Normalized defining polynomial
\( x^{26} + 158 x^{24} + 10112 x^{22} + 341912 x^{20} + 6687824 x^{18} + 78249184 x^{16} + 550482112 x^{14} + \cdots + 647168 \)
Invariants
| Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-151651640667491538463656289407733022385317053604971635802112\)
\(\medspace = -\,2^{39}\cdot 79^{25}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(188.88\) | 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: | $2^{3/2}79^{25/26}\approx 188.88019717124266$ | ||
| Ramified primes: |
\(2\), \(79\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-158}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $26$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(632=2^{3}\cdot 79\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{632}(1,·)$, $\chi_{632}(97,·)$, $\chi_{632}(453,·)$, $\chi_{632}(65,·)$, $\chi_{632}(457,·)$, $\chi_{632}(333,·)$, $\chi_{632}(417,·)$, $\chi_{632}(337,·)$, $\chi_{632}(89,·)$, $\chi_{632}(157,·)$, $\chi_{632}(69,·)$, $\chi_{632}(289,·)$, $\chi_{632}(229,·)$, $\chi_{632}(373,·)$, $\chi_{632}(357,·)$, $\chi_{632}(561,·)$, $\chi_{632}(617,·)$, $\chi_{632}(225,·)$, $\chi_{632}(173,·)$, $\chi_{632}(93,·)$, $\chi_{632}(433,·)$, $\chi_{632}(565,·)$, $\chi_{632}(349,·)$, $\chi_{632}(441,·)$, $\chi_{632}(61,·)$, $\chi_{632}(501,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{4096}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{11776}a^{18}-\frac{1}{1472}a^{16}-\frac{9}{2944}a^{14}-\frac{1}{736}a^{12}+\frac{3}{736}a^{10}-\frac{1}{46}a^{8}+\frac{3}{184}a^{6}+\frac{1}{23}a^{4}+\frac{1}{46}a^{2}+\frac{11}{23}$, $\frac{1}{11776}a^{19}-\frac{1}{1472}a^{17}-\frac{9}{2944}a^{15}-\frac{1}{736}a^{13}+\frac{3}{736}a^{11}-\frac{1}{46}a^{9}+\frac{3}{184}a^{7}+\frac{1}{23}a^{5}+\frac{1}{46}a^{3}+\frac{11}{23}a$, $\frac{1}{23552}a^{20}-\frac{1}{2944}a^{16}+\frac{1}{368}a^{14}-\frac{5}{1472}a^{12}+\frac{1}{184}a^{10}-\frac{3}{184}a^{8}-\frac{7}{184}a^{6}-\frac{3}{46}a^{4}-\frac{4}{23}a^{2}-\frac{2}{23}$, $\frac{1}{23552}a^{21}-\frac{1}{2944}a^{17}+\frac{1}{368}a^{15}-\frac{5}{1472}a^{13}+\frac{1}{184}a^{11}-\frac{3}{184}a^{9}-\frac{7}{184}a^{7}-\frac{3}{46}a^{5}-\frac{4}{23}a^{3}-\frac{2}{23}a$, $\frac{1}{4851712}a^{22}+\frac{9}{606464}a^{20}-\frac{15}{1212928}a^{18}-\frac{29}{303232}a^{16}-\frac{1003}{303232}a^{14}+\frac{43}{37904}a^{12}+\frac{153}{37904}a^{10}+\frac{1077}{37904}a^{8}-\frac{321}{9476}a^{6}-\frac{9}{412}a^{4}-\frac{579}{4738}a^{2}-\frac{376}{2369}$, $\frac{1}{4851712}a^{23}+\frac{9}{606464}a^{21}-\frac{15}{1212928}a^{19}-\frac{29}{303232}a^{17}-\frac{1003}{303232}a^{15}+\frac{43}{37904}a^{13}+\frac{153}{37904}a^{11}+\frac{1077}{37904}a^{9}-\frac{321}{9476}a^{7}-\frac{9}{412}a^{5}-\frac{579}{4738}a^{3}-\frac{376}{2369}a$, $\frac{1}{25\!\cdots\!72}a^{24}-\frac{48\!\cdots\!69}{62\!\cdots\!68}a^{22}-\frac{43\!\cdots\!61}{31\!\cdots\!84}a^{20}-\frac{20\!\cdots\!51}{39\!\cdots\!48}a^{18}-\frac{25\!\cdots\!87}{39\!\cdots\!48}a^{16}+\frac{68\!\cdots\!49}{24\!\cdots\!28}a^{14}+\frac{95\!\cdots\!07}{19\!\cdots\!24}a^{12}-\frac{35\!\cdots\!13}{19\!\cdots\!24}a^{10}-\frac{10\!\cdots\!01}{49\!\cdots\!56}a^{8}-\frac{26\!\cdots\!61}{49\!\cdots\!56}a^{6}+\frac{10\!\cdots\!95}{12\!\cdots\!14}a^{4}+\frac{45\!\cdots\!51}{61\!\cdots\!07}a^{2}+\frac{38\!\cdots\!87}{61\!\cdots\!07}$, $\frac{1}{25\!\cdots\!72}a^{25}-\frac{48\!\cdots\!69}{62\!\cdots\!68}a^{23}-\frac{43\!\cdots\!61}{31\!\cdots\!84}a^{21}-\frac{20\!\cdots\!51}{39\!\cdots\!48}a^{19}-\frac{25\!\cdots\!87}{39\!\cdots\!48}a^{17}+\frac{68\!\cdots\!49}{24\!\cdots\!28}a^{15}+\frac{95\!\cdots\!07}{19\!\cdots\!24}a^{13}-\frac{35\!\cdots\!13}{19\!\cdots\!24}a^{11}-\frac{10\!\cdots\!01}{49\!\cdots\!56}a^{9}-\frac{26\!\cdots\!61}{49\!\cdots\!56}a^{7}+\frac{10\!\cdots\!95}{12\!\cdots\!14}a^{5}+\frac{45\!\cdots\!51}{61\!\cdots\!07}a^{3}+\frac{38\!\cdots\!87}{61\!\cdots\!07}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $23$ |
Class group and class number
$C_{217593176}$, which has order $217593176$ (assuming GRH)
Unit group
| Rank: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{26\!\cdots\!31}{62\!\cdots\!68}a^{24}+\frac{82\!\cdots\!51}{12\!\cdots\!36}a^{22}+\frac{26\!\cdots\!03}{62\!\cdots\!68}a^{20}+\frac{44\!\cdots\!33}{31\!\cdots\!84}a^{18}+\frac{10\!\cdots\!03}{39\!\cdots\!48}a^{16}+\frac{63\!\cdots\!01}{19\!\cdots\!24}a^{14}+\frac{11\!\cdots\!97}{49\!\cdots\!56}a^{12}+\frac{17\!\cdots\!19}{19\!\cdots\!24}a^{10}+\frac{10\!\cdots\!21}{49\!\cdots\!56}a^{8}+\frac{26\!\cdots\!77}{12\!\cdots\!14}a^{6}+\frac{25\!\cdots\!81}{24\!\cdots\!28}a^{4}+\frac{22\!\cdots\!19}{12\!\cdots\!14}a^{2}+\frac{53\!\cdots\!12}{61\!\cdots\!07}$, $\frac{18\!\cdots\!95}{25\!\cdots\!72}a^{24}+\frac{14\!\cdots\!63}{12\!\cdots\!36}a^{22}+\frac{47\!\cdots\!37}{62\!\cdots\!68}a^{20}+\frac{40\!\cdots\!21}{15\!\cdots\!92}a^{18}+\frac{78\!\cdots\!13}{15\!\cdots\!92}a^{16}+\frac{14\!\cdots\!63}{24\!\cdots\!28}a^{14}+\frac{16\!\cdots\!97}{39\!\cdots\!48}a^{12}+\frac{33\!\cdots\!97}{19\!\cdots\!24}a^{10}+\frac{18\!\cdots\!59}{49\!\cdots\!56}a^{8}+\frac{26\!\cdots\!20}{61\!\cdots\!07}a^{6}+\frac{26\!\cdots\!99}{12\!\cdots\!14}a^{4}+\frac{24\!\cdots\!34}{61\!\cdots\!07}a^{2}+\frac{96\!\cdots\!83}{61\!\cdots\!07}$, $\frac{62\!\cdots\!17}{25\!\cdots\!72}a^{24}+\frac{12\!\cdots\!35}{31\!\cdots\!84}a^{22}+\frac{19\!\cdots\!91}{78\!\cdots\!96}a^{20}+\frac{65\!\cdots\!71}{78\!\cdots\!96}a^{18}+\frac{12\!\cdots\!89}{78\!\cdots\!96}a^{16}+\frac{71\!\cdots\!15}{39\!\cdots\!48}a^{14}+\frac{47\!\cdots\!73}{39\!\cdots\!48}a^{12}+\frac{45\!\cdots\!31}{98\!\cdots\!12}a^{10}+\frac{42\!\cdots\!33}{49\!\cdots\!56}a^{8}+\frac{27\!\cdots\!97}{49\!\cdots\!56}a^{6}-\frac{85\!\cdots\!75}{12\!\cdots\!14}a^{4}-\frac{95\!\cdots\!10}{61\!\cdots\!07}a^{2}-\frac{18\!\cdots\!49}{61\!\cdots\!07}$, $\frac{16\!\cdots\!85}{25\!\cdots\!72}a^{24}+\frac{65\!\cdots\!97}{62\!\cdots\!68}a^{22}+\frac{20\!\cdots\!29}{31\!\cdots\!84}a^{20}+\frac{17\!\cdots\!45}{78\!\cdots\!96}a^{18}+\frac{34\!\cdots\!77}{78\!\cdots\!96}a^{16}+\frac{20\!\cdots\!29}{39\!\cdots\!48}a^{14}+\frac{14\!\cdots\!69}{39\!\cdots\!48}a^{12}+\frac{29\!\cdots\!07}{19\!\cdots\!24}a^{10}+\frac{20\!\cdots\!70}{61\!\cdots\!07}a^{8}+\frac{18\!\cdots\!35}{49\!\cdots\!56}a^{6}+\frac{12\!\cdots\!59}{61\!\cdots\!07}a^{4}+\frac{25\!\cdots\!42}{61\!\cdots\!07}a^{2}+\frac{16\!\cdots\!06}{61\!\cdots\!07}$, $\frac{33\!\cdots\!81}{12\!\cdots\!36}a^{24}+\frac{52\!\cdots\!49}{12\!\cdots\!36}a^{22}+\frac{83\!\cdots\!69}{31\!\cdots\!84}a^{20}+\frac{69\!\cdots\!23}{78\!\cdots\!96}a^{18}+\frac{13\!\cdots\!91}{78\!\cdots\!96}a^{16}+\frac{38\!\cdots\!77}{19\!\cdots\!24}a^{14}+\frac{12\!\cdots\!21}{98\!\cdots\!12}a^{12}+\frac{48\!\cdots\!05}{98\!\cdots\!12}a^{10}+\frac{91\!\cdots\!63}{98\!\cdots\!12}a^{8}+\frac{28\!\cdots\!87}{49\!\cdots\!56}a^{6}-\frac{41\!\cdots\!23}{24\!\cdots\!28}a^{4}-\frac{24\!\cdots\!35}{12\!\cdots\!14}a^{2}-\frac{91\!\cdots\!76}{61\!\cdots\!07}$, $\frac{97\!\cdots\!31}{62\!\cdots\!68}a^{24}+\frac{15\!\cdots\!31}{62\!\cdots\!68}a^{22}+\frac{49\!\cdots\!51}{31\!\cdots\!84}a^{20}+\frac{10\!\cdots\!79}{19\!\cdots\!24}a^{18}+\frac{16\!\cdots\!61}{15\!\cdots\!92}a^{16}+\frac{24\!\cdots\!83}{19\!\cdots\!24}a^{14}+\frac{85\!\cdots\!89}{98\!\cdots\!12}a^{12}+\frac{17\!\cdots\!71}{49\!\cdots\!56}a^{10}+\frac{50\!\cdots\!15}{61\!\cdots\!07}a^{8}+\frac{22\!\cdots\!37}{24\!\cdots\!28}a^{6}+\frac{54\!\cdots\!21}{12\!\cdots\!14}a^{4}+\frac{44\!\cdots\!75}{61\!\cdots\!07}a^{2}+\frac{38\!\cdots\!26}{61\!\cdots\!07}$, $\frac{16\!\cdots\!51}{24\!\cdots\!24}a^{24}+\frac{65\!\cdots\!23}{62\!\cdots\!68}a^{22}+\frac{42\!\cdots\!09}{62\!\cdots\!68}a^{20}+\frac{35\!\cdots\!37}{15\!\cdots\!92}a^{18}+\frac{87\!\cdots\!45}{19\!\cdots\!24}a^{16}+\frac{40\!\cdots\!53}{78\!\cdots\!96}a^{14}+\frac{17\!\cdots\!63}{49\!\cdots\!56}a^{12}+\frac{29\!\cdots\!93}{19\!\cdots\!24}a^{10}+\frac{32\!\cdots\!19}{98\!\cdots\!12}a^{8}+\frac{44\!\cdots\!01}{12\!\cdots\!14}a^{6}+\frac{11\!\cdots\!38}{61\!\cdots\!07}a^{4}+\frac{42\!\cdots\!89}{12\!\cdots\!14}a^{2}+\frac{10\!\cdots\!88}{61\!\cdots\!07}$, $\frac{41\!\cdots\!21}{25\!\cdots\!72}a^{24}+\frac{16\!\cdots\!65}{62\!\cdots\!68}a^{22}+\frac{10\!\cdots\!09}{62\!\cdots\!68}a^{20}+\frac{87\!\cdots\!95}{15\!\cdots\!92}a^{18}+\frac{17\!\cdots\!15}{15\!\cdots\!92}a^{16}+\frac{99\!\cdots\!19}{78\!\cdots\!96}a^{14}+\frac{34\!\cdots\!35}{39\!\cdots\!48}a^{12}+\frac{70\!\cdots\!63}{19\!\cdots\!24}a^{10}+\frac{78\!\cdots\!97}{98\!\cdots\!12}a^{8}+\frac{51\!\cdots\!77}{61\!\cdots\!07}a^{6}+\frac{23\!\cdots\!94}{61\!\cdots\!07}a^{4}+\frac{79\!\cdots\!41}{12\!\cdots\!14}a^{2}+\frac{68\!\cdots\!43}{61\!\cdots\!07}$, $\frac{76\!\cdots\!17}{12\!\cdots\!36}a^{24}+\frac{29\!\cdots\!95}{31\!\cdots\!84}a^{22}+\frac{47\!\cdots\!13}{78\!\cdots\!96}a^{20}+\frac{80\!\cdots\!03}{39\!\cdots\!48}a^{18}+\frac{97\!\cdots\!47}{24\!\cdots\!28}a^{16}+\frac{36\!\cdots\!71}{78\!\cdots\!96}a^{14}+\frac{62\!\cdots\!25}{19\!\cdots\!24}a^{12}+\frac{12\!\cdots\!91}{98\!\cdots\!12}a^{10}+\frac{66\!\cdots\!67}{24\!\cdots\!28}a^{8}+\frac{32\!\cdots\!89}{12\!\cdots\!14}a^{6}+\frac{64\!\cdots\!08}{61\!\cdots\!07}a^{4}+\frac{91\!\cdots\!23}{61\!\cdots\!07}a^{2}+\frac{21\!\cdots\!28}{61\!\cdots\!07}$, $\frac{20\!\cdots\!79}{25\!\cdots\!72}a^{24}+\frac{16\!\cdots\!29}{12\!\cdots\!36}a^{22}+\frac{65\!\cdots\!71}{78\!\cdots\!96}a^{20}+\frac{22\!\cdots\!63}{78\!\cdots\!96}a^{18}+\frac{43\!\cdots\!51}{78\!\cdots\!96}a^{16}+\frac{25\!\cdots\!27}{39\!\cdots\!48}a^{14}+\frac{17\!\cdots\!21}{39\!\cdots\!48}a^{12}+\frac{37\!\cdots\!87}{19\!\cdots\!24}a^{10}+\frac{20\!\cdots\!97}{49\!\cdots\!56}a^{8}+\frac{57\!\cdots\!37}{12\!\cdots\!14}a^{6}+\frac{56\!\cdots\!71}{24\!\cdots\!28}a^{4}+\frac{49\!\cdots\!39}{12\!\cdots\!14}a^{2}+\frac{45\!\cdots\!31}{61\!\cdots\!07}$, $\frac{37\!\cdots\!27}{25\!\cdots\!72}a^{24}+\frac{29\!\cdots\!67}{12\!\cdots\!36}a^{22}+\frac{47\!\cdots\!67}{31\!\cdots\!84}a^{20}+\frac{79\!\cdots\!51}{15\!\cdots\!92}a^{18}+\frac{77\!\cdots\!71}{78\!\cdots\!96}a^{16}+\frac{70\!\cdots\!71}{61\!\cdots\!07}a^{14}+\frac{31\!\cdots\!19}{39\!\cdots\!48}a^{12}+\frac{63\!\cdots\!29}{19\!\cdots\!24}a^{10}+\frac{69\!\cdots\!03}{98\!\cdots\!12}a^{8}+\frac{35\!\cdots\!31}{49\!\cdots\!56}a^{6}+\frac{81\!\cdots\!05}{24\!\cdots\!28}a^{4}+\frac{68\!\cdots\!99}{12\!\cdots\!14}a^{2}+\frac{38\!\cdots\!98}{61\!\cdots\!07}$, $\frac{40\!\cdots\!13}{25\!\cdots\!72}a^{24}+\frac{32\!\cdots\!91}{12\!\cdots\!36}a^{22}+\frac{10\!\cdots\!21}{62\!\cdots\!68}a^{20}+\frac{46\!\cdots\!69}{78\!\cdots\!96}a^{18}+\frac{18\!\cdots\!81}{15\!\cdots\!92}a^{16}+\frac{11\!\cdots\!19}{78\!\cdots\!96}a^{14}+\frac{10\!\cdots\!87}{98\!\cdots\!12}a^{12}+\frac{98\!\cdots\!55}{19\!\cdots\!24}a^{10}+\frac{66\!\cdots\!87}{49\!\cdots\!56}a^{8}+\frac{11\!\cdots\!37}{49\!\cdots\!56}a^{6}+\frac{73\!\cdots\!73}{24\!\cdots\!28}a^{4}+\frac{10\!\cdots\!00}{61\!\cdots\!07}a^{2}+\frac{65\!\cdots\!91}{61\!\cdots\!07}$
| 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: | \( 57529828940.82975 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 57529828940.82975 \cdot 217593176}{2\cdot\sqrt{151651640667491538463656289407733022385317053604971635802112}}\cr\approx \mathstrut & 0.382316542811987 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 26 |
| The 26 conjugacy class representatives for $C_{26}$ |
| Character table for $C_{26}$ |
Intermediate fields
| \(\Q(\sqrt{-158}) \), 13.13.59091511031674153381441.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.13.0.1}{13} }^{2}$ | $26$ | $26$ | $26$ | $26$ | $26$ | $26$ | ${\href{/padicField/23.1.0.1}{1} }^{26}$ | ${\href{/padicField/29.13.0.1}{13} }^{2}$ | ${\href{/padicField/31.13.0.1}{13} }^{2}$ | ${\href{/padicField/37.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/43.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/53.13.0.1}{13} }^{2}$ | ${\href{/padicField/59.13.0.1}{13} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $26$ | $2$ | $13$ | $39$ | |||
|
\(79\)
| Deg $26$ | $26$ | $1$ | $25$ |