Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 14*x^30 + 109*x^28 - 698*x^26 + 3468*x^24 - 10052*x^22 + 23637*x^20 - 66932*x^18 + 81853*x^16 + 83068*x^14 - 23268*x^12 - 308752*x^10 + 118928*x^8 + 217472*x^6 - 106496*x^4 + 24576*x^2 + 65536)
gp: K = bnfinit(y^32 - 14*y^30 + 109*y^28 - 698*y^26 + 3468*y^24 - 10052*y^22 + 23637*y^20 - 66932*y^18 + 81853*y^16 + 83068*y^14 - 23268*y^12 - 308752*y^10 + 118928*y^8 + 217472*y^6 - 106496*y^4 + 24576*y^2 + 65536, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 14*x^30 + 109*x^28 - 698*x^26 + 3468*x^24 - 10052*x^22 + 23637*x^20 - 66932*x^18 + 81853*x^16 + 83068*x^14 - 23268*x^12 - 308752*x^10 + 118928*x^8 + 217472*x^6 - 106496*x^4 + 24576*x^2 + 65536);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 14*x^30 + 109*x^28 - 698*x^26 + 3468*x^24 - 10052*x^22 + 23637*x^20 - 66932*x^18 + 81853*x^16 + 83068*x^14 - 23268*x^12 - 308752*x^10 + 118928*x^8 + 217472*x^6 - 106496*x^4 + 24576*x^2 + 65536)
\( x^{32} - 14 x^{30} + 109 x^{28} - 698 x^{26} + 3468 x^{24} - 10052 x^{22} + 23637 x^{20} - 66932 x^{18} + \cdots + 65536 \)
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: |
\(1720706227830984913861923492628167351336960000000000000000\)
\(\medspace = 2^{56}\cdot 3^{16}\cdot 5^{16}\cdot 31^{4}\cdot 89^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(61.46\) | 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: | $2^{9/4}3^{1/2}5^{1/2}31^{1/2}89^{1/2}\approx 967.6960537591174$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(31\), \(89\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{7}$, $\frac{1}{20}a^{20}+\frac{1}{10}a^{18}+\frac{3}{20}a^{16}+\frac{1}{5}a^{14}-\frac{1}{5}a^{10}+\frac{1}{4}a^{8}+\frac{1}{5}a^{6}+\frac{3}{20}a^{4}+\frac{1}{10}a^{2}-\frac{1}{5}$, $\frac{1}{40}a^{21}-\frac{1}{5}a^{19}-\frac{7}{40}a^{17}+\frac{1}{10}a^{15}-\frac{1}{10}a^{11}+\frac{1}{8}a^{9}-\frac{3}{20}a^{7}+\frac{13}{40}a^{5}+\frac{1}{20}a^{3}-\frac{1}{10}a$, $\frac{1}{160}a^{22}+\frac{9}{160}a^{18}-\frac{3}{40}a^{16}+\frac{1}{5}a^{14}+\frac{19}{40}a^{12}-\frac{67}{160}a^{10}-\frac{23}{80}a^{8}+\frac{1}{32}a^{6}-\frac{27}{80}a^{4}+\frac{13}{40}a^{2}-\frac{1}{5}$, $\frac{1}{320}a^{23}-\frac{71}{320}a^{19}+\frac{17}{80}a^{17}-\frac{3}{20}a^{15}-\frac{21}{80}a^{13}-\frac{67}{320}a^{11}+\frac{57}{160}a^{9}+\frac{17}{64}a^{7}+\frac{13}{160}a^{5}-\frac{7}{80}a^{3}+\frac{2}{5}a$, $\frac{1}{19200}a^{24}-\frac{1}{1200}a^{22}+\frac{313}{19200}a^{20}+\frac{271}{1600}a^{18}-\frac{209}{1200}a^{16}-\frac{161}{960}a^{14}+\frac{4157}{19200}a^{12}-\frac{803}{1920}a^{10}+\frac{6901}{19200}a^{8}-\frac{59}{9600}a^{6}-\frac{781}{1600}a^{4}-\frac{1}{75}a^{2}+\frac{1}{75}$, $\frac{1}{19200}a^{25}-\frac{1}{1200}a^{23}-\frac{167}{19200}a^{21}-\frac{209}{1600}a^{19}+\frac{1}{1200}a^{17}+\frac{223}{960}a^{15}+\frac{4157}{19200}a^{13}-\frac{611}{1920}a^{11}+\frac{4501}{19200}a^{9}-\frac{3419}{9600}a^{7}+\frac{299}{1600}a^{5}+\frac{131}{300}a^{3}+\frac{17}{150}a$, $\frac{1}{38400}a^{26}+\frac{19}{12800}a^{22}+\frac{29}{1920}a^{20}+\frac{283}{2400}a^{18}-\frac{247}{3200}a^{16}-\frac{1283}{38400}a^{14}-\frac{3053}{6400}a^{12}-\frac{14059}{38400}a^{10}-\frac{817}{6400}a^{8}-\frac{659}{1920}a^{6}+\frac{293}{600}a^{4}-\frac{1}{2}a^{2}-\frac{7}{75}$, $\frac{1}{38400}a^{27}+\frac{19}{12800}a^{23}-\frac{19}{1920}a^{21}-\frac{437}{2400}a^{19}+\frac{313}{3200}a^{17}-\frac{5123}{38400}a^{15}-\frac{3053}{6400}a^{13}-\frac{10219}{38400}a^{11}-\frac{1617}{6400}a^{9}+\frac{589}{1920}a^{7}+\frac{49}{300}a^{5}+\frac{9}{20}a^{3}+\frac{1}{150}a$, $\frac{1}{5283763200}a^{28}-\frac{2537}{220156800}a^{26}-\frac{50491}{5283763200}a^{24}+\frac{2655683}{1320940800}a^{22}-\frac{69097}{23174400}a^{20}-\frac{15122169}{146771200}a^{18}-\frac{379151393}{1761254400}a^{16}-\frac{269265787}{2641881600}a^{14}+\frac{123190763}{1761254400}a^{12}-\frac{6120587}{105675264}a^{10}+\frac{158881453}{330235200}a^{8}-\frac{1434011}{220156800}a^{6}-\frac{3401573}{7338560}a^{4}-\frac{707431}{2428200}a^{2}-\frac{1151299}{5159925}$, $\frac{1}{10567526400}a^{29}+\frac{2557}{352250880}a^{27}-\frac{50491}{10567526400}a^{25}-\frac{7278851}{5283763200}a^{23}+\frac{280933}{46348800}a^{21}+\frac{40388593}{176125440}a^{19}-\frac{383004137}{3522508800}a^{17}-\frac{110738903}{660470400}a^{15}+\frac{41535187}{704501760}a^{13}-\frac{1746487}{660470400}a^{11}+\frac{846669943}{2641881600}a^{9}+\frac{4366709}{73385600}a^{7}+\frac{39882817}{220156800}a^{5}-\frac{17656}{303525}a^{3}-\frac{600907}{10319850}a$, $\frac{1}{22\!\cdots\!00}a^{30}-\frac{11\!\cdots\!51}{12\!\cdots\!00}a^{28}-\frac{91\!\cdots\!51}{22\!\cdots\!00}a^{26}-\frac{85\!\cdots\!73}{11\!\cdots\!00}a^{24}+\frac{61\!\cdots\!21}{18\!\cdots\!00}a^{22}-\frac{10\!\cdots\!13}{62\!\cdots\!00}a^{20}-\frac{10\!\cdots\!33}{14\!\cdots\!20}a^{18}-\frac{42\!\cdots\!99}{32\!\cdots\!00}a^{16}-\frac{36\!\cdots\!57}{74\!\cdots\!00}a^{14}-\frac{38\!\cdots\!07}{32\!\cdots\!00}a^{12}+\frac{10\!\cdots\!07}{55\!\cdots\!00}a^{10}+\frac{67\!\cdots\!31}{46\!\cdots\!00}a^{8}+\frac{33\!\cdots\!81}{15\!\cdots\!00}a^{6}-\frac{60\!\cdots\!49}{21\!\cdots\!20}a^{4}+\frac{26\!\cdots\!41}{88\!\cdots\!00}a^{2}+\frac{56\!\cdots\!97}{91\!\cdots\!00}$, $\frac{1}{44\!\cdots\!00}a^{31}-\frac{11\!\cdots\!51}{24\!\cdots\!00}a^{29}-\frac{91\!\cdots\!51}{44\!\cdots\!00}a^{27}-\frac{85\!\cdots\!73}{22\!\cdots\!00}a^{25}+\frac{61\!\cdots\!21}{37\!\cdots\!00}a^{23}-\frac{10\!\cdots\!13}{12\!\cdots\!00}a^{21}-\frac{10\!\cdots\!33}{29\!\cdots\!40}a^{19}-\frac{42\!\cdots\!99}{65\!\cdots\!00}a^{17}-\frac{36\!\cdots\!57}{14\!\cdots\!00}a^{15}+\frac{29\!\cdots\!93}{65\!\cdots\!00}a^{13}+\frac{10\!\cdots\!07}{11\!\cdots\!00}a^{11}+\frac{67\!\cdots\!31}{93\!\cdots\!00}a^{9}-\frac{12\!\cdots\!19}{31\!\cdots\!00}a^{7}-\frac{60\!\cdots\!49}{43\!\cdots\!40}a^{5}+\frac{26\!\cdots\!41}{17\!\cdots\!00}a^{3}-\frac{85\!\cdots\!03}{18\!\cdots\!00}a$
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
$C_{4}\times C_{12}\times C_{24}$, which has order $1152$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $1152$ (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: |
\( -\frac{1152046779458820366599}{12369511581471107161128960} a^{30} + \frac{2831889170634392449047}{2061585263578517860188160} a^{28} - \frac{138160136568185756445163}{12369511581471107161128960} a^{26} + \frac{452846500346247522927439}{6184755790735553580564480} a^{24} - \frac{388230083447283646367587}{1030792631789258930094080} a^{22} + \frac{1247845767175440180368901}{1030792631789258930094080} a^{20} - \frac{12657322983928827713331793}{4123170527157035720376320} a^{18} + \frac{5208177694216287499639097}{618475579073555358056448} a^{16} - \frac{56306653966049508039682393}{4123170527157035720376320} a^{14} + \frac{1061101141024489668287605}{618475579073555358056448} a^{12} + \frac{1142997953076588582446987}{618475579073555358056448} a^{10} + \frac{7192582415604303556036863}{257698157947314732523520} a^{8} - \frac{430698357625068994652463}{13563060944595512238080} a^{6} + \frac{3207414617402730074789}{3019900288445094521760} a^{4} + \frac{531005013003058932739}{48708069168469266480} a^{2} - \frac{4646381574826178373247}{503316714740849086960} \)
(order $6$)
| 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: |
$\frac{18263705546347}{35\!\cdots\!52}a^{30}-\frac{612028578677109}{89\!\cdots\!80}a^{28}+\frac{92\!\cdots\!27}{17\!\cdots\!60}a^{26}-\frac{29\!\cdots\!83}{89\!\cdots\!80}a^{24}+\frac{71\!\cdots\!49}{44\!\cdots\!40}a^{22}-\frac{19\!\cdots\!47}{44\!\cdots\!40}a^{20}+\frac{18\!\cdots\!23}{17\!\cdots\!60}a^{18}-\frac{26\!\cdots\!69}{89\!\cdots\!88}a^{16}+\frac{48\!\cdots\!31}{17\!\cdots\!60}a^{14}+\frac{22\!\cdots\!47}{44\!\cdots\!40}a^{12}+\frac{19\!\cdots\!77}{89\!\cdots\!88}a^{10}-\frac{98\!\cdots\!01}{11\!\cdots\!60}a^{8}-\frac{35\!\cdots\!83}{58\!\cdots\!40}a^{6}+\frac{10\!\cdots\!03}{87\!\cdots\!20}a^{4}-\frac{611978134559791}{21\!\cdots\!80}a^{2}+\frac{847428887109837}{21\!\cdots\!80}$, $\frac{12050053552679}{64\!\cdots\!00}a^{31}-\frac{88596984175013}{32\!\cdots\!00}a^{29}+\frac{14\!\cdots\!03}{64\!\cdots\!00}a^{27}-\frac{47\!\cdots\!59}{32\!\cdots\!00}a^{25}+\frac{711675287680121}{943041082982400}a^{23}-\frac{12\!\cdots\!01}{53\!\cdots\!00}a^{21}+\frac{7233180229433}{1180646113280}a^{19}-\frac{10\!\cdots\!73}{641267936428032}a^{17}+\frac{17\!\cdots\!27}{64\!\cdots\!00}a^{15}-\frac{48\!\cdots\!29}{16\!\cdots\!00}a^{13}-\frac{15\!\cdots\!49}{53\!\cdots\!00}a^{11}-\frac{22\!\cdots\!73}{40\!\cdots\!00}a^{9}+\frac{27\!\cdots\!99}{445324955852800}a^{7}-\frac{28692035698949}{15655955479200}a^{5}-\frac{15841862005289}{782797773960}a^{3}+\frac{155995162911917}{7827977739600}a-1$, $\frac{36\!\cdots\!51}{44\!\cdots\!00}a^{31}+\frac{15\!\cdots\!25}{99\!\cdots\!48}a^{30}-\frac{27\!\cdots\!57}{22\!\cdots\!00}a^{29}-\frac{17\!\cdots\!13}{74\!\cdots\!60}a^{28}+\frac{44\!\cdots\!71}{44\!\cdots\!00}a^{27}+\frac{18\!\cdots\!25}{99\!\cdots\!48}a^{26}-\frac{86\!\cdots\!87}{13\!\cdots\!00}a^{25}-\frac{18\!\cdots\!95}{14\!\cdots\!72}a^{24}+\frac{37\!\cdots\!29}{11\!\cdots\!00}a^{23}+\frac{27\!\cdots\!17}{43\!\cdots\!08}a^{22}-\frac{13\!\cdots\!51}{12\!\cdots\!00}a^{21}-\frac{24\!\cdots\!69}{12\!\cdots\!60}a^{20}+\frac{84\!\cdots\!53}{29\!\cdots\!40}a^{19}+\frac{20\!\cdots\!79}{41\!\cdots\!20}a^{18}-\frac{86\!\cdots\!93}{11\!\cdots\!00}a^{17}-\frac{17\!\cdots\!71}{12\!\cdots\!60}a^{16}+\frac{57\!\cdots\!51}{44\!\cdots\!00}a^{15}+\frac{33\!\cdots\!43}{14\!\cdots\!20}a^{14}-\frac{30\!\cdots\!97}{11\!\cdots\!00}a^{13}-\frac{60\!\cdots\!51}{24\!\cdots\!12}a^{12}-\frac{64\!\cdots\!29}{37\!\cdots\!00}a^{11}-\frac{83\!\cdots\!87}{37\!\cdots\!80}a^{10}-\frac{68\!\cdots\!01}{27\!\cdots\!00}a^{9}-\frac{85\!\cdots\!93}{18\!\cdots\!84}a^{8}+\frac{28\!\cdots\!49}{93\!\cdots\!00}a^{7}+\frac{83\!\cdots\!19}{16\!\cdots\!60}a^{6}-\frac{73\!\cdots\!27}{21\!\cdots\!00}a^{5}-\frac{35\!\cdots\!51}{24\!\cdots\!88}a^{4}-\frac{17\!\cdots\!27}{17\!\cdots\!00}a^{3}-\frac{10\!\cdots\!73}{58\!\cdots\!60}a^{2}+\frac{49\!\cdots\!17}{54\!\cdots\!00}a+\frac{29\!\cdots\!61}{18\!\cdots\!60}$, $\frac{46\!\cdots\!29}{55\!\cdots\!00}a^{30}-\frac{34\!\cdots\!27}{27\!\cdots\!00}a^{28}+\frac{55\!\cdots\!73}{55\!\cdots\!00}a^{26}-\frac{35\!\cdots\!43}{54\!\cdots\!00}a^{24}+\frac{94\!\cdots\!91}{27\!\cdots\!60}a^{22}-\frac{50\!\cdots\!11}{46\!\cdots\!00}a^{20}+\frac{17\!\cdots\!57}{62\!\cdots\!00}a^{18}-\frac{10\!\cdots\!67}{13\!\cdots\!00}a^{16}+\frac{68\!\cdots\!41}{55\!\cdots\!00}a^{14}-\frac{18\!\cdots\!47}{13\!\cdots\!00}a^{12}-\frac{34\!\cdots\!33}{13\!\cdots\!00}a^{10}-\frac{17\!\cdots\!31}{69\!\cdots\!40}a^{8}+\frac{11\!\cdots\!09}{38\!\cdots\!00}a^{6}-\frac{26\!\cdots\!09}{21\!\cdots\!20}a^{4}-\frac{24\!\cdots\!37}{24\!\cdots\!00}a^{2}+\frac{56\!\cdots\!91}{68\!\cdots\!00}$, $\frac{21\!\cdots\!49}{22\!\cdots\!00}a^{30}-\frac{47\!\cdots\!21}{37\!\cdots\!00}a^{28}+\frac{21\!\cdots\!37}{22\!\cdots\!00}a^{26}-\frac{68\!\cdots\!17}{11\!\cdots\!00}a^{24}+\frac{55\!\cdots\!21}{18\!\cdots\!00}a^{22}-\frac{30\!\cdots\!27}{37\!\cdots\!80}a^{20}+\frac{28\!\cdots\!87}{14\!\cdots\!20}a^{18}-\frac{31\!\cdots\!23}{55\!\cdots\!00}a^{16}+\frac{76\!\cdots\!87}{14\!\cdots\!20}a^{14}+\frac{10\!\cdots\!37}{11\!\cdots\!40}a^{12}+\frac{22\!\cdots\!39}{55\!\cdots\!00}a^{10}-\frac{76\!\cdots\!09}{46\!\cdots\!00}a^{8}-\frac{52\!\cdots\!57}{46\!\cdots\!00}a^{6}+\frac{25\!\cdots\!57}{10\!\cdots\!00}a^{4}-\frac{46\!\cdots\!07}{88\!\cdots\!00}a^{2}+\frac{91\!\cdots\!19}{30\!\cdots\!00}$, $\frac{51\!\cdots\!19}{65\!\cdots\!00}a^{30}-\frac{64\!\cdots\!81}{55\!\cdots\!00}a^{28}+\frac{10\!\cdots\!47}{11\!\cdots\!00}a^{26}-\frac{11\!\cdots\!97}{18\!\cdots\!00}a^{24}+\frac{87\!\cdots\!41}{27\!\cdots\!00}a^{22}-\frac{31\!\cdots\!47}{31\!\cdots\!00}a^{20}+\frac{31\!\cdots\!59}{12\!\cdots\!00}a^{18}-\frac{78\!\cdots\!57}{11\!\cdots\!04}a^{16}+\frac{12\!\cdots\!91}{11\!\cdots\!00}a^{14}-\frac{14\!\cdots\!07}{14\!\cdots\!00}a^{12}-\frac{70\!\cdots\!63}{55\!\cdots\!20}a^{10}-\frac{16\!\cdots\!89}{69\!\cdots\!00}a^{8}+\frac{23\!\cdots\!77}{93\!\cdots\!92}a^{6}+\frac{23\!\cdots\!83}{13\!\cdots\!00}a^{4}-\frac{36\!\cdots\!91}{43\!\cdots\!00}a^{2}+\frac{10\!\cdots\!89}{13\!\cdots\!00}$, $\frac{44\!\cdots\!19}{74\!\cdots\!00}a^{31}+\frac{47\!\cdots\!81}{13\!\cdots\!00}a^{30}-\frac{97\!\cdots\!91}{11\!\cdots\!00}a^{29}-\frac{35\!\cdots\!17}{65\!\cdots\!00}a^{28}+\frac{52\!\cdots\!03}{74\!\cdots\!00}a^{27}+\frac{53\!\cdots\!91}{11\!\cdots\!00}a^{26}-\frac{51\!\cdots\!69}{11\!\cdots\!00}a^{25}-\frac{33\!\cdots\!69}{11\!\cdots\!00}a^{24}+\frac{15\!\cdots\!79}{65\!\cdots\!20}a^{23}+\frac{29\!\cdots\!93}{18\!\cdots\!00}a^{22}-\frac{13\!\cdots\!23}{18\!\cdots\!00}a^{21}-\frac{19\!\cdots\!87}{37\!\cdots\!80}a^{20}+\frac{82\!\cdots\!39}{43\!\cdots\!00}a^{19}+\frac{13\!\cdots\!93}{99\!\cdots\!48}a^{18}-\frac{97\!\cdots\!17}{18\!\cdots\!00}a^{17}-\frac{20\!\cdots\!71}{55\!\cdots\!00}a^{16}+\frac{18\!\cdots\!73}{22\!\cdots\!00}a^{15}+\frac{15\!\cdots\!21}{24\!\cdots\!00}a^{14}-\frac{11\!\cdots\!17}{18\!\cdots\!00}a^{13}-\frac{11\!\cdots\!87}{55\!\cdots\!00}a^{12}+\frac{12\!\cdots\!31}{55\!\cdots\!00}a^{11}-\frac{91\!\cdots\!73}{55\!\cdots\!00}a^{10}-\frac{47\!\cdots\!63}{27\!\cdots\!60}a^{9}-\frac{75\!\cdots\!07}{65\!\cdots\!00}a^{8}+\frac{11\!\cdots\!61}{65\!\cdots\!00}a^{7}+\frac{75\!\cdots\!39}{46\!\cdots\!00}a^{6}-\frac{22\!\cdots\!83}{14\!\cdots\!80}a^{5}-\frac{27\!\cdots\!57}{13\!\cdots\!00}a^{4}-\frac{57\!\cdots\!59}{88\!\cdots\!00}a^{3}-\frac{24\!\cdots\!61}{51\!\cdots\!00}a^{2}+\frac{15\!\cdots\!43}{27\!\cdots\!00}a+\frac{38\!\cdots\!31}{60\!\cdots\!20}$, $\frac{46\!\cdots\!93}{62\!\cdots\!00}a^{31}-\frac{33\!\cdots\!43}{10\!\cdots\!00}a^{30}-\frac{30\!\cdots\!11}{27\!\cdots\!00}a^{29}+\frac{10\!\cdots\!09}{22\!\cdots\!80}a^{28}+\frac{15\!\cdots\!03}{18\!\cdots\!00}a^{27}-\frac{93\!\cdots\!51}{24\!\cdots\!00}a^{26}-\frac{15\!\cdots\!69}{27\!\cdots\!00}a^{25}+\frac{27\!\cdots\!19}{11\!\cdots\!00}a^{24}+\frac{15\!\cdots\!51}{55\!\cdots\!52}a^{23}-\frac{70\!\cdots\!93}{55\!\cdots\!00}a^{22}-\frac{39\!\cdots\!63}{46\!\cdots\!00}a^{21}+\frac{15\!\cdots\!29}{37\!\cdots\!80}a^{20}+\frac{38\!\cdots\!43}{18\!\cdots\!00}a^{19}-\frac{10\!\cdots\!51}{99\!\cdots\!48}a^{18}-\frac{26\!\cdots\!37}{46\!\cdots\!00}a^{17}+\frac{17\!\cdots\!69}{62\!\cdots\!00}a^{16}+\frac{45\!\cdots\!33}{55\!\cdots\!00}a^{15}-\frac{53\!\cdots\!89}{11\!\cdots\!00}a^{14}+\frac{18\!\cdots\!03}{46\!\cdots\!00}a^{13}+\frac{30\!\cdots\!37}{62\!\cdots\!00}a^{12}-\frac{55\!\cdots\!09}{13\!\cdots\!00}a^{11}+\frac{13\!\cdots\!49}{29\!\cdots\!00}a^{10}-\frac{13\!\cdots\!51}{69\!\cdots\!40}a^{9}+\frac{13\!\cdots\!97}{13\!\cdots\!00}a^{8}+\frac{77\!\cdots\!51}{11\!\cdots\!00}a^{7}-\frac{47\!\cdots\!73}{46\!\cdots\!00}a^{6}+\frac{19\!\cdots\!97}{14\!\cdots\!80}a^{5}+\frac{98\!\cdots\!53}{91\!\cdots\!00}a^{4}-\frac{29\!\cdots\!73}{44\!\cdots\!00}a^{3}+\frac{30\!\cdots\!21}{88\!\cdots\!00}a^{2}+\frac{52\!\cdots\!63}{68\!\cdots\!00}a-\frac{85\!\cdots\!49}{27\!\cdots\!00}$, $\frac{17\!\cdots\!81}{49\!\cdots\!00}a^{31}-\frac{18\!\cdots\!51}{12\!\cdots\!00}a^{30}-\frac{23\!\cdots\!11}{44\!\cdots\!60}a^{29}+\frac{12\!\cdots\!77}{55\!\cdots\!00}a^{28}+\frac{64\!\cdots\!59}{14\!\cdots\!00}a^{27}-\frac{66\!\cdots\!81}{37\!\cdots\!00}a^{26}-\frac{38\!\cdots\!99}{13\!\cdots\!20}a^{25}+\frac{13\!\cdots\!91}{11\!\cdots\!40}a^{24}+\frac{63\!\cdots\!59}{44\!\cdots\!16}a^{23}-\frac{16\!\cdots\!21}{27\!\cdots\!00}a^{22}-\frac{55\!\cdots\!81}{12\!\cdots\!00}a^{21}+\frac{17\!\cdots\!97}{93\!\cdots\!00}a^{20}+\frac{99\!\cdots\!33}{99\!\cdots\!80}a^{19}-\frac{12\!\cdots\!83}{24\!\cdots\!20}a^{18}-\frac{10\!\cdots\!89}{37\!\cdots\!00}a^{17}+\frac{41\!\cdots\!37}{31\!\cdots\!00}a^{16}+\frac{19\!\cdots\!17}{44\!\cdots\!00}a^{15}-\frac{24\!\cdots\!11}{11\!\cdots\!00}a^{14}+\frac{27\!\cdots\!51}{74\!\cdots\!60}a^{13}+\frac{97\!\cdots\!41}{31\!\cdots\!00}a^{12}-\frac{54\!\cdots\!89}{11\!\cdots\!00}a^{11}+\frac{55\!\cdots\!23}{27\!\cdots\!00}a^{10}-\frac{11\!\cdots\!07}{55\!\cdots\!20}a^{9}+\frac{30\!\cdots\!49}{69\!\cdots\!00}a^{8}-\frac{22\!\cdots\!81}{18\!\cdots\!40}a^{7}-\frac{39\!\cdots\!87}{77\!\cdots\!00}a^{6}+\frac{31\!\cdots\!01}{24\!\cdots\!00}a^{5}+\frac{40\!\cdots\!37}{12\!\cdots\!00}a^{4}+\frac{91\!\cdots\!13}{35\!\cdots\!60}a^{3}+\frac{79\!\cdots\!41}{44\!\cdots\!00}a^{2}+\frac{27\!\cdots\!47}{54\!\cdots\!00}a-\frac{21\!\cdots\!69}{13\!\cdots\!00}$, $\frac{91\!\cdots\!47}{44\!\cdots\!60}a^{30}-\frac{33\!\cdots\!57}{11\!\cdots\!00}a^{28}+\frac{21\!\cdots\!15}{89\!\cdots\!32}a^{26}-\frac{17\!\cdots\!83}{11\!\cdots\!00}a^{24}+\frac{45\!\cdots\!37}{55\!\cdots\!00}a^{22}-\frac{48\!\cdots\!33}{18\!\cdots\!00}a^{20}+\frac{99\!\cdots\!33}{14\!\cdots\!20}a^{18}-\frac{10\!\cdots\!29}{55\!\cdots\!00}a^{16}+\frac{65\!\cdots\!67}{22\!\cdots\!00}a^{14}-\frac{37\!\cdots\!21}{11\!\cdots\!40}a^{12}-\frac{13\!\cdots\!67}{62\!\cdots\!00}a^{10}-\frac{85\!\cdots\!33}{13\!\cdots\!00}a^{8}+\frac{31\!\cdots\!37}{46\!\cdots\!00}a^{6}-\frac{18\!\cdots\!83}{10\!\cdots\!00}a^{4}-\frac{17\!\cdots\!73}{88\!\cdots\!00}a^{2}+\frac{53\!\cdots\!61}{27\!\cdots\!00}$, $\frac{10\!\cdots\!77}{31\!\cdots\!00}a^{31}+\frac{55\!\cdots\!89}{44\!\cdots\!60}a^{30}-\frac{36\!\cdots\!33}{73\!\cdots\!00}a^{29}-\frac{14\!\cdots\!29}{74\!\cdots\!60}a^{28}+\frac{90\!\cdots\!11}{22\!\cdots\!00}a^{27}+\frac{34\!\cdots\!17}{22\!\cdots\!00}a^{26}-\frac{29\!\cdots\!83}{11\!\cdots\!00}a^{25}-\frac{11\!\cdots\!77}{11\!\cdots\!00}a^{24}+\frac{16\!\cdots\!13}{12\!\cdots\!60}a^{23}+\frac{31\!\cdots\!63}{62\!\cdots\!00}a^{22}-\frac{27\!\cdots\!47}{62\!\cdots\!00}a^{21}-\frac{17\!\cdots\!23}{10\!\cdots\!00}a^{20}+\frac{27\!\cdots\!43}{24\!\cdots\!00}a^{19}+\frac{28\!\cdots\!31}{74\!\cdots\!00}a^{18}-\frac{17\!\cdots\!83}{58\!\cdots\!60}a^{17}-\frac{56\!\cdots\!23}{55\!\cdots\!00}a^{16}+\frac{73\!\cdots\!41}{14\!\cdots\!20}a^{15}+\frac{12\!\cdots\!63}{74\!\cdots\!00}a^{14}-\frac{33\!\cdots\!17}{55\!\cdots\!00}a^{13}+\frac{45\!\cdots\!69}{55\!\cdots\!00}a^{12}-\frac{14\!\cdots\!99}{32\!\cdots\!00}a^{11}-\frac{12\!\cdots\!57}{55\!\cdots\!00}a^{10}-\frac{92\!\cdots\!43}{93\!\cdots\!20}a^{9}-\frac{28\!\cdots\!01}{46\!\cdots\!00}a^{8}+\frac{17\!\cdots\!31}{15\!\cdots\!00}a^{7}+\frac{34\!\cdots\!59}{46\!\cdots\!00}a^{6}-\frac{21\!\cdots\!81}{21\!\cdots\!00}a^{5}+\frac{73\!\cdots\!19}{10\!\cdots\!00}a^{4}-\frac{19\!\cdots\!09}{55\!\cdots\!75}a^{3}-\frac{63\!\cdots\!03}{88\!\cdots\!00}a^{2}+\frac{35\!\cdots\!67}{91\!\cdots\!00}a-\frac{14\!\cdots\!47}{91\!\cdots\!00}$, $\frac{91\!\cdots\!19}{44\!\cdots\!00}a^{31}+\frac{17\!\cdots\!13}{55\!\cdots\!00}a^{30}-\frac{88\!\cdots\!35}{29\!\cdots\!44}a^{29}-\frac{57\!\cdots\!79}{12\!\cdots\!56}a^{28}+\frac{10\!\cdots\!67}{44\!\cdots\!00}a^{27}+\frac{20\!\cdots\!37}{55\!\cdots\!00}a^{26}-\frac{34\!\cdots\!47}{22\!\cdots\!00}a^{25}-\frac{13\!\cdots\!61}{55\!\cdots\!20}a^{24}+\frac{98\!\cdots\!69}{12\!\cdots\!00}a^{23}+\frac{57\!\cdots\!73}{46\!\cdots\!00}a^{22}-\frac{91\!\cdots\!69}{37\!\cdots\!00}a^{21}-\frac{60\!\cdots\!17}{15\!\cdots\!00}a^{20}+\frac{30\!\cdots\!23}{49\!\cdots\!00}a^{19}+\frac{18\!\cdots\!83}{18\!\cdots\!00}a^{18}-\frac{19\!\cdots\!33}{11\!\cdots\!00}a^{17}-\frac{22\!\cdots\!91}{82\!\cdots\!00}a^{16}+\frac{12\!\cdots\!63}{49\!\cdots\!00}a^{15}+\frac{15\!\cdots\!87}{37\!\cdots\!80}a^{14}+\frac{16\!\cdots\!23}{11\!\cdots\!00}a^{13}+\frac{86\!\cdots\!83}{43\!\cdots\!00}a^{12}-\frac{84\!\cdots\!59}{11\!\cdots\!00}a^{11}-\frac{47\!\cdots\!61}{13\!\cdots\!00}a^{10}-\frac{57\!\cdots\!83}{93\!\cdots\!00}a^{9}-\frac{10\!\cdots\!97}{11\!\cdots\!00}a^{8}+\frac{29\!\cdots\!77}{54\!\cdots\!00}a^{7}+\frac{69\!\cdots\!89}{77\!\cdots\!60}a^{6}+\frac{15\!\cdots\!53}{43\!\cdots\!40}a^{5}+\frac{12\!\cdots\!21}{21\!\cdots\!20}a^{4}-\frac{27\!\cdots\!43}{17\!\cdots\!00}a^{3}-\frac{28\!\cdots\!86}{11\!\cdots\!55}a^{2}+\frac{26\!\cdots\!37}{12\!\cdots\!40}a+\frac{64\!\cdots\!21}{22\!\cdots\!00}$, $\frac{55\!\cdots\!61}{99\!\cdots\!80}a^{31}-\frac{18\!\cdots\!87}{24\!\cdots\!00}a^{30}-\frac{40\!\cdots\!47}{49\!\cdots\!40}a^{29}+\frac{80\!\cdots\!89}{73\!\cdots\!00}a^{28}+\frac{19\!\cdots\!79}{29\!\cdots\!40}a^{27}-\frac{67\!\cdots\!49}{74\!\cdots\!00}a^{26}-\frac{10\!\cdots\!53}{24\!\cdots\!00}a^{25}+\frac{21\!\cdots\!73}{37\!\cdots\!00}a^{24}+\frac{27\!\cdots\!79}{12\!\cdots\!00}a^{23}-\frac{56\!\cdots\!03}{18\!\cdots\!00}a^{22}-\frac{26\!\cdots\!11}{37\!\cdots\!00}a^{21}+\frac{18\!\cdots\!77}{18\!\cdots\!00}a^{20}+\frac{26\!\cdots\!79}{14\!\cdots\!00}a^{19}-\frac{36\!\cdots\!49}{14\!\cdots\!20}a^{18}-\frac{61\!\cdots\!99}{12\!\cdots\!00}a^{17}+\frac{12\!\cdots\!63}{18\!\cdots\!00}a^{16}+\frac{47\!\cdots\!83}{59\!\cdots\!88}a^{15}-\frac{26\!\cdots\!63}{24\!\cdots\!00}a^{14}-\frac{77\!\cdots\!43}{12\!\cdots\!00}a^{13}+\frac{22\!\cdots\!79}{18\!\cdots\!00}a^{12}-\frac{86\!\cdots\!67}{74\!\cdots\!60}a^{11}+\frac{44\!\cdots\!83}{36\!\cdots\!00}a^{10}-\frac{30\!\cdots\!83}{18\!\cdots\!00}a^{9}+\frac{10\!\cdots\!67}{46\!\cdots\!00}a^{8}+\frac{89\!\cdots\!81}{49\!\cdots\!00}a^{7}-\frac{11\!\cdots\!29}{46\!\cdots\!00}a^{6}+\frac{11\!\cdots\!23}{18\!\cdots\!00}a^{5}+\frac{32\!\cdots\!77}{72\!\cdots\!40}a^{4}-\frac{13\!\cdots\!87}{19\!\cdots\!00}a^{3}+\frac{24\!\cdots\!39}{29\!\cdots\!00}a^{2}+\frac{14\!\cdots\!91}{25\!\cdots\!00}a-\frac{67\!\cdots\!11}{91\!\cdots\!00}$, $\frac{39\!\cdots\!73}{29\!\cdots\!40}a^{31}-\frac{38\!\cdots\!63}{37\!\cdots\!00}a^{30}-\frac{14\!\cdots\!71}{74\!\cdots\!00}a^{29}+\frac{28\!\cdots\!09}{18\!\cdots\!00}a^{28}+\frac{77\!\cdots\!91}{49\!\cdots\!00}a^{27}-\frac{45\!\cdots\!31}{37\!\cdots\!00}a^{26}-\frac{13\!\cdots\!37}{13\!\cdots\!00}a^{25}+\frac{14\!\cdots\!47}{18\!\cdots\!00}a^{24}+\frac{38\!\cdots\!11}{74\!\cdots\!60}a^{23}-\frac{38\!\cdots\!13}{93\!\cdots\!00}a^{22}-\frac{61\!\cdots\!77}{37\!\cdots\!00}a^{21}+\frac{24\!\cdots\!43}{18\!\cdots\!40}a^{20}+\frac{62\!\cdots\!49}{14\!\cdots\!00}a^{19}-\frac{40\!\cdots\!89}{12\!\cdots\!00}a^{18}-\frac{85\!\cdots\!19}{74\!\cdots\!60}a^{17}+\frac{49\!\cdots\!61}{54\!\cdots\!00}a^{16}+\frac{90\!\cdots\!19}{49\!\cdots\!00}a^{15}-\frac{17\!\cdots\!69}{12\!\cdots\!00}a^{14}-\frac{43\!\cdots\!11}{37\!\cdots\!00}a^{13}+\frac{73\!\cdots\!13}{10\!\cdots\!20}a^{12}-\frac{70\!\cdots\!77}{37\!\cdots\!00}a^{11}+\frac{17\!\cdots\!71}{93\!\cdots\!00}a^{10}-\frac{71\!\cdots\!79}{18\!\cdots\!40}a^{9}+\frac{98\!\cdots\!47}{32\!\cdots\!00}a^{8}+\frac{36\!\cdots\!13}{93\!\cdots\!00}a^{7}-\frac{75\!\cdots\!83}{23\!\cdots\!00}a^{6}+\frac{63\!\cdots\!99}{36\!\cdots\!00}a^{5}-\frac{12\!\cdots\!69}{91\!\cdots\!00}a^{4}-\frac{27\!\cdots\!13}{19\!\cdots\!00}a^{3}+\frac{99\!\cdots\!23}{97\!\cdots\!60}a^{2}+\frac{23\!\cdots\!07}{18\!\cdots\!00}a-\frac{14\!\cdots\!07}{15\!\cdots\!00}$, $\frac{31\!\cdots\!75}{44\!\cdots\!16}a^{31}-\frac{44\!\cdots\!81}{14\!\cdots\!20}a^{30}-\frac{57\!\cdots\!49}{55\!\cdots\!00}a^{29}+\frac{54\!\cdots\!49}{12\!\cdots\!00}a^{28}+\frac{93\!\cdots\!39}{11\!\cdots\!00}a^{27}-\frac{89\!\cdots\!67}{24\!\cdots\!00}a^{26}-\frac{94\!\cdots\!69}{17\!\cdots\!00}a^{25}+\frac{87\!\cdots\!73}{37\!\cdots\!00}a^{24}+\frac{78\!\cdots\!49}{27\!\cdots\!00}a^{23}-\frac{45\!\cdots\!23}{37\!\cdots\!80}a^{22}-\frac{27\!\cdots\!83}{31\!\cdots\!00}a^{21}+\frac{24\!\cdots\!03}{62\!\cdots\!00}a^{20}+\frac{84\!\cdots\!81}{37\!\cdots\!00}a^{19}-\frac{73\!\cdots\!13}{74\!\cdots\!00}a^{18}-\frac{34\!\cdots\!53}{55\!\cdots\!20}a^{17}+\frac{33\!\cdots\!29}{12\!\cdots\!60}a^{16}+\frac{11\!\cdots\!67}{11\!\cdots\!00}a^{15}-\frac{17\!\cdots\!31}{39\!\cdots\!00}a^{14}-\frac{22\!\cdots\!89}{27\!\cdots\!00}a^{13}+\frac{37\!\cdots\!29}{62\!\cdots\!00}a^{12}-\frac{25\!\cdots\!97}{18\!\cdots\!40}a^{11}+\frac{56\!\cdots\!51}{98\!\cdots\!00}a^{10}-\frac{14\!\cdots\!41}{69\!\cdots\!00}a^{9}+\frac{83\!\cdots\!67}{93\!\cdots\!20}a^{8}+\frac{53\!\cdots\!81}{23\!\cdots\!00}a^{7}-\frac{16\!\cdots\!07}{15\!\cdots\!00}a^{6}+\frac{89\!\cdots\!91}{10\!\cdots\!00}a^{5}+\frac{62\!\cdots\!77}{18\!\cdots\!00}a^{4}-\frac{71\!\cdots\!87}{88\!\cdots\!40}a^{3}+\frac{34\!\cdots\!71}{97\!\cdots\!00}a^{2}+\frac{89\!\cdots\!73}{13\!\cdots\!00}a-\frac{81\!\cdots\!93}{30\!\cdots\!00}$
| 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: | \( 46284791623985.1 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 46284791623985.1 \cdot 1152}{6\cdot\sqrt{1720706227830984913861923492628167351336960000000000000000}}\cr\approx \mathstrut & 1.26404946744971 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 14*x^30 + 109*x^28 - 698*x^26 + 3468*x^24 - 10052*x^22 + 23637*x^20 - 66932*x^18 + 81853*x^16 + 83068*x^14 - 23268*x^12 - 308752*x^10 + 118928*x^8 + 217472*x^6 - 106496*x^4 + 24576*x^2 + 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 - 14*x^30 + 109*x^28 - 698*x^26 + 3468*x^24 - 10052*x^22 + 23637*x^20 - 66932*x^18 + 81853*x^16 + 83068*x^14 - 23268*x^12 - 308752*x^10 + 118928*x^8 + 217472*x^6 - 106496*x^4 + 24576*x^2 + 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 - 14*x^30 + 109*x^28 - 698*x^26 + 3468*x^24 - 10052*x^22 + 23637*x^20 - 66932*x^18 + 81853*x^16 + 83068*x^14 - 23268*x^12 - 308752*x^10 + 118928*x^8 + 217472*x^6 - 106496*x^4 + 24576*x^2 + 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 - 14*x^30 + 109*x^28 - 698*x^26 + 3468*x^24 - 10052*x^22 + 23637*x^20 - 66932*x^18 + 81853*x^16 + 83068*x^14 - 23268*x^12 - 308752*x^10 + 118928*x^8 + 217472*x^6 - 106496*x^4 + 24576*x^2 + 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
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 | ${\href{/padicField/7.8.0.1}{8} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.8.0.1}{8} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{16}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{16}$ | R | ${\href{/padicField/37.8.0.1}{8} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{16}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.8.16.13 | $x^{8} + 8 x^{7} + 18 x^{6} + 4 x^{5} - 8 x^{4} + 24 x^{3} + 20 x^{2} - 8 x + 4$ | $4$ | $2$ | $16$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ | |
| 2.8.16.13 | $x^{8} + 8 x^{7} + 18 x^{6} + 4 x^{5} - 8 x^{4} + 24 x^{3} + 20 x^{2} - 8 x + 4$ | $4$ | $2$ | $16$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ | |
|
\(3\)
| 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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(89\)
| 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 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 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 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |