Normalized defining polynomial
\( x^{26} + 121 x^{24} + 6052 x^{22} + 163895 x^{20} + 2652975 x^{18} + 26945411 x^{16} + 177180824 x^{14} + \cdots + 245454889 \)
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: |
\(-43781534893766029378911430108761129396314054312752152838144\)
\(\medspace = -\,2^{26}\cdot 131^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(180.07\) | 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: | $2\cdot 131^{12/13}\approx 180.06705783723962$ | ||
| Ramified primes: |
\(2\), \(131\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $26$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(524=2^{2}\cdot 131\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{524}(1,·)$, $\chi_{524}(107,·)$, $\chi_{524}(325,·)$, $\chi_{524}(263,·)$, $\chi_{524}(375,·)$, $\chi_{524}(369,·)$, $\chi_{524}(45,·)$, $\chi_{524}(211,·)$, $\chi_{524}(215,·)$, $\chi_{524}(473,·)$, $\chi_{524}(477,·)$, $\chi_{524}(453,·)$, $\chi_{524}(99,·)$, $\chi_{524}(243,·)$, $\chi_{524}(39,·)$, $\chi_{524}(361,·)$, $\chi_{524}(193,·)$, $\chi_{524}(455,·)$, $\chi_{524}(301,·)$, $\chi_{524}(113,·)$, $\chi_{524}(307,·)$, $\chi_{524}(183,·)$, $\chi_{524}(505,·)$, $\chi_{524}(191,·)$, $\chi_{524}(445,·)$, $\chi_{524}(63,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{4096}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{89}a^{18}+\frac{4}{89}a^{16}+\frac{25}{89}a^{14}+\frac{4}{89}a^{12}+\frac{9}{89}a^{10}-\frac{40}{89}a^{8}-\frac{20}{89}a^{6}-\frac{20}{89}a^{4}-\frac{43}{89}a^{2}+\frac{4}{89}$, $\frac{1}{89}a^{19}+\frac{4}{89}a^{17}+\frac{25}{89}a^{15}+\frac{4}{89}a^{13}+\frac{9}{89}a^{11}-\frac{40}{89}a^{9}-\frac{20}{89}a^{7}-\frac{20}{89}a^{5}-\frac{43}{89}a^{3}+\frac{4}{89}a$, $\frac{1}{4717}a^{20}+\frac{12}{4717}a^{18}+\frac{947}{4717}a^{16}+\frac{115}{4717}a^{14}-\frac{2095}{4717}a^{12}+\frac{9}{89}a^{10}-\frac{1675}{4717}a^{8}+\frac{443}{4717}a^{6}-\frac{292}{4717}a^{4}+\frac{2330}{4717}a^{2}+\frac{1278}{4717}$, $\frac{1}{4717}a^{21}+\frac{12}{4717}a^{19}+\frac{947}{4717}a^{17}+\frac{115}{4717}a^{15}-\frac{2095}{4717}a^{13}+\frac{9}{89}a^{11}-\frac{1675}{4717}a^{9}+\frac{443}{4717}a^{7}-\frac{292}{4717}a^{5}+\frac{2330}{4717}a^{3}+\frac{1278}{4717}a$, $\frac{1}{134302975889}a^{22}+\frac{10725326}{134302975889}a^{20}-\frac{389869598}{134302975889}a^{18}-\frac{5125378554}{134302975889}a^{16}-\frac{27434544643}{134302975889}a^{14}-\frac{18111369913}{134302975889}a^{12}+\frac{27517959732}{134302975889}a^{10}+\frac{38379855929}{134302975889}a^{8}+\frac{34447188688}{134302975889}a^{6}+\frac{1798184570}{134302975889}a^{4}-\frac{19598688248}{134302975889}a^{2}-\frac{33406148756}{134302975889}$, $\frac{1}{134302975889}a^{23}+\frac{10725326}{134302975889}a^{21}-\frac{389869598}{134302975889}a^{19}-\frac{5125378554}{134302975889}a^{17}-\frac{27434544643}{134302975889}a^{15}-\frac{18111369913}{134302975889}a^{13}+\frac{27517959732}{134302975889}a^{11}+\frac{38379855929}{134302975889}a^{9}+\frac{34447188688}{134302975889}a^{7}+\frac{1798184570}{134302975889}a^{5}-\frac{19598688248}{134302975889}a^{3}-\frac{33406148756}{134302975889}a$, $\frac{1}{14\!\cdots\!53}a^{24}+\frac{32\!\cdots\!05}{14\!\cdots\!53}a^{22}+\frac{19\!\cdots\!33}{14\!\cdots\!53}a^{20}+\frac{64\!\cdots\!53}{14\!\cdots\!53}a^{18}+\frac{65\!\cdots\!42}{14\!\cdots\!53}a^{16}-\frac{44\!\cdots\!36}{14\!\cdots\!53}a^{14}+\frac{57\!\cdots\!05}{14\!\cdots\!53}a^{12}+\frac{27\!\cdots\!85}{14\!\cdots\!53}a^{10}+\frac{50\!\cdots\!36}{14\!\cdots\!53}a^{8}-\frac{10\!\cdots\!32}{23\!\cdots\!73}a^{6}+\frac{22\!\cdots\!00}{14\!\cdots\!53}a^{4}+\frac{36\!\cdots\!08}{14\!\cdots\!53}a^{2}-\frac{36\!\cdots\!01}{14\!\cdots\!53}$, $\frac{1}{22\!\cdots\!51}a^{25}+\frac{14\!\cdots\!28}{22\!\cdots\!51}a^{23}-\frac{84\!\cdots\!23}{22\!\cdots\!51}a^{21}-\frac{11\!\cdots\!23}{22\!\cdots\!51}a^{19}+\frac{37\!\cdots\!29}{22\!\cdots\!51}a^{17}+\frac{15\!\cdots\!33}{22\!\cdots\!51}a^{15}-\frac{27\!\cdots\!00}{22\!\cdots\!51}a^{13}+\frac{83\!\cdots\!82}{22\!\cdots\!51}a^{11}+\frac{97\!\cdots\!20}{22\!\cdots\!51}a^{9}+\frac{96\!\cdots\!25}{36\!\cdots\!91}a^{7}+\frac{94\!\cdots\!00}{22\!\cdots\!51}a^{5}+\frac{27\!\cdots\!97}{22\!\cdots\!51}a^{3}-\frac{27\!\cdots\!52}{22\!\cdots\!51}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{3}\times C_{3}\times C_{1620687}$, which has order $14586183$ (assuming GRH)
Unit group
| Rank: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{18742303651753981654}{5268580110437032626801293} a^{25} - \frac{2232702633907726946086}{5268580110437032626801293} a^{23} - \frac{109246311084833357631699}{5268580110437032626801293} a^{21} - \frac{2867217040610101824896010}{5268580110437032626801293} a^{19} - \frac{44357186238362705592131616}{5268580110437032626801293} a^{17} - \frac{422073619611879790622887207}{5268580110437032626801293} a^{15} - \frac{2532387269702891181856705091}{5268580110437032626801293} a^{13} - \frac{9704055717554598099271574919}{5268580110437032626801293} a^{11} - \frac{23673835294114779851195236185}{5268580110437032626801293} a^{9} - \frac{35836841376669367226742312082}{5268580110437032626801293} a^{7} - \frac{31694352450296996928587506755}{5268580110437032626801293} a^{5} - \frac{14463258486750102097348665865}{5268580110437032626801293} a^{3} - \frac{2543925526041780354828639082}{5268580110437032626801293} a \)
(order $4$)
| 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: |
$\frac{50\!\cdots\!83}{14\!\cdots\!53}a^{24}+\frac{60\!\cdots\!30}{14\!\cdots\!53}a^{22}+\frac{29\!\cdots\!78}{14\!\cdots\!53}a^{20}+\frac{78\!\cdots\!25}{14\!\cdots\!53}a^{18}+\frac{12\!\cdots\!79}{14\!\cdots\!53}a^{16}+\frac{11\!\cdots\!86}{14\!\cdots\!53}a^{14}+\frac{69\!\cdots\!08}{14\!\cdots\!53}a^{12}+\frac{26\!\cdots\!36}{14\!\cdots\!53}a^{10}+\frac{90\!\cdots\!81}{19\!\cdots\!61}a^{8}+\frac{16\!\cdots\!53}{23\!\cdots\!73}a^{6}+\frac{91\!\cdots\!61}{14\!\cdots\!53}a^{4}+\frac{43\!\cdots\!77}{14\!\cdots\!53}a^{2}+\frac{80\!\cdots\!53}{14\!\cdots\!53}$, $\frac{40\!\cdots\!90}{14\!\cdots\!53}a^{24}+\frac{42\!\cdots\!89}{14\!\cdots\!53}a^{22}+\frac{17\!\cdots\!40}{14\!\cdots\!53}a^{20}+\frac{33\!\cdots\!70}{14\!\cdots\!53}a^{18}+\frac{24\!\cdots\!72}{14\!\cdots\!53}a^{16}-\frac{14\!\cdots\!77}{14\!\cdots\!53}a^{14}-\frac{39\!\cdots\!05}{14\!\cdots\!53}a^{12}-\frac{41\!\cdots\!37}{19\!\cdots\!61}a^{10}-\frac{11\!\cdots\!66}{14\!\cdots\!53}a^{8}-\frac{43\!\cdots\!73}{23\!\cdots\!73}a^{6}-\frac{31\!\cdots\!12}{14\!\cdots\!53}a^{4}-\frac{18\!\cdots\!39}{14\!\cdots\!53}a^{2}-\frac{37\!\cdots\!93}{14\!\cdots\!53}$, $\frac{80\!\cdots\!17}{14\!\cdots\!53}a^{24}+\frac{95\!\cdots\!57}{14\!\cdots\!53}a^{22}+\frac{46\!\cdots\!17}{14\!\cdots\!53}a^{20}+\frac{12\!\cdots\!28}{14\!\cdots\!53}a^{18}+\frac{18\!\cdots\!42}{14\!\cdots\!53}a^{16}+\frac{18\!\cdots\!36}{14\!\cdots\!53}a^{14}+\frac{10\!\cdots\!58}{14\!\cdots\!53}a^{12}+\frac{41\!\cdots\!27}{14\!\cdots\!53}a^{10}+\frac{10\!\cdots\!94}{14\!\cdots\!53}a^{8}+\frac{24\!\cdots\!04}{23\!\cdots\!73}a^{6}+\frac{13\!\cdots\!33}{14\!\cdots\!53}a^{4}+\frac{59\!\cdots\!87}{14\!\cdots\!53}a^{2}+\frac{10\!\cdots\!22}{14\!\cdots\!53}$, $\frac{21\!\cdots\!68}{14\!\cdots\!53}a^{24}+\frac{25\!\cdots\!46}{14\!\cdots\!53}a^{22}+\frac{12\!\cdots\!30}{14\!\cdots\!53}a^{20}+\frac{32\!\cdots\!52}{14\!\cdots\!53}a^{18}+\frac{49\!\cdots\!59}{14\!\cdots\!53}a^{16}+\frac{45\!\cdots\!45}{14\!\cdots\!53}a^{14}+\frac{26\!\cdots\!98}{14\!\cdots\!53}a^{12}+\frac{96\!\cdots\!26}{14\!\cdots\!53}a^{10}+\frac{22\!\cdots\!06}{14\!\cdots\!53}a^{8}+\frac{52\!\cdots\!49}{23\!\cdots\!73}a^{6}+\frac{26\!\cdots\!35}{14\!\cdots\!53}a^{4}+\frac{11\!\cdots\!29}{14\!\cdots\!53}a^{2}+\frac{19\!\cdots\!24}{14\!\cdots\!53}$, $\frac{66\!\cdots\!70}{14\!\cdots\!53}a^{24}+\frac{79\!\cdots\!08}{14\!\cdots\!53}a^{22}+\frac{39\!\cdots\!91}{14\!\cdots\!53}a^{20}+\frac{10\!\cdots\!70}{14\!\cdots\!53}a^{18}+\frac{16\!\cdots\!15}{14\!\cdots\!53}a^{16}+\frac{15\!\cdots\!41}{14\!\cdots\!53}a^{14}+\frac{97\!\cdots\!04}{14\!\cdots\!53}a^{12}+\frac{38\!\cdots\!00}{14\!\cdots\!53}a^{10}+\frac{96\!\cdots\!93}{14\!\cdots\!53}a^{8}+\frac{24\!\cdots\!55}{23\!\cdots\!73}a^{6}+\frac{13\!\cdots\!76}{14\!\cdots\!53}a^{4}+\frac{63\!\cdots\!51}{14\!\cdots\!53}a^{2}+\frac{11\!\cdots\!77}{14\!\cdots\!53}$, $\frac{19\!\cdots\!84}{14\!\cdots\!53}a^{24}+\frac{22\!\cdots\!20}{14\!\cdots\!53}a^{22}+\frac{11\!\cdots\!53}{14\!\cdots\!53}a^{20}+\frac{29\!\cdots\!83}{14\!\cdots\!53}a^{18}+\frac{44\!\cdots\!33}{14\!\cdots\!53}a^{16}+\frac{42\!\cdots\!87}{14\!\cdots\!53}a^{14}+\frac{25\!\cdots\!00}{14\!\cdots\!53}a^{12}+\frac{95\!\cdots\!76}{14\!\cdots\!53}a^{10}+\frac{23\!\cdots\!88}{14\!\cdots\!53}a^{8}+\frac{56\!\cdots\!27}{23\!\cdots\!73}a^{6}+\frac{30\!\cdots\!67}{14\!\cdots\!53}a^{4}+\frac{13\!\cdots\!66}{14\!\cdots\!53}a^{2}+\frac{23\!\cdots\!69}{14\!\cdots\!53}$, $\frac{34\!\cdots\!78}{19\!\cdots\!61}a^{24}+\frac{29\!\cdots\!80}{14\!\cdots\!53}a^{22}+\frac{14\!\cdots\!45}{14\!\cdots\!53}a^{20}+\frac{38\!\cdots\!55}{14\!\cdots\!53}a^{18}+\frac{59\!\cdots\!97}{14\!\cdots\!53}a^{16}+\frac{57\!\cdots\!77}{14\!\cdots\!53}a^{14}+\frac{34\!\cdots\!15}{14\!\cdots\!53}a^{12}+\frac{13\!\cdots\!74}{14\!\cdots\!53}a^{10}+\frac{34\!\cdots\!61}{14\!\cdots\!53}a^{8}+\frac{87\!\cdots\!47}{23\!\cdots\!73}a^{6}+\frac{49\!\cdots\!16}{14\!\cdots\!53}a^{4}+\frac{23\!\cdots\!81}{14\!\cdots\!53}a^{2}+\frac{43\!\cdots\!72}{14\!\cdots\!53}$, $\frac{18\!\cdots\!25}{14\!\cdots\!53}a^{24}+\frac{21\!\cdots\!33}{14\!\cdots\!53}a^{22}+\frac{10\!\cdots\!03}{14\!\cdots\!53}a^{20}+\frac{27\!\cdots\!60}{14\!\cdots\!53}a^{18}+\frac{42\!\cdots\!40}{14\!\cdots\!53}a^{16}+\frac{40\!\cdots\!16}{14\!\cdots\!53}a^{14}+\frac{23\!\cdots\!38}{14\!\cdots\!53}a^{12}+\frac{90\!\cdots\!12}{14\!\cdots\!53}a^{10}+\frac{21\!\cdots\!89}{14\!\cdots\!53}a^{8}+\frac{53\!\cdots\!40}{23\!\cdots\!73}a^{6}+\frac{28\!\cdots\!73}{14\!\cdots\!53}a^{4}+\frac{17\!\cdots\!47}{19\!\cdots\!61}a^{2}+\frac{22\!\cdots\!43}{14\!\cdots\!53}$, $\frac{26\!\cdots\!76}{14\!\cdots\!53}a^{24}+\frac{32\!\cdots\!59}{14\!\cdots\!53}a^{22}+\frac{15\!\cdots\!95}{14\!\cdots\!53}a^{20}+\frac{56\!\cdots\!10}{19\!\cdots\!61}a^{18}+\frac{63\!\cdots\!31}{14\!\cdots\!53}a^{16}+\frac{60\!\cdots\!13}{14\!\cdots\!53}a^{14}+\frac{36\!\cdots\!09}{14\!\cdots\!53}a^{12}+\frac{13\!\cdots\!47}{14\!\cdots\!53}a^{10}+\frac{33\!\cdots\!88}{14\!\cdots\!53}a^{8}+\frac{82\!\cdots\!45}{23\!\cdots\!73}a^{6}+\frac{44\!\cdots\!88}{14\!\cdots\!53}a^{4}+\frac{19\!\cdots\!17}{14\!\cdots\!53}a^{2}+\frac{34\!\cdots\!86}{14\!\cdots\!53}$, $\frac{19\!\cdots\!65}{14\!\cdots\!53}a^{24}+\frac{23\!\cdots\!91}{14\!\cdots\!53}a^{22}+\frac{11\!\cdots\!57}{14\!\cdots\!53}a^{20}+\frac{30\!\cdots\!65}{14\!\cdots\!53}a^{18}+\frac{46\!\cdots\!34}{14\!\cdots\!53}a^{16}+\frac{44\!\cdots\!08}{14\!\cdots\!53}a^{14}+\frac{27\!\cdots\!99}{14\!\cdots\!53}a^{12}+\frac{10\!\cdots\!73}{14\!\cdots\!53}a^{10}+\frac{26\!\cdots\!27}{14\!\cdots\!53}a^{8}+\frac{65\!\cdots\!60}{23\!\cdots\!73}a^{6}+\frac{36\!\cdots\!01}{14\!\cdots\!53}a^{4}+\frac{16\!\cdots\!41}{14\!\cdots\!53}a^{2}+\frac{29\!\cdots\!57}{14\!\cdots\!53}$, $\frac{26\!\cdots\!06}{14\!\cdots\!53}a^{24}+\frac{34\!\cdots\!59}{14\!\cdots\!53}a^{22}+\frac{18\!\cdots\!49}{14\!\cdots\!53}a^{20}+\frac{54\!\cdots\!14}{14\!\cdots\!53}a^{18}+\frac{96\!\cdots\!25}{14\!\cdots\!53}a^{16}+\frac{10\!\cdots\!52}{14\!\cdots\!53}a^{14}+\frac{77\!\cdots\!42}{14\!\cdots\!53}a^{12}+\frac{35\!\cdots\!60}{14\!\cdots\!53}a^{10}+\frac{99\!\cdots\!37}{14\!\cdots\!53}a^{8}+\frac{28\!\cdots\!66}{23\!\cdots\!73}a^{6}+\frac{16\!\cdots\!58}{14\!\cdots\!53}a^{4}+\frac{82\!\cdots\!37}{14\!\cdots\!53}a^{2}+\frac{14\!\cdots\!43}{14\!\cdots\!53}$, $\frac{82\!\cdots\!24}{14\!\cdots\!53}a^{24}+\frac{98\!\cdots\!81}{14\!\cdots\!53}a^{22}+\frac{47\!\cdots\!04}{14\!\cdots\!53}a^{20}+\frac{12\!\cdots\!00}{14\!\cdots\!53}a^{18}+\frac{19\!\cdots\!37}{14\!\cdots\!53}a^{16}+\frac{17\!\cdots\!16}{14\!\cdots\!53}a^{14}+\frac{10\!\cdots\!44}{14\!\cdots\!53}a^{12}+\frac{39\!\cdots\!34}{14\!\cdots\!53}a^{10}+\frac{93\!\cdots\!50}{14\!\cdots\!53}a^{8}+\frac{22\!\cdots\!27}{23\!\cdots\!73}a^{6}+\frac{11\!\cdots\!55}{14\!\cdots\!53}a^{4}+\frac{50\!\cdots\!94}{14\!\cdots\!53}a^{2}+\frac{81\!\cdots\!20}{14\!\cdots\!53}$
| 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: | \( 1197545162478.713 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 1197545162478.713 \cdot 14586183}{4\cdot\sqrt{43781534893766029378911430108761129396314054312752152838144}}\cr\approx \mathstrut & 0.496439540020851 \end{aligned}\] (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
| \(\Q(\sqrt{-1}) \), 13.13.25542038069936263923006961.1 |
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 | $26$ | ${\href{/padicField/5.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/padicField/13.13.0.1}{13} }^{2}$ | ${\href{/padicField/17.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/padicField/29.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/37.13.0.1}{13} }^{2}$ | ${\href{/padicField/41.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/padicField/53.1.0.1}{1} }^{26}$ | $26$ |
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 $26$ | $2$ | $13$ | $26$ | |||
|
\(131\)
| Deg $26$ | $13$ | $2$ | $24$ |