Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 5*x^30 - 2*x^28 + 29*x^26 + 210*x^24 - 901*x^22 - 4431*x^20 + 4007*x^18 + 45913*x^16 + 21602*x^14 - 46434*x^12 + 2224*x^10 + 76721*x^8 + 36772*x^6 + 12160*x^4 + 2112*x^2 + 256)
gp: K = bnfinit(y^32 + 5*y^30 - 2*y^28 + 29*y^26 + 210*y^24 - 901*y^22 - 4431*y^20 + 4007*y^18 + 45913*y^16 + 21602*y^14 - 46434*y^12 + 2224*y^10 + 76721*y^8 + 36772*y^6 + 12160*y^4 + 2112*y^2 + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 5*x^30 - 2*x^28 + 29*x^26 + 210*x^24 - 901*x^22 - 4431*x^20 + 4007*x^18 + 45913*x^16 + 21602*x^14 - 46434*x^12 + 2224*x^10 + 76721*x^8 + 36772*x^6 + 12160*x^4 + 2112*x^2 + 256);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 + 5*x^30 - 2*x^28 + 29*x^26 + 210*x^24 - 901*x^22 - 4431*x^20 + 4007*x^18 + 45913*x^16 + 21602*x^14 - 46434*x^12 + 2224*x^10 + 76721*x^8 + 36772*x^6 + 12160*x^4 + 2112*x^2 + 256)
\( x^{32} + 5 x^{30} - 2 x^{28} + 29 x^{26} + 210 x^{24} - 901 x^{22} - 4431 x^{20} + 4007 x^{18} + \cdots + 256 \)
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: |
\(119193990224161520045746372986484162560000000000000000\)
\(\medspace = 2^{32}\cdot 3^{16}\cdot 5^{16}\cdot 89^{8}\cdot 181^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(45.57\) | 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: | $2\cdot 3^{1/2}5^{1/2}89^{1/2}181^{1/2}\approx 983.1276621070124$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(89\), \(181\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{5}a^{20}-\frac{2}{5}a^{18}-\frac{2}{5}a^{16}-\frac{1}{5}a^{14}-\frac{1}{5}a^{12}-\frac{2}{5}a^{10}+\frac{2}{5}a^{8}-\frac{2}{5}a^{6}+\frac{2}{5}a^{4}+\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{21}-\frac{2}{5}a^{19}-\frac{2}{5}a^{17}-\frac{1}{5}a^{15}-\frac{1}{5}a^{13}-\frac{2}{5}a^{11}+\frac{2}{5}a^{9}-\frac{2}{5}a^{7}+\frac{2}{5}a^{5}+\frac{2}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{25}a^{22}-\frac{11}{25}a^{18}-\frac{1}{5}a^{16}-\frac{8}{25}a^{14}+\frac{1}{25}a^{12}-\frac{12}{25}a^{10}-\frac{3}{25}a^{8}+\frac{3}{25}a^{6}-\frac{9}{25}a^{4}-\frac{2}{25}a^{2}+\frac{3}{25}$, $\frac{1}{50}a^{23}+\frac{7}{25}a^{19}-\frac{1}{10}a^{17}+\frac{17}{50}a^{15}-\frac{12}{25}a^{13}+\frac{13}{50}a^{11}+\frac{11}{25}a^{9}+\frac{3}{50}a^{7}-\frac{9}{50}a^{5}+\frac{23}{50}a^{3}+\frac{3}{50}a$, $\frac{1}{900}a^{24}+\frac{4}{225}a^{22}+\frac{17}{450}a^{20}-\frac{7}{300}a^{18}-\frac{53}{900}a^{16}+\frac{82}{225}a^{14}-\frac{91}{900}a^{12}-\frac{1}{90}a^{10}+\frac{59}{180}a^{8}+\frac{199}{900}a^{6}+\frac{73}{300}a^{4}+\frac{61}{900}a^{2}+\frac{82}{225}$, $\frac{1}{1800}a^{25}+\frac{2}{225}a^{23}-\frac{73}{900}a^{21}-\frac{187}{600}a^{19}-\frac{593}{1800}a^{17}-\frac{49}{225}a^{15}+\frac{89}{1800}a^{13}-\frac{11}{36}a^{11}-\frac{13}{360}a^{9}+\frac{559}{1800}a^{7}-\frac{47}{600}a^{5}+\frac{601}{1800}a^{3}+\frac{127}{450}a$, $\frac{1}{1800}a^{26}+\frac{1}{60}a^{22}-\frac{1}{72}a^{20}+\frac{31}{1800}a^{18}-\frac{11}{75}a^{16}-\frac{139}{360}a^{14}+\frac{43}{300}a^{12}-\frac{49}{1800}a^{10}+\frac{101}{600}a^{8}-\frac{589}{1800}a^{6}+\frac{769}{1800}a^{4}+\frac{23}{50}a^{2}+\frac{46}{225}$, $\frac{1}{1800}a^{27}-\frac{1}{300}a^{23}-\frac{1}{72}a^{21}-\frac{473}{1800}a^{19}-\frac{7}{150}a^{17}+\frac{493}{1800}a^{15}-\frac{113}{300}a^{13}-\frac{517}{1800}a^{11}-\frac{163}{600}a^{9}-\frac{697}{1800}a^{7}-\frac{707}{1800}a^{5}+\frac{13}{90}a$, $\frac{1}{28717200}a^{28}-\frac{497}{3190800}a^{26}+\frac{7417}{14358600}a^{24}-\frac{312359}{28717200}a^{22}+\frac{100573}{1196550}a^{20}+\frac{153791}{1063600}a^{18}-\frac{338381}{1914480}a^{16}-\frac{12104791}{28717200}a^{14}-\frac{261041}{1914480}a^{12}+\frac{15077}{3589650}a^{10}+\frac{564977}{3589650}a^{8}-\frac{2109397}{4786200}a^{6}+\frac{1428053}{9572400}a^{4}-\frac{268106}{1794825}a^{2}-\frac{10771}{1794825}$, $\frac{1}{57434400}a^{29}+\frac{3827}{19144800}a^{27}+\frac{7417}{28717200}a^{25}+\frac{166261}{57434400}a^{23}+\frac{37313}{1063600}a^{21}+\frac{1548977}{19144800}a^{19}+\frac{6476543}{19144800}a^{17}+\frac{5524379}{57434400}a^{15}-\frac{9505561}{19144800}a^{13}-\frac{66113}{5743440}a^{11}+\frac{4676939}{28717200}a^{9}-\frac{918887}{2393100}a^{7}-\frac{810961}{3828960}a^{5}-\frac{2474623}{7179300}a^{3}+\frac{356171}{3589650}a$, $\frac{1}{34\!\cdots\!00}a^{30}+\frac{12\!\cdots\!01}{12\!\cdots\!20}a^{28}-\frac{17\!\cdots\!11}{17\!\cdots\!00}a^{26}-\frac{34\!\cdots\!39}{69\!\cdots\!40}a^{24}-\frac{29\!\cdots\!21}{19\!\cdots\!00}a^{22}+\frac{36\!\cdots\!19}{38\!\cdots\!00}a^{20}-\frac{54\!\cdots\!81}{11\!\cdots\!00}a^{18}-\frac{24\!\cdots\!69}{34\!\cdots\!00}a^{16}+\frac{30\!\cdots\!71}{11\!\cdots\!00}a^{14}-\frac{51\!\cdots\!59}{90\!\cdots\!00}a^{12}+\frac{65\!\cdots\!51}{17\!\cdots\!00}a^{10}-\frac{14\!\cdots\!19}{28\!\cdots\!60}a^{8}-\frac{11\!\cdots\!91}{24\!\cdots\!00}a^{6}-\frac{14\!\cdots\!07}{10\!\cdots\!00}a^{4}+\frac{10\!\cdots\!37}{13\!\cdots\!50}a^{2}+\frac{15\!\cdots\!11}{17\!\cdots\!00}$, $\frac{1}{69\!\cdots\!00}a^{31}+\frac{12\!\cdots\!01}{24\!\cdots\!40}a^{29}+\frac{78\!\cdots\!01}{34\!\cdots\!00}a^{27}-\frac{34\!\cdots\!39}{13\!\cdots\!80}a^{25}+\frac{65\!\cdots\!73}{12\!\cdots\!00}a^{23}-\frac{13\!\cdots\!23}{23\!\cdots\!00}a^{21}+\frac{10\!\cdots\!27}{23\!\cdots\!00}a^{19}-\frac{11\!\cdots\!41}{27\!\cdots\!56}a^{17}+\frac{96\!\cdots\!91}{23\!\cdots\!00}a^{15}+\frac{52\!\cdots\!81}{36\!\cdots\!60}a^{13}+\frac{70\!\cdots\!03}{34\!\cdots\!00}a^{11}-\frac{10\!\cdots\!87}{28\!\cdots\!00}a^{9}-\frac{96\!\cdots\!07}{48\!\cdots\!00}a^{7}+\frac{14\!\cdots\!11}{21\!\cdots\!00}a^{5}+\frac{47\!\cdots\!11}{67\!\cdots\!25}a^{3}-\frac{30\!\cdots\!91}{11\!\cdots\!00}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{2}\times C_{42}$, which has order $168$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $168$ (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: |
\( \frac{101519717266549589329344049}{69008118812713427092356326400} a^{31} + \frac{9814756809491871509067481}{1210668751100235563023795200} a^{29} + \frac{18259287081873723642833149}{34504059406356713546178163200} a^{27} + \frac{548699673241276062906837857}{13801623762542685418471265280} a^{25} + \frac{254118574121350280829841403}{766756875696815856581736960} a^{23} - \frac{27049248516575068805183189263}{23002706270904475697452108800} a^{21} - \frac{55627107382565054493165108979}{7667568756968158565817369600} a^{19} + \frac{39027232426174851649098459943}{13801623762542685418471265280} a^{17} + \frac{1651657932490947478613200167839}{23002706270904475697452108800} a^{15} + \frac{118060818167691810266604766733}{1816003126650353344535692800} a^{13} - \frac{2248599575579577228628362750877}{34504059406356713546178163200} a^{11} - \frac{105309918353601354814717931159}{2875338283863059462181513600} a^{9} + \frac{63268428325154320660849747781}{489419282359669695690470400} a^{7} + \frac{7388526630409816921471024519}{67390741028040456144879225} a^{5} + \frac{12114834843541589171118251011}{539125928224323649159033800} a^{3} + \frac{1424031690588536696250051451}{359417285482882432772689200} a \)
(order $12$)
| 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: |
$\frac{229538629707331}{10\!\cdots\!00}a^{30}+\frac{780836543602403}{10\!\cdots\!00}a^{28}-\frac{203871257104773}{10\!\cdots\!60}a^{26}+\frac{82\!\cdots\!47}{10\!\cdots\!00}a^{24}+\frac{35\!\cdots\!93}{10\!\cdots\!60}a^{22}-\frac{27\!\cdots\!83}{10\!\cdots\!00}a^{20}-\frac{64\!\cdots\!61}{10\!\cdots\!00}a^{18}+\frac{22\!\cdots\!57}{10\!\cdots\!00}a^{16}+\frac{33\!\cdots\!59}{41\!\cdots\!64}a^{14}-\frac{46\!\cdots\!67}{52\!\cdots\!00}a^{12}-\frac{45\!\cdots\!27}{52\!\cdots\!00}a^{10}+\frac{73\!\cdots\!97}{13\!\cdots\!00}a^{8}+\frac{75\!\cdots\!37}{22\!\cdots\!00}a^{6}+\frac{10\!\cdots\!29}{81\!\cdots\!00}a^{4}+\frac{499432533738121}{20\!\cdots\!25}a^{2}-\frac{10\!\cdots\!97}{16\!\cdots\!00}$, $\frac{10\!\cdots\!49}{69\!\cdots\!00}a^{31}+\frac{98\!\cdots\!81}{12\!\cdots\!00}a^{29}+\frac{18\!\cdots\!49}{34\!\cdots\!00}a^{27}+\frac{54\!\cdots\!57}{13\!\cdots\!80}a^{25}+\frac{25\!\cdots\!03}{76\!\cdots\!60}a^{23}-\frac{27\!\cdots\!63}{23\!\cdots\!00}a^{21}-\frac{55\!\cdots\!79}{76\!\cdots\!00}a^{19}+\frac{39\!\cdots\!43}{13\!\cdots\!80}a^{17}+\frac{16\!\cdots\!39}{23\!\cdots\!00}a^{15}+\frac{11\!\cdots\!33}{18\!\cdots\!00}a^{13}-\frac{22\!\cdots\!77}{34\!\cdots\!00}a^{11}-\frac{10\!\cdots\!59}{28\!\cdots\!00}a^{9}+\frac{63\!\cdots\!81}{48\!\cdots\!00}a^{7}+\frac{73\!\cdots\!19}{67\!\cdots\!25}a^{5}+\frac{12\!\cdots\!11}{53\!\cdots\!00}a^{3}+\frac{14\!\cdots\!51}{35\!\cdots\!00}a-1$, $\frac{37\!\cdots\!01}{43\!\cdots\!00}a^{30}+\frac{30\!\cdots\!33}{75\!\cdots\!00}a^{28}-\frac{70\!\cdots\!61}{21\!\cdots\!00}a^{26}+\frac{11\!\cdots\!73}{43\!\cdots\!00}a^{24}+\frac{12\!\cdots\!47}{71\!\cdots\!00}a^{22}-\frac{12\!\cdots\!67}{14\!\cdots\!00}a^{20}-\frac{50\!\cdots\!09}{14\!\cdots\!00}a^{18}+\frac{20\!\cdots\!67}{43\!\cdots\!00}a^{16}+\frac{61\!\cdots\!47}{15\!\cdots\!00}a^{14}+\frac{48\!\cdots\!51}{11\!\cdots\!00}a^{12}-\frac{97\!\cdots\!17}{21\!\cdots\!00}a^{10}+\frac{13\!\cdots\!73}{71\!\cdots\!84}a^{8}+\frac{19\!\cdots\!01}{30\!\cdots\!00}a^{6}+\frac{80\!\cdots\!47}{10\!\cdots\!00}a^{4}+\frac{27\!\cdots\!87}{13\!\cdots\!50}a^{2}-\frac{38\!\cdots\!59}{22\!\cdots\!75}$, $\frac{95\!\cdots\!23}{86\!\cdots\!00}a^{30}+\frac{82\!\cdots\!59}{15\!\cdots\!00}a^{28}-\frac{11\!\cdots\!53}{43\!\cdots\!00}a^{26}+\frac{27\!\cdots\!79}{86\!\cdots\!00}a^{24}+\frac{33\!\cdots\!31}{14\!\cdots\!00}a^{22}-\frac{29\!\cdots\!41}{28\!\cdots\!00}a^{20}-\frac{13\!\cdots\!07}{28\!\cdots\!00}a^{18}+\frac{42\!\cdots\!41}{86\!\cdots\!00}a^{16}+\frac{48\!\cdots\!43}{95\!\cdots\!00}a^{14}+\frac{43\!\cdots\!23}{22\!\cdots\!00}a^{12}-\frac{23\!\cdots\!91}{43\!\cdots\!00}a^{10}+\frac{10\!\cdots\!51}{14\!\cdots\!68}a^{8}+\frac{53\!\cdots\!23}{61\!\cdots\!00}a^{6}+\frac{33\!\cdots\!03}{10\!\cdots\!00}a^{4}+\frac{85\!\cdots\!13}{13\!\cdots\!50}a^{2}+\frac{87\!\cdots\!31}{14\!\cdots\!50}$, $\frac{15\!\cdots\!83}{95\!\cdots\!20}a^{30}+\frac{17\!\cdots\!77}{22\!\cdots\!00}a^{28}-\frac{31\!\cdots\!49}{71\!\cdots\!00}a^{26}+\frac{20\!\cdots\!83}{43\!\cdots\!00}a^{24}+\frac{71\!\cdots\!51}{21\!\cdots\!00}a^{22}-\frac{86\!\cdots\!13}{57\!\cdots\!72}a^{20}-\frac{99\!\cdots\!71}{14\!\cdots\!00}a^{18}+\frac{36\!\cdots\!93}{47\!\cdots\!00}a^{16}+\frac{31\!\cdots\!31}{43\!\cdots\!00}a^{14}+\frac{89\!\cdots\!43}{37\!\cdots\!00}a^{12}-\frac{16\!\cdots\!19}{21\!\cdots\!00}a^{10}+\frac{11\!\cdots\!26}{67\!\cdots\!25}a^{8}+\frac{36\!\cdots\!67}{30\!\cdots\!00}a^{6}+\frac{17\!\cdots\!23}{39\!\cdots\!00}a^{4}+\frac{12\!\cdots\!18}{67\!\cdots\!25}a^{2}+\frac{21\!\cdots\!31}{67\!\cdots\!25}$, $\frac{61\!\cdots\!03}{17\!\cdots\!00}a^{30}+\frac{17\!\cdots\!33}{90\!\cdots\!00}a^{28}+\frac{11\!\cdots\!11}{86\!\cdots\!00}a^{26}+\frac{16\!\cdots\!63}{17\!\cdots\!00}a^{24}+\frac{69\!\cdots\!41}{86\!\cdots\!00}a^{22}-\frac{54\!\cdots\!47}{19\!\cdots\!00}a^{20}-\frac{20\!\cdots\!19}{11\!\cdots\!40}a^{18}+\frac{11\!\cdots\!37}{17\!\cdots\!00}a^{16}+\frac{60\!\cdots\!47}{34\!\cdots\!20}a^{14}+\frac{14\!\cdots\!03}{90\!\cdots\!40}a^{12}-\frac{44\!\cdots\!61}{28\!\cdots\!00}a^{10}-\frac{19\!\cdots\!31}{21\!\cdots\!00}a^{8}+\frac{12\!\cdots\!73}{40\!\cdots\!00}a^{6}+\frac{28\!\cdots\!91}{10\!\cdots\!00}a^{4}+\frac{73\!\cdots\!21}{13\!\cdots\!50}a^{2}-a+\frac{25\!\cdots\!07}{26\!\cdots\!00}$, $\frac{89\!\cdots\!43}{13\!\cdots\!80}a^{31}-\frac{76\!\cdots\!69}{63\!\cdots\!00}a^{30}+\frac{22\!\cdots\!53}{72\!\cdots\!20}a^{29}-\frac{59\!\cdots\!77}{10\!\cdots\!00}a^{28}-\frac{65\!\cdots\!29}{34\!\cdots\!00}a^{27}+\frac{20\!\cdots\!71}{63\!\cdots\!80}a^{26}+\frac{13\!\cdots\!23}{69\!\cdots\!00}a^{25}-\frac{68\!\cdots\!07}{19\!\cdots\!00}a^{24}+\frac{18\!\cdots\!69}{13\!\cdots\!28}a^{23}-\frac{23\!\cdots\!61}{95\!\cdots\!00}a^{22}-\frac{13\!\cdots\!41}{23\!\cdots\!00}a^{21}+\frac{71\!\cdots\!89}{63\!\cdots\!00}a^{20}-\frac{21\!\cdots\!57}{76\!\cdots\!00}a^{19}+\frac{33\!\cdots\!19}{63\!\cdots\!00}a^{18}+\frac{21\!\cdots\!13}{69\!\cdots\!00}a^{17}-\frac{69\!\cdots\!51}{12\!\cdots\!60}a^{16}+\frac{20\!\cdots\!23}{69\!\cdots\!00}a^{15}-\frac{10\!\cdots\!23}{19\!\cdots\!00}a^{14}+\frac{30\!\cdots\!19}{36\!\cdots\!60}a^{13}-\frac{12\!\cdots\!29}{67\!\cdots\!64}a^{12}-\frac{37\!\cdots\!57}{11\!\cdots\!00}a^{11}+\frac{10\!\cdots\!79}{19\!\cdots\!40}a^{10}+\frac{25\!\cdots\!17}{34\!\cdots\!32}a^{9}-\frac{26\!\cdots\!09}{23\!\cdots\!00}a^{8}+\frac{48\!\cdots\!51}{97\!\cdots\!80}a^{7}-\frac{24\!\cdots\!27}{27\!\cdots\!80}a^{6}+\frac{37\!\cdots\!27}{26\!\cdots\!00}a^{5}-\frac{12\!\cdots\!97}{39\!\cdots\!00}a^{4}+\frac{30\!\cdots\!79}{10\!\cdots\!60}a^{3}-\frac{40\!\cdots\!69}{29\!\cdots\!10}a^{2}+\frac{25\!\cdots\!59}{10\!\cdots\!00}a-\frac{81\!\cdots\!79}{59\!\cdots\!20}$, $\frac{27\!\cdots\!57}{17\!\cdots\!00}a^{31}+\frac{22\!\cdots\!99}{34\!\cdots\!00}a^{30}+\frac{12\!\cdots\!31}{90\!\cdots\!00}a^{29}+\frac{58\!\cdots\!97}{18\!\cdots\!00}a^{28}+\frac{20\!\cdots\!29}{86\!\cdots\!00}a^{27}-\frac{27\!\cdots\!53}{17\!\cdots\!00}a^{26}+\frac{12\!\cdots\!31}{57\!\cdots\!00}a^{25}+\frac{12\!\cdots\!11}{69\!\cdots\!40}a^{24}+\frac{43\!\cdots\!99}{86\!\cdots\!00}a^{23}+\frac{23\!\cdots\!49}{17\!\cdots\!00}a^{22}-\frac{15\!\cdots\!79}{57\!\cdots\!00}a^{21}-\frac{68\!\cdots\!01}{11\!\cdots\!00}a^{20}-\frac{72\!\cdots\!33}{57\!\cdots\!00}a^{19}-\frac{32\!\cdots\!39}{11\!\cdots\!00}a^{18}-\frac{59\!\cdots\!49}{34\!\cdots\!20}a^{17}+\frac{98\!\cdots\!01}{34\!\cdots\!00}a^{16}+\frac{18\!\cdots\!49}{17\!\cdots\!00}a^{15}+\frac{10\!\cdots\!91}{34\!\cdots\!00}a^{14}+\frac{13\!\cdots\!33}{45\!\cdots\!00}a^{13}+\frac{10\!\cdots\!67}{90\!\cdots\!00}a^{12}-\frac{49\!\cdots\!77}{86\!\cdots\!00}a^{11}-\frac{19\!\cdots\!61}{57\!\cdots\!00}a^{10}-\frac{65\!\cdots\!11}{21\!\cdots\!00}a^{9}+\frac{16\!\cdots\!53}{43\!\cdots\!00}a^{8}+\frac{30\!\cdots\!81}{12\!\cdots\!00}a^{7}+\frac{26\!\cdots\!83}{48\!\cdots\!40}a^{6}+\frac{41\!\cdots\!65}{86\!\cdots\!08}a^{5}+\frac{20\!\cdots\!73}{10\!\cdots\!00}a^{4}+\frac{24\!\cdots\!77}{44\!\cdots\!50}a^{3}+\frac{30\!\cdots\!01}{26\!\cdots\!00}a^{2}+\frac{26\!\cdots\!13}{26\!\cdots\!00}a+\frac{23\!\cdots\!71}{53\!\cdots\!00}$, $\frac{56\!\cdots\!61}{69\!\cdots\!40}a^{30}+\frac{69\!\cdots\!59}{18\!\cdots\!00}a^{28}-\frac{46\!\cdots\!59}{17\!\cdots\!00}a^{26}+\frac{27\!\cdots\!79}{11\!\cdots\!00}a^{24}+\frac{56\!\cdots\!63}{34\!\cdots\!20}a^{22}-\frac{89\!\cdots\!87}{11\!\cdots\!00}a^{20}-\frac{39\!\cdots\!49}{11\!\cdots\!00}a^{18}+\frac{14\!\cdots\!71}{34\!\cdots\!00}a^{16}+\frac{25\!\cdots\!17}{69\!\cdots\!40}a^{14}+\frac{68\!\cdots\!01}{90\!\cdots\!00}a^{12}-\frac{71\!\cdots\!73}{17\!\cdots\!00}a^{10}+\frac{62\!\cdots\!71}{43\!\cdots\!00}a^{8}+\frac{47\!\cdots\!11}{81\!\cdots\!00}a^{6}+\frac{70\!\cdots\!63}{53\!\cdots\!00}a^{4}+\frac{44\!\cdots\!79}{89\!\cdots\!30}a^{2}+\frac{18\!\cdots\!49}{53\!\cdots\!00}$, $\frac{28\!\cdots\!23}{18\!\cdots\!00}a^{30}+\frac{28\!\cdots\!63}{36\!\cdots\!60}a^{28}-\frac{26\!\cdots\!61}{90\!\cdots\!00}a^{26}+\frac{27\!\cdots\!13}{60\!\cdots\!00}a^{24}+\frac{30\!\cdots\!73}{90\!\cdots\!00}a^{22}-\frac{84\!\cdots\!37}{60\!\cdots\!00}a^{20}-\frac{28\!\cdots\!53}{40\!\cdots\!40}a^{18}+\frac{44\!\cdots\!41}{72\!\cdots\!12}a^{16}+\frac{13\!\cdots\!03}{18\!\cdots\!00}a^{14}+\frac{33\!\cdots\!69}{90\!\cdots\!00}a^{12}-\frac{68\!\cdots\!11}{90\!\cdots\!00}a^{10}-\frac{56\!\cdots\!87}{22\!\cdots\!00}a^{8}+\frac{21\!\cdots\!53}{17\!\cdots\!44}a^{6}+\frac{34\!\cdots\!63}{56\!\cdots\!00}a^{4}+\frac{61\!\cdots\!31}{47\!\cdots\!70}a^{2}+\frac{84\!\cdots\!67}{28\!\cdots\!00}$, $\frac{11\!\cdots\!39}{86\!\cdots\!80}a^{31}+\frac{10\!\cdots\!79}{69\!\cdots\!40}a^{30}+\frac{18\!\cdots\!41}{75\!\cdots\!00}a^{29}+\frac{13\!\cdots\!41}{18\!\cdots\!00}a^{28}+\frac{43\!\cdots\!03}{53\!\cdots\!00}a^{27}-\frac{10\!\cdots\!09}{34\!\cdots\!20}a^{26}-\frac{10\!\cdots\!93}{43\!\cdots\!00}a^{25}+\frac{15\!\cdots\!31}{34\!\cdots\!00}a^{24}+\frac{31\!\cdots\!09}{39\!\cdots\!00}a^{23}+\frac{55\!\cdots\!69}{17\!\cdots\!00}a^{22}+\frac{13\!\cdots\!23}{57\!\cdots\!72}a^{21}-\frac{17\!\cdots\!01}{12\!\cdots\!00}a^{20}-\frac{10\!\cdots\!79}{47\!\cdots\!00}a^{19}-\frac{78\!\cdots\!83}{11\!\cdots\!00}a^{18}-\frac{29\!\cdots\!77}{43\!\cdots\!00}a^{17}+\frac{20\!\cdots\!37}{34\!\cdots\!00}a^{16}+\frac{24\!\cdots\!51}{15\!\cdots\!00}a^{15}+\frac{24\!\cdots\!19}{34\!\cdots\!00}a^{14}+\frac{46\!\cdots\!07}{56\!\cdots\!00}a^{13}+\frac{61\!\cdots\!87}{18\!\cdots\!80}a^{12}+\frac{37\!\cdots\!63}{43\!\cdots\!40}a^{11}-\frac{42\!\cdots\!97}{57\!\cdots\!00}a^{10}-\frac{32\!\cdots\!83}{35\!\cdots\!00}a^{9}+\frac{34\!\cdots\!01}{43\!\cdots\!00}a^{8}+\frac{15\!\cdots\!93}{40\!\cdots\!92}a^{7}+\frac{33\!\cdots\!79}{27\!\cdots\!00}a^{6}+\frac{60\!\cdots\!77}{43\!\cdots\!40}a^{5}+\frac{60\!\cdots\!79}{10\!\cdots\!00}a^{4}+\frac{26\!\cdots\!61}{10\!\cdots\!60}a^{3}+\frac{18\!\cdots\!79}{13\!\cdots\!50}a^{2}+\frac{15\!\cdots\!34}{22\!\cdots\!75}a+\frac{19\!\cdots\!31}{53\!\cdots\!00}$, $\frac{12\!\cdots\!93}{73\!\cdots\!60}a^{31}+\frac{93\!\cdots\!71}{69\!\cdots\!40}a^{30}+\frac{57\!\cdots\!83}{19\!\cdots\!00}a^{29}+\frac{13\!\cdots\!69}{20\!\cdots\!00}a^{28}+\frac{17\!\cdots\!33}{18\!\cdots\!00}a^{27}-\frac{79\!\cdots\!17}{17\!\cdots\!00}a^{26}-\frac{64\!\cdots\!71}{40\!\cdots\!00}a^{25}+\frac{13\!\cdots\!67}{34\!\cdots\!00}a^{24}+\frac{17\!\cdots\!39}{18\!\cdots\!00}a^{23}+\frac{15\!\cdots\!39}{57\!\cdots\!00}a^{22}+\frac{11\!\cdots\!19}{40\!\cdots\!00}a^{21}-\frac{15\!\cdots\!21}{11\!\cdots\!00}a^{20}-\frac{22\!\cdots\!43}{81\!\cdots\!40}a^{19}-\frac{65\!\cdots\!83}{11\!\cdots\!00}a^{18}-\frac{30\!\cdots\!77}{36\!\cdots\!00}a^{17}+\frac{49\!\cdots\!41}{69\!\cdots\!40}a^{16}+\frac{67\!\cdots\!93}{36\!\cdots\!00}a^{15}+\frac{70\!\cdots\!09}{11\!\cdots\!00}a^{14}+\frac{95\!\cdots\!57}{96\!\cdots\!00}a^{13}+\frac{11\!\cdots\!19}{90\!\cdots\!00}a^{12}+\frac{29\!\cdots\!39}{18\!\cdots\!00}a^{11}-\frac{12\!\cdots\!23}{17\!\cdots\!00}a^{10}-\frac{50\!\cdots\!57}{45\!\cdots\!00}a^{9}+\frac{85\!\cdots\!13}{47\!\cdots\!00}a^{8}+\frac{46\!\cdots\!63}{12\!\cdots\!00}a^{7}+\frac{26\!\cdots\!71}{24\!\cdots\!00}a^{6}+\frac{38\!\cdots\!51}{22\!\cdots\!00}a^{5}+\frac{25\!\cdots\!13}{10\!\cdots\!00}a^{4}+\frac{17\!\cdots\!11}{38\!\cdots\!00}a^{3}-\frac{10\!\cdots\!39}{26\!\cdots\!00}a^{2}+\frac{48\!\cdots\!63}{57\!\cdots\!00}a-\frac{81\!\cdots\!47}{35\!\cdots\!20}$, $\frac{15\!\cdots\!39}{34\!\cdots\!00}a^{31}+\frac{28\!\cdots\!09}{34\!\cdots\!00}a^{30}+\frac{36\!\cdots\!49}{18\!\cdots\!00}a^{29}+\frac{92\!\cdots\!99}{18\!\cdots\!00}a^{28}-\frac{30\!\cdots\!73}{17\!\cdots\!00}a^{27}+\frac{47\!\cdots\!33}{17\!\cdots\!00}a^{26}+\frac{45\!\cdots\!27}{34\!\cdots\!00}a^{25}+\frac{13\!\cdots\!57}{69\!\cdots\!40}a^{24}+\frac{30\!\cdots\!21}{34\!\cdots\!20}a^{23}+\frac{34\!\cdots\!99}{17\!\cdots\!00}a^{22}-\frac{50\!\cdots\!29}{11\!\cdots\!00}a^{21}-\frac{28\!\cdots\!83}{51\!\cdots\!64}a^{20}-\frac{22\!\cdots\!83}{12\!\cdots\!00}a^{19}-\frac{52\!\cdots\!69}{11\!\cdots\!00}a^{18}+\frac{17\!\cdots\!01}{69\!\cdots\!40}a^{17}-\frac{20\!\cdots\!17}{34\!\cdots\!00}a^{16}+\frac{67\!\cdots\!91}{34\!\cdots\!00}a^{15}+\frac{29\!\cdots\!37}{69\!\cdots\!40}a^{14}+\frac{12\!\cdots\!19}{90\!\cdots\!00}a^{13}+\frac{10\!\cdots\!89}{18\!\cdots\!80}a^{12}-\frac{14\!\cdots\!53}{63\!\cdots\!00}a^{11}-\frac{40\!\cdots\!87}{11\!\cdots\!40}a^{10}+\frac{41\!\cdots\!89}{43\!\cdots\!00}a^{9}-\frac{20\!\cdots\!89}{43\!\cdots\!00}a^{8}+\frac{15\!\cdots\!79}{48\!\cdots\!40}a^{7}+\frac{23\!\cdots\!41}{27\!\cdots\!00}a^{6}+\frac{72\!\cdots\!41}{26\!\cdots\!00}a^{5}+\frac{10\!\cdots\!69}{10\!\cdots\!00}a^{4}+\frac{33\!\cdots\!37}{67\!\cdots\!25}a^{3}+\frac{10\!\cdots\!29}{67\!\cdots\!25}a^{2}-\frac{14\!\cdots\!93}{10\!\cdots\!60}a+\frac{11\!\cdots\!97}{21\!\cdots\!52}$, $\frac{44\!\cdots\!29}{69\!\cdots\!00}a^{31}+\frac{10\!\cdots\!79}{20\!\cdots\!00}a^{30}+\frac{11\!\cdots\!19}{36\!\cdots\!00}a^{29}+\frac{26\!\cdots\!81}{10\!\cdots\!00}a^{28}-\frac{11\!\cdots\!07}{69\!\cdots\!40}a^{27}-\frac{37\!\cdots\!49}{20\!\cdots\!60}a^{26}+\frac{26\!\cdots\!49}{13\!\cdots\!80}a^{25}+\frac{32\!\cdots\!23}{20\!\cdots\!00}a^{24}+\frac{92\!\cdots\!03}{69\!\cdots\!40}a^{23}+\frac{10\!\cdots\!29}{10\!\cdots\!00}a^{22}-\frac{27\!\cdots\!19}{46\!\cdots\!60}a^{21}-\frac{13\!\cdots\!57}{27\!\cdots\!28}a^{20}-\frac{12\!\cdots\!37}{46\!\cdots\!60}a^{19}-\frac{33\!\cdots\!31}{15\!\cdots\!60}a^{18}+\frac{20\!\cdots\!07}{69\!\cdots\!00}a^{17}+\frac{57\!\cdots\!13}{20\!\cdots\!00}a^{16}+\frac{20\!\cdots\!89}{69\!\cdots\!00}a^{15}+\frac{47\!\cdots\!67}{20\!\cdots\!00}a^{14}+\frac{18\!\cdots\!93}{18\!\cdots\!00}a^{13}+\frac{22\!\cdots\!03}{53\!\cdots\!00}a^{12}-\frac{24\!\cdots\!33}{76\!\cdots\!60}a^{11}-\frac{91\!\cdots\!21}{33\!\cdots\!00}a^{10}+\frac{39\!\cdots\!43}{86\!\cdots\!00}a^{9}+\frac{23\!\cdots\!49}{25\!\cdots\!00}a^{8}+\frac{24\!\cdots\!89}{48\!\cdots\!00}a^{7}+\frac{56\!\cdots\!63}{14\!\cdots\!00}a^{6}+\frac{38\!\cdots\!43}{21\!\cdots\!00}a^{5}+\frac{11\!\cdots\!49}{15\!\cdots\!00}a^{4}+\frac{19\!\cdots\!17}{53\!\cdots\!00}a^{3}+\frac{41\!\cdots\!91}{15\!\cdots\!00}a^{2}+\frac{44\!\cdots\!77}{10\!\cdots\!00}a+\frac{36\!\cdots\!03}{31\!\cdots\!00}$, $\frac{41\!\cdots\!83}{69\!\cdots\!00}a^{31}-\frac{23\!\cdots\!71}{12\!\cdots\!00}a^{30}+\frac{10\!\cdots\!09}{36\!\cdots\!00}a^{29}-\frac{13\!\cdots\!03}{18\!\cdots\!00}a^{28}-\frac{66\!\cdots\!61}{34\!\cdots\!00}a^{27}+\frac{23\!\cdots\!51}{19\!\cdots\!00}a^{26}+\frac{40\!\cdots\!49}{23\!\cdots\!00}a^{25}-\frac{20\!\cdots\!37}{34\!\cdots\!00}a^{24}+\frac{42\!\cdots\!93}{34\!\cdots\!00}a^{23}-\frac{57\!\cdots\!31}{17\!\cdots\!00}a^{22}-\frac{14\!\cdots\!89}{25\!\cdots\!00}a^{21}+\frac{77\!\cdots\!77}{38\!\cdots\!00}a^{20}-\frac{58\!\cdots\!43}{23\!\cdots\!00}a^{19}+\frac{82\!\cdots\!33}{12\!\cdots\!00}a^{18}+\frac{21\!\cdots\!21}{69\!\cdots\!00}a^{17}-\frac{56\!\cdots\!83}{38\!\cdots\!00}a^{16}+\frac{18\!\cdots\!79}{69\!\cdots\!00}a^{15}-\frac{52\!\cdots\!57}{69\!\cdots\!40}a^{14}+\frac{11\!\cdots\!71}{18\!\cdots\!00}a^{13}+\frac{23\!\cdots\!17}{60\!\cdots\!60}a^{12}-\frac{10\!\cdots\!39}{34\!\cdots\!00}a^{11}+\frac{17\!\cdots\!13}{17\!\cdots\!00}a^{10}+\frac{72\!\cdots\!93}{86\!\cdots\!00}a^{9}-\frac{39\!\cdots\!59}{43\!\cdots\!00}a^{8}+\frac{22\!\cdots\!87}{48\!\cdots\!00}a^{7}-\frac{28\!\cdots\!13}{24\!\cdots\!00}a^{6}+\frac{11\!\cdots\!77}{10\!\cdots\!00}a^{5}+\frac{23\!\cdots\!79}{35\!\cdots\!00}a^{4}+\frac{23\!\cdots\!89}{14\!\cdots\!50}a^{3}+\frac{14\!\cdots\!17}{13\!\cdots\!50}a^{2}-\frac{14\!\cdots\!53}{10\!\cdots\!00}a+\frac{23\!\cdots\!11}{53\!\cdots\!00}$
| 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: | \( 3049689592222.7627 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 3049689592222.7627 \cdot 168}{12\cdot\sqrt{119193990224161520045746372986484162560000000000000000}}\cr\approx \mathstrut & 0.729683267780638 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 + 5*x^30 - 2*x^28 + 29*x^26 + 210*x^24 - 901*x^22 - 4431*x^20 + 4007*x^18 + 45913*x^16 + 21602*x^14 - 46434*x^12 + 2224*x^10 + 76721*x^8 + 36772*x^6 + 12160*x^4 + 2112*x^2 + 256)
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 + 5*x^30 - 2*x^28 + 29*x^26 + 210*x^24 - 901*x^22 - 4431*x^20 + 4007*x^18 + 45913*x^16 + 21602*x^14 - 46434*x^12 + 2224*x^10 + 76721*x^8 + 36772*x^6 + 12160*x^4 + 2112*x^2 + 256, 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 + 5*x^30 - 2*x^28 + 29*x^26 + 210*x^24 - 901*x^22 - 4431*x^20 + 4007*x^18 + 45913*x^16 + 21602*x^14 - 46434*x^12 + 2224*x^10 + 76721*x^8 + 36772*x^6 + 12160*x^4 + 2112*x^2 + 256);
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 + 5*x^30 - 2*x^28 + 29*x^26 + 210*x^24 - 901*x^22 - 4431*x^20 + 4007*x^18 + 45913*x^16 + 21602*x^14 - 46434*x^12 + 2224*x^10 + 76721*x^8 + 36772*x^6 + 12160*x^4 + 2112*x^2 + 256);
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
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 | R | R | ${\href{/padicField/7.8.0.1}{8} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(89\)
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(181\)
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |