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
$C_{2}\times C_{2}\times C_{6}\times C_{48}$, 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: |
\( -\frac{1152046779458820366599}{12369511581471107161128960} a^{30} - \frac{2831889170634392449047}{2061585263578517860188160} a^{28} - \frac{138160136568185756445163}{12369511581471107161128960} a^{26} - \frac{452846500346247522927439}{6184755790735553580564480} a^{24} - \frac{388230083447283646367587}{1030792631789258930094080} a^{22} - \frac{1247845767175440180368901}{1030792631789258930094080} a^{20} - \frac{12657322983928827713331793}{4123170527157035720376320} a^{18} - \frac{5208177694216287499639097}{618475579073555358056448} a^{16} - \frac{56306653966049508039682393}{4123170527157035720376320} a^{14} - \frac{1061101141024489668287605}{618475579073555358056448} a^{12} + \frac{1142997953076588582446987}{618475579073555358056448} a^{10} - \frac{7192582415604303556036863}{257698157947314732523520} a^{8} - \frac{430698357625068994652463}{13563060944595512238080} a^{6} - \frac{3207414617402730074789}{3019900288445094521760} a^{4} + \frac{531005013003058932739}{48708069168469266480} a^{2} + \frac{5149698289567027460207}{503316714740849086960} \)
(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: |
$\frac{85\!\cdots\!37}{14\!\cdots\!20}a^{30}+\frac{63\!\cdots\!03}{74\!\cdots\!60}a^{28}+\frac{20\!\cdots\!53}{29\!\cdots\!44}a^{26}+\frac{65\!\cdots\!51}{14\!\cdots\!60}a^{24}+\frac{17\!\cdots\!03}{74\!\cdots\!36}a^{22}+\frac{92\!\cdots\!59}{12\!\cdots\!60}a^{20}+\frac{18\!\cdots\!55}{99\!\cdots\!48}a^{18}+\frac{19\!\cdots\!03}{37\!\cdots\!80}a^{16}+\frac{12\!\cdots\!33}{14\!\cdots\!20}a^{14}+\frac{45\!\cdots\!99}{37\!\cdots\!80}a^{12}-\frac{22\!\cdots\!37}{37\!\cdots\!80}a^{10}+\frac{31\!\cdots\!19}{18\!\cdots\!84}a^{8}+\frac{64\!\cdots\!49}{32\!\cdots\!12}a^{6}+\frac{73\!\cdots\!05}{14\!\cdots\!28}a^{4}-\frac{12\!\cdots\!11}{19\!\cdots\!20}a^{2}-\frac{20\!\cdots\!95}{36\!\cdots\!32}$, $\frac{18\!\cdots\!29}{22\!\cdots\!00}a^{30}+\frac{24\!\cdots\!43}{22\!\cdots\!80}a^{28}+\frac{18\!\cdots\!37}{22\!\cdots\!00}a^{26}+\frac{12\!\cdots\!93}{24\!\cdots\!20}a^{24}+\frac{28\!\cdots\!39}{11\!\cdots\!40}a^{22}+\frac{12\!\cdots\!83}{18\!\cdots\!00}a^{20}+\frac{24\!\cdots\!83}{14\!\cdots\!20}a^{18}+\frac{26\!\cdots\!07}{55\!\cdots\!00}a^{16}+\frac{98\!\cdots\!37}{22\!\cdots\!00}a^{14}-\frac{88\!\cdots\!37}{11\!\cdots\!40}a^{12}+\frac{19\!\cdots\!51}{55\!\cdots\!00}a^{10}+\frac{39\!\cdots\!23}{27\!\cdots\!60}a^{8}-\frac{29\!\cdots\!23}{31\!\cdots\!40}a^{6}-\frac{21\!\cdots\!49}{10\!\cdots\!00}a^{4}-\frac{26\!\cdots\!97}{58\!\cdots\!60}a^{2}+\frac{34\!\cdots\!73}{27\!\cdots\!00}$, $\frac{92\!\cdots\!51}{27\!\cdots\!00}a^{30}+\frac{22\!\cdots\!39}{46\!\cdots\!00}a^{28}+\frac{10\!\cdots\!07}{27\!\cdots\!00}a^{26}+\frac{36\!\cdots\!71}{13\!\cdots\!00}a^{24}+\frac{18\!\cdots\!63}{13\!\cdots\!00}a^{22}+\frac{32\!\cdots\!83}{77\!\cdots\!00}a^{20}+\frac{87\!\cdots\!77}{81\!\cdots\!80}a^{18}+\frac{41\!\cdots\!77}{13\!\cdots\!80}a^{16}+\frac{14\!\cdots\!07}{31\!\cdots\!00}a^{14}+\frac{36\!\cdots\!21}{69\!\cdots\!00}a^{12}-\frac{34\!\cdots\!43}{69\!\cdots\!00}a^{10}+\frac{18\!\cdots\!73}{19\!\cdots\!00}a^{8}+\frac{20\!\cdots\!71}{19\!\cdots\!00}a^{6}+\frac{11\!\cdots\!99}{35\!\cdots\!00}a^{4}-\frac{35\!\cdots\!03}{92\!\cdots\!32}a^{2}-\frac{39\!\cdots\!91}{11\!\cdots\!50}$, $\frac{38\!\cdots\!87}{22\!\cdots\!00}a^{30}+\frac{49\!\cdots\!89}{22\!\cdots\!80}a^{28}+\frac{36\!\cdots\!31}{22\!\cdots\!00}a^{26}+\frac{38\!\cdots\!33}{37\!\cdots\!00}a^{24}+\frac{27\!\cdots\!57}{55\!\cdots\!00}a^{22}+\frac{22\!\cdots\!21}{18\!\cdots\!00}a^{20}+\frac{36\!\cdots\!09}{13\!\cdots\!00}a^{18}+\frac{49\!\cdots\!33}{55\!\cdots\!00}a^{16}+\frac{13\!\cdots\!11}{22\!\cdots\!00}a^{14}-\frac{10\!\cdots\!71}{55\!\cdots\!00}a^{12}+\frac{15\!\cdots\!93}{55\!\cdots\!00}a^{10}+\frac{75\!\cdots\!93}{13\!\cdots\!00}a^{8}-\frac{19\!\cdots\!87}{46\!\cdots\!00}a^{6}-\frac{62\!\cdots\!01}{57\!\cdots\!00}a^{4}+\frac{68\!\cdots\!67}{15\!\cdots\!00}a^{2}-\frac{67\!\cdots\!57}{27\!\cdots\!00}$, $\frac{36\!\cdots\!51}{44\!\cdots\!00}a^{31}+\frac{11\!\cdots\!99}{12\!\cdots\!60}a^{30}+\frac{27\!\cdots\!57}{22\!\cdots\!00}a^{29}+\frac{28\!\cdots\!47}{20\!\cdots\!60}a^{28}+\frac{44\!\cdots\!71}{44\!\cdots\!00}a^{27}+\frac{13\!\cdots\!63}{12\!\cdots\!60}a^{26}+\frac{86\!\cdots\!87}{13\!\cdots\!00}a^{25}+\frac{45\!\cdots\!39}{61\!\cdots\!80}a^{24}+\frac{37\!\cdots\!29}{11\!\cdots\!00}a^{23}+\frac{38\!\cdots\!87}{10\!\cdots\!80}a^{22}+\frac{13\!\cdots\!51}{12\!\cdots\!00}a^{21}+\frac{12\!\cdots\!01}{10\!\cdots\!80}a^{20}+\frac{84\!\cdots\!53}{29\!\cdots\!40}a^{19}+\frac{12\!\cdots\!93}{41\!\cdots\!20}a^{18}+\frac{86\!\cdots\!93}{11\!\cdots\!00}a^{17}+\frac{52\!\cdots\!97}{61\!\cdots\!48}a^{16}+\frac{57\!\cdots\!51}{44\!\cdots\!00}a^{15}+\frac{56\!\cdots\!93}{41\!\cdots\!20}a^{14}+\frac{30\!\cdots\!97}{11\!\cdots\!00}a^{13}+\frac{10\!\cdots\!05}{61\!\cdots\!48}a^{12}-\frac{64\!\cdots\!29}{37\!\cdots\!00}a^{11}-\frac{11\!\cdots\!87}{61\!\cdots\!48}a^{10}+\frac{68\!\cdots\!01}{27\!\cdots\!00}a^{9}+\frac{71\!\cdots\!63}{25\!\cdots\!20}a^{8}+\frac{28\!\cdots\!49}{93\!\cdots\!00}a^{7}+\frac{43\!\cdots\!63}{13\!\cdots\!80}a^{6}+\frac{73\!\cdots\!27}{21\!\cdots\!00}a^{5}+\frac{32\!\cdots\!89}{30\!\cdots\!60}a^{4}-\frac{17\!\cdots\!27}{17\!\cdots\!00}a^{3}-\frac{53\!\cdots\!39}{48\!\cdots\!80}a^{2}-\frac{49\!\cdots\!17}{54\!\cdots\!00}a-\frac{41\!\cdots\!87}{50\!\cdots\!60}$, $\frac{14\!\cdots\!93}{22\!\cdots\!80}a^{30}+\frac{51\!\cdots\!23}{55\!\cdots\!00}a^{28}+\frac{82\!\cdots\!37}{11\!\cdots\!00}a^{26}+\frac{53\!\cdots\!73}{11\!\cdots\!40}a^{24}+\frac{40\!\cdots\!99}{16\!\cdots\!00}a^{22}+\frac{70\!\cdots\!91}{93\!\cdots\!00}a^{20}+\frac{40\!\cdots\!39}{21\!\cdots\!00}a^{18}+\frac{14\!\cdots\!07}{27\!\cdots\!00}a^{16}+\frac{85\!\cdots\!77}{11\!\cdots\!00}a^{14}-\frac{40\!\cdots\!73}{27\!\cdots\!00}a^{12}-\frac{61\!\cdots\!03}{93\!\cdots\!00}a^{10}+\frac{14\!\cdots\!47}{69\!\cdots\!00}a^{8}+\frac{11\!\cdots\!93}{77\!\cdots\!00}a^{6}-\frac{57\!\cdots\!23}{10\!\cdots\!60}a^{4}-\frac{21\!\cdots\!13}{55\!\cdots\!75}a^{2}-\frac{57\!\cdots\!27}{13\!\cdots\!00}$, $\frac{74\!\cdots\!53}{37\!\cdots\!00}a^{30}+\frac{54\!\cdots\!31}{18\!\cdots\!00}a^{28}+\frac{88\!\cdots\!49}{37\!\cdots\!00}a^{26}+\frac{28\!\cdots\!93}{18\!\cdots\!00}a^{24}+\frac{74\!\cdots\!23}{93\!\cdots\!00}a^{22}+\frac{23\!\cdots\!21}{93\!\cdots\!00}a^{20}+\frac{79\!\cdots\!23}{12\!\cdots\!00}a^{18}+\frac{54\!\cdots\!21}{31\!\cdots\!00}a^{16}+\frac{69\!\cdots\!87}{24\!\cdots\!20}a^{14}+\frac{31\!\cdots\!07}{16\!\cdots\!00}a^{12}-\frac{32\!\cdots\!77}{93\!\cdots\!00}a^{10}+\frac{13\!\cdots\!27}{23\!\cdots\!00}a^{8}+\frac{15\!\cdots\!37}{23\!\cdots\!00}a^{6}-\frac{52\!\cdots\!11}{18\!\cdots\!00}a^{4}-\frac{65\!\cdots\!39}{29\!\cdots\!80}a^{2}-\frac{30\!\cdots\!21}{15\!\cdots\!00}$, $\frac{28\!\cdots\!73}{14\!\cdots\!00}a^{31}-\frac{16\!\cdots\!23}{14\!\cdots\!00}a^{30}+\frac{20\!\cdots\!63}{74\!\cdots\!00}a^{29}-\frac{12\!\cdots\!01}{72\!\cdots\!00}a^{28}+\frac{32\!\cdots\!53}{14\!\cdots\!00}a^{27}-\frac{20\!\cdots\!03}{14\!\cdots\!00}a^{26}+\frac{35\!\cdots\!51}{24\!\cdots\!00}a^{25}-\frac{66\!\cdots\!31}{72\!\cdots\!00}a^{24}+\frac{89\!\cdots\!61}{12\!\cdots\!00}a^{23}-\frac{17\!\cdots\!69}{36\!\cdots\!00}a^{22}+\frac{83\!\cdots\!93}{37\!\cdots\!00}a^{21}-\frac{18\!\cdots\!41}{12\!\cdots\!00}a^{20}+\frac{27\!\cdots\!51}{49\!\cdots\!00}a^{19}-\frac{18\!\cdots\!73}{48\!\cdots\!00}a^{18}+\frac{19\!\cdots\!69}{12\!\cdots\!00}a^{17}-\frac{38\!\cdots\!21}{36\!\cdots\!00}a^{16}+\frac{41\!\cdots\!73}{17\!\cdots\!20}a^{15}-\frac{25\!\cdots\!63}{14\!\cdots\!00}a^{14}-\frac{11\!\cdots\!53}{65\!\cdots\!00}a^{13}-\frac{21\!\cdots\!73}{72\!\cdots\!40}a^{12}+\frac{43\!\cdots\!51}{37\!\cdots\!00}a^{11}+\frac{35\!\cdots\!57}{12\!\cdots\!00}a^{10}+\frac{17\!\cdots\!89}{31\!\cdots\!00}a^{9}-\frac{31\!\cdots\!41}{90\!\cdots\!00}a^{8}+\frac{88\!\cdots\!93}{18\!\cdots\!40}a^{7}-\frac{52\!\cdots\!61}{12\!\cdots\!24}a^{6}-\frac{13\!\cdots\!03}{18\!\cdots\!60}a^{5}-\frac{11\!\cdots\!19}{70\!\cdots\!00}a^{4}-\frac{79\!\cdots\!41}{58\!\cdots\!00}a^{3}+\frac{27\!\cdots\!27}{17\!\cdots\!00}a^{2}-\frac{23\!\cdots\!39}{18\!\cdots\!00}a+\frac{17\!\cdots\!77}{17\!\cdots\!00}$, $\frac{12\!\cdots\!67}{16\!\cdots\!00}a^{31}+\frac{38\!\cdots\!63}{37\!\cdots\!00}a^{30}+\frac{50\!\cdots\!59}{48\!\cdots\!00}a^{29}+\frac{28\!\cdots\!09}{18\!\cdots\!00}a^{28}+\frac{24\!\cdots\!79}{28\!\cdots\!00}a^{27}+\frac{45\!\cdots\!31}{37\!\cdots\!00}a^{26}+\frac{81\!\cdots\!59}{14\!\cdots\!00}a^{25}+\frac{14\!\cdots\!47}{18\!\cdots\!00}a^{24}+\frac{69\!\cdots\!87}{24\!\cdots\!00}a^{23}+\frac{38\!\cdots\!13}{93\!\cdots\!00}a^{22}+\frac{74\!\cdots\!03}{80\!\cdots\!00}a^{21}+\frac{24\!\cdots\!43}{18\!\cdots\!40}a^{20}+\frac{22\!\cdots\!49}{96\!\cdots\!00}a^{19}+\frac{40\!\cdots\!89}{12\!\cdots\!00}a^{18}+\frac{46\!\cdots\!01}{72\!\cdots\!00}a^{17}+\frac{49\!\cdots\!61}{54\!\cdots\!00}a^{16}+\frac{33\!\cdots\!07}{32\!\cdots\!00}a^{15}+\frac{17\!\cdots\!69}{12\!\cdots\!00}a^{14}+\frac{32\!\cdots\!97}{28\!\cdots\!76}a^{13}+\frac{73\!\cdots\!13}{10\!\cdots\!20}a^{12}-\frac{13\!\cdots\!39}{72\!\cdots\!00}a^{11}-\frac{17\!\cdots\!71}{93\!\cdots\!00}a^{10}+\frac{41\!\cdots\!01}{20\!\cdots\!00}a^{9}+\frac{98\!\cdots\!47}{32\!\cdots\!00}a^{8}+\frac{48\!\cdots\!43}{20\!\cdots\!00}a^{7}+\frac{75\!\cdots\!83}{23\!\cdots\!00}a^{6}+\frac{16\!\cdots\!61}{17\!\cdots\!00}a^{5}-\frac{12\!\cdots\!69}{91\!\cdots\!00}a^{4}-\frac{15\!\cdots\!01}{20\!\cdots\!00}a^{3}-\frac{99\!\cdots\!23}{97\!\cdots\!60}a^{2}-\frac{86\!\cdots\!71}{11\!\cdots\!00}a-\frac{14\!\cdots\!07}{15\!\cdots\!00}$, $\frac{16\!\cdots\!21}{74\!\cdots\!00}a^{31}+\frac{83\!\cdots\!01}{22\!\cdots\!00}a^{30}+\frac{35\!\cdots\!09}{11\!\cdots\!00}a^{29}+\frac{14\!\cdots\!03}{24\!\cdots\!20}a^{28}+\frac{18\!\cdots\!73}{74\!\cdots\!00}a^{27}+\frac{10\!\cdots\!69}{22\!\cdots\!00}a^{26}+\frac{17\!\cdots\!23}{11\!\cdots\!00}a^{25}+\frac{34\!\cdots\!29}{11\!\cdots\!00}a^{24}+\frac{90\!\cdots\!81}{11\!\cdots\!40}a^{23}+\frac{83\!\cdots\!23}{52\!\cdots\!80}a^{22}+\frac{45\!\cdots\!73}{18\!\cdots\!00}a^{21}+\frac{96\!\cdots\!99}{18\!\cdots\!00}a^{20}+\frac{14\!\cdots\!83}{24\!\cdots\!00}a^{19}+\frac{51\!\cdots\!41}{39\!\cdots\!00}a^{18}+\frac{31\!\cdots\!31}{18\!\cdots\!00}a^{17}+\frac{19\!\cdots\!91}{55\!\cdots\!00}a^{16}+\frac{52\!\cdots\!63}{22\!\cdots\!00}a^{15}+\frac{89\!\cdots\!11}{14\!\cdots\!20}a^{14}-\frac{13\!\cdots\!21}{18\!\cdots\!00}a^{13}+\frac{33\!\cdots\!31}{55\!\cdots\!00}a^{12}+\frac{38\!\cdots\!41}{55\!\cdots\!00}a^{11}-\frac{13\!\cdots\!77}{55\!\cdots\!00}a^{10}+\frac{10\!\cdots\!09}{16\!\cdots\!80}a^{9}+\frac{12\!\cdots\!33}{93\!\cdots\!20}a^{8}+\frac{13\!\cdots\!73}{46\!\cdots\!00}a^{7}+\frac{67\!\cdots\!43}{46\!\cdots\!00}a^{6}+\frac{84\!\cdots\!33}{72\!\cdots\!00}a^{5}-\frac{15\!\cdots\!83}{57\!\cdots\!00}a^{4}-\frac{93\!\cdots\!49}{88\!\cdots\!00}a^{3}-\frac{18\!\cdots\!89}{46\!\cdots\!00}a^{2}-\frac{64\!\cdots\!49}{27\!\cdots\!00}a-\frac{34\!\cdots\!29}{91\!\cdots\!00}$, $\frac{16\!\cdots\!43}{44\!\cdots\!00}a^{31}+\frac{52\!\cdots\!55}{29\!\cdots\!44}a^{30}+\frac{24\!\cdots\!57}{49\!\cdots\!40}a^{29}+\frac{26\!\cdots\!79}{11\!\cdots\!00}a^{28}+\frac{15\!\cdots\!87}{44\!\cdots\!00}a^{27}+\frac{43\!\cdots\!77}{24\!\cdots\!00}a^{26}+\frac{48\!\cdots\!59}{22\!\cdots\!00}a^{25}+\frac{11\!\cdots\!49}{11\!\cdots\!00}a^{24}+\frac{37\!\cdots\!27}{37\!\cdots\!00}a^{23}+\frac{28\!\cdots\!87}{55\!\cdots\!00}a^{22}+\frac{87\!\cdots\!53}{37\!\cdots\!00}a^{21}+\frac{44\!\cdots\!91}{37\!\cdots\!80}a^{20}+\frac{70\!\cdots\!21}{14\!\cdots\!00}a^{19}+\frac{54\!\cdots\!73}{24\!\cdots\!00}a^{18}+\frac{16\!\cdots\!69}{11\!\cdots\!00}a^{17}+\frac{26\!\cdots\!81}{37\!\cdots\!80}a^{16}+\frac{18\!\cdots\!09}{59\!\cdots\!88}a^{15}+\frac{30\!\cdots\!25}{89\!\cdots\!32}a^{14}-\frac{27\!\cdots\!15}{44\!\cdots\!16}a^{13}-\frac{59\!\cdots\!87}{18\!\cdots\!00}a^{12}+\frac{30\!\cdots\!09}{11\!\cdots\!00}a^{11}+\frac{78\!\cdots\!51}{55\!\cdots\!00}a^{10}+\frac{48\!\cdots\!61}{31\!\cdots\!00}a^{9}+\frac{20\!\cdots\!83}{13\!\cdots\!00}a^{8}+\frac{39\!\cdots\!13}{93\!\cdots\!00}a^{7}+\frac{12\!\cdots\!71}{16\!\cdots\!60}a^{6}-\frac{30\!\cdots\!69}{32\!\cdots\!00}a^{5}-\frac{21\!\cdots\!17}{18\!\cdots\!60}a^{4}-\frac{86\!\cdots\!71}{17\!\cdots\!00}a^{3}-\frac{10\!\cdots\!53}{88\!\cdots\!00}a^{2}-\frac{15\!\cdots\!13}{35\!\cdots\!00}a+\frac{33\!\cdots\!97}{27\!\cdots\!00}$, $\frac{41\!\cdots\!53}{44\!\cdots\!00}a^{31}-\frac{18\!\cdots\!17}{22\!\cdots\!00}a^{30}+\frac{10\!\cdots\!61}{74\!\cdots\!00}a^{29}-\frac{13\!\cdots\!11}{11\!\cdots\!00}a^{28}+\frac{49\!\cdots\!13}{44\!\cdots\!00}a^{27}-\frac{21\!\cdots\!13}{22\!\cdots\!00}a^{26}+\frac{16\!\cdots\!29}{22\!\cdots\!00}a^{25}-\frac{69\!\cdots\!57}{11\!\cdots\!00}a^{24}+\frac{46\!\cdots\!59}{12\!\cdots\!00}a^{23}-\frac{17\!\cdots\!11}{55\!\cdots\!00}a^{22}+\frac{44\!\cdots\!79}{37\!\cdots\!00}a^{21}-\frac{63\!\cdots\!89}{62\!\cdots\!00}a^{20}+\frac{45\!\cdots\!43}{14\!\cdots\!00}a^{19}-\frac{64\!\cdots\!41}{24\!\cdots\!00}a^{18}+\frac{18\!\cdots\!99}{22\!\cdots\!80}a^{17}-\frac{39\!\cdots\!99}{55\!\cdots\!00}a^{16}+\frac{70\!\cdots\!59}{52\!\cdots\!20}a^{15}-\frac{25\!\cdots\!01}{22\!\cdots\!00}a^{14}+\frac{33\!\cdots\!71}{22\!\cdots\!80}a^{13}-\frac{52\!\cdots\!67}{55\!\cdots\!00}a^{12}-\frac{80\!\cdots\!31}{58\!\cdots\!00}a^{11}+\frac{17\!\cdots\!31}{18\!\cdots\!00}a^{10}+\frac{25\!\cdots\!33}{93\!\cdots\!00}a^{9}-\frac{33\!\cdots\!79}{13\!\cdots\!00}a^{8}+\frac{28\!\cdots\!47}{93\!\cdots\!00}a^{7}-\frac{11\!\cdots\!47}{46\!\cdots\!00}a^{6}+\frac{11\!\cdots\!57}{13\!\cdots\!00}a^{5}+\frac{60\!\cdots\!03}{21\!\cdots\!20}a^{4}-\frac{35\!\cdots\!09}{35\!\cdots\!60}a^{3}+\frac{71\!\cdots\!51}{88\!\cdots\!00}a^{2}-\frac{17\!\cdots\!81}{18\!\cdots\!00}a+\frac{20\!\cdots\!47}{27\!\cdots\!00}$, $\frac{16\!\cdots\!91}{14\!\cdots\!00}a^{31}-\frac{12\!\cdots\!11}{18\!\cdots\!00}a^{30}+\frac{38\!\cdots\!79}{24\!\cdots\!00}a^{29}-\frac{28\!\cdots\!19}{27\!\cdots\!00}a^{28}+\frac{18\!\cdots\!99}{14\!\cdots\!00}a^{27}-\frac{52\!\cdots\!21}{62\!\cdots\!00}a^{26}+\frac{24\!\cdots\!79}{29\!\cdots\!44}a^{25}-\frac{15\!\cdots\!53}{27\!\cdots\!00}a^{24}+\frac{15\!\cdots\!61}{37\!\cdots\!00}a^{23}-\frac{80\!\cdots\!39}{27\!\cdots\!60}a^{22}+\frac{48\!\cdots\!03}{37\!\cdots\!00}a^{21}-\frac{14\!\cdots\!41}{15\!\cdots\!00}a^{20}+\frac{98\!\cdots\!03}{29\!\cdots\!40}a^{19}-\frac{80\!\cdots\!07}{32\!\cdots\!00}a^{18}+\frac{11\!\cdots\!23}{12\!\cdots\!00}a^{17}-\frac{31\!\cdots\!21}{46\!\cdots\!00}a^{16}+\frac{69\!\cdots\!69}{49\!\cdots\!00}a^{15}-\frac{64\!\cdots\!93}{55\!\cdots\!00}a^{14}-\frac{66\!\cdots\!77}{12\!\cdots\!00}a^{13}-\frac{15\!\cdots\!89}{46\!\cdots\!00}a^{12}-\frac{34\!\cdots\!79}{74\!\cdots\!60}a^{11}+\frac{36\!\cdots\!89}{13\!\cdots\!00}a^{10}+\frac{29\!\cdots\!89}{93\!\cdots\!00}a^{9}-\frac{29\!\cdots\!51}{13\!\cdots\!88}a^{8}+\frac{27\!\cdots\!83}{93\!\cdots\!00}a^{7}-\frac{33\!\cdots\!63}{11\!\cdots\!00}a^{6}-\frac{43\!\cdots\!71}{24\!\cdots\!00}a^{5}-\frac{60\!\cdots\!33}{17\!\cdots\!25}a^{4}-\frac{17\!\cdots\!37}{19\!\cdots\!00}a^{3}+\frac{19\!\cdots\!57}{23\!\cdots\!00}a^{2}-\frac{41\!\cdots\!57}{36\!\cdots\!20}a+\frac{45\!\cdots\!27}{40\!\cdots\!00}$, $\frac{10\!\cdots\!81}{41\!\cdots\!32}a^{31}-\frac{18263705546347}{17\!\cdots\!76}a^{30}+\frac{33\!\cdots\!43}{10\!\cdots\!80}a^{29}-\frac{612028578677109}{44\!\cdots\!40}a^{28}+\frac{86\!\cdots\!87}{34\!\cdots\!00}a^{27}-\frac{92\!\cdots\!27}{89\!\cdots\!80}a^{26}+\frac{82\!\cdots\!17}{51\!\cdots\!00}a^{25}-\frac{29\!\cdots\!83}{44\!\cdots\!40}a^{24}+\frac{20\!\cdots\!79}{25\!\cdots\!00}a^{23}-\frac{71\!\cdots\!49}{22\!\cdots\!20}a^{22}+\frac{55\!\cdots\!73}{25\!\cdots\!00}a^{21}-\frac{19\!\cdots\!47}{22\!\cdots\!20}a^{20}+\frac{10\!\cdots\!17}{20\!\cdots\!60}a^{19}-\frac{18\!\cdots\!23}{89\!\cdots\!80}a^{18}+\frac{37\!\cdots\!07}{25\!\cdots\!00}a^{17}-\frac{26\!\cdots\!69}{44\!\cdots\!44}a^{16}+\frac{15\!\cdots\!37}{10\!\cdots\!00}a^{15}-\frac{48\!\cdots\!31}{89\!\cdots\!80}a^{14}-\frac{62\!\cdots\!21}{25\!\cdots\!00}a^{13}+\frac{22\!\cdots\!47}{22\!\cdots\!20}a^{12}+\frac{44\!\cdots\!73}{86\!\cdots\!00}a^{11}-\frac{19\!\cdots\!77}{44\!\cdots\!44}a^{10}+\frac{43\!\cdots\!11}{64\!\cdots\!00}a^{9}-\frac{98\!\cdots\!01}{55\!\cdots\!80}a^{8}+\frac{37\!\cdots\!57}{64\!\cdots\!00}a^{7}+\frac{35\!\cdots\!83}{29\!\cdots\!20}a^{6}-\frac{19\!\cdots\!37}{40\!\cdots\!80}a^{5}+\frac{10\!\cdots\!03}{43\!\cdots\!60}a^{4}-\frac{41\!\cdots\!69}{54\!\cdots\!00}a^{3}+\frac{611978134559791}{10\!\cdots\!40}a^{2}-\frac{23\!\cdots\!31}{12\!\cdots\!00}a+\frac{19\!\cdots\!77}{10\!\cdots\!40}$, $\frac{12\!\cdots\!89}{22\!\cdots\!00}a^{30}+\frac{54\!\cdots\!67}{65\!\cdots\!00}a^{28}+\frac{15\!\cdots\!61}{22\!\cdots\!00}a^{26}+\frac{16\!\cdots\!19}{37\!\cdots\!00}a^{24}+\frac{12\!\cdots\!39}{55\!\cdots\!00}a^{22}+\frac{19\!\cdots\!37}{26\!\cdots\!00}a^{20}+\frac{45\!\cdots\!37}{24\!\cdots\!00}a^{18}+\frac{28\!\cdots\!67}{55\!\cdots\!00}a^{16}+\frac{18\!\cdots\!09}{22\!\cdots\!00}a^{14}+\frac{35\!\cdots\!99}{55\!\cdots\!00}a^{12}-\frac{68\!\cdots\!37}{32\!\cdots\!00}a^{10}+\frac{22\!\cdots\!11}{13\!\cdots\!00}a^{8}+\frac{30\!\cdots\!37}{15\!\cdots\!00}a^{6}+\frac{13\!\cdots\!23}{10\!\cdots\!00}a^{4}-\frac{39\!\cdots\!93}{58\!\cdots\!60}a^{2}-\frac{30\!\cdots\!43}{54\!\cdots\!80}$
| 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: | \( 51163178113729.83 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 51163178113729.83 \cdot 1152}{6\cdot\sqrt{1720706227830984913861923492628167351336960000000000000000}}\cr\approx \mathstrut & 1.39727944706099 \end{aligned}\] (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 | ${\href{/padicField/7.8.0.1}{8} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.8.0.1}{8} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{12}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{16}$ | 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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.8.16.11 | $x^{8} + 2 x^{6} - 4 x^{5} + 20 x^{4} + 8 x^{3} + 44 x^{2} - 8 x + 76$ | $4$ | $2$ | $16$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ | |
| 2.8.16.11 | $x^{8} + 2 x^{6} - 4 x^{5} + 20 x^{4} + 8 x^{3} + 44 x^{2} - 8 x + 76$ | $4$ | $2$ | $16$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ | |
|
\(3\)
| 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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(31\)
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(89\)
| 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 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |