cdot\sqrt{170408552095468407540899423358812160000000000000000}}\cr\approx \mathstrut & 0.660374793448188 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 24*x^30 + 407*x^28 - 3048*x^26 + 15745*x^24 - 51120*x^22 + 120578*x^20 - 196608*x^18 + 236974*x^16 - 196608*x^14 + 120578*x^12 - 51120*x^10 + 15745*x^8 - 3048*x^6 + 407*x^4 - 24*x^2 + 1)
 
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 - 24*x^30 + 407*x^28 - 3048*x^26 + 15745*x^24 - 51120*x^22 + 120578*x^20 - 196608*x^18 + 236974*x^16 - 196608*x^14 + 120578*x^12 - 51120*x^10 + 15745*x^8 - 3048*x^6 + 407*x^4 - 24*x^2 + 1, 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 - 24*x^30 + 407*x^28 - 3048*x^26 + 15745*x^24 - 51120*x^22 + 120578*x^20 - 196608*x^18 + 236974*x^16 - 196608*x^14 + 120578*x^12 - 51120*x^10 + 15745*x^8 - 3048*x^6 + 407*x^4 - 24*x^2 + 1);
 
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 - 24*x^30 + 407*x^28 - 3048*x^26 + 15745*x^24 - 51120*x^22 + 120578*x^20 - 196608*x^18 + 236974*x^16 - 196608*x^14 + 120578*x^12 - 51120*x^10 + 15745*x^8 - 3048*x^6 + 407*x^4 - 24*x^2 + 1);
 
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

$D_4\times C_2^3$ (as 32T273):

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 $D_4\times C_2^3$
Character table for $D_4\times C_2^3$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{15}) \), 4.0.4752.1, 4.4.1900800.1, 4.4.76032.1, 4.0.118800.1, 4.0.1900800.3, 4.4.4752.1, 4.4.118800.1, 4.0.76032.2, \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(i, \sqrt{30})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{-5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{6})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(\sqrt{3}, \sqrt{5})\), 8.0.3317760000.4, 8.0.3317760000.8, 8.0.3317760000.7, 8.0.3317760000.2, 8.0.3317760000.9, 8.0.207360000.2, 8.0.3317760000.1, 8.0.207360000.1, 8.0.3317760000.6, \(\Q(\zeta_{24})\), 8.0.40960000.1, 8.0.3317760000.3, 8.8.3317760000.1, 8.0.12960000.1, 8.0.3317760000.5, 8.0.3613040640000.40, 8.0.3613040640000.62, 8.0.3613040640000.79, 8.0.3613040640000.52, 8.0.5780865024.3, 8.0.3613040640000.29, 8.0.3613040640000.21, 8.0.5780865024.13, 8.0.3613040640000.41, 8.8.3613040640000.7, 8.8.3613040640000.5, 8.0.3613040640000.63, 8.0.3613040640000.30, 8.8.5780865024.1, 8.8.3613040640000.4, 8.0.5780865024.4, 8.0.3613040640000.19, 8.0.14113440000.11, 8.0.14113440000.12, 8.0.3613040640000.4, 8.0.3613040640000.51, 8.0.22581504.2, 8.0.14113440000.3, 8.0.5780865024.5, 8.8.3613040640000.6, 8.0.14113440000.9, 8.8.14113440000.1, 8.0.3613040640000.26, 16.0.11007531417600000000.1, 16.0.13054062666291609600000000.6, 16.0.13054062666291609600000000.4, 16.0.13054062666291609600000000.8, 16.0.13054062666291609600000000.5, 16.0.13054062666291609600000000.2, 16.0.13054062666291609600000000.9, 16.0.13054062666291609600000000.10, 16.0.13054062666291609600000000.3, 16.0.13054062666291609600000000.12, 16.0.33418400425706520576.1, 16.16.13054062666291609600000000.1, 16.0.13054062666291609600000000.7, 16.0.13054062666291609600000000.11, 16.0.199189188633600000000.2

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: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.4.0.1}{4} }^{8}$ R ${\href{/padicField/13.2.0.1}{2} }^{16}$ ${\href{/padicField/17.2.0.1}{2} }^{16}$ ${\href{/padicField/19.4.0.1}{4} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{16}$ ${\href{/padicField/29.2.0.1}{2} }^{16}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.2.0.1}{2} }^{16}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{16}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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 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}$
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 3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
\(5\) 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}$
\(11\) Copy content Toggle raw display 11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$