Normalized defining polynomial
\( x^{26} + 53 x^{24} + 1166 x^{22} + 13939 x^{20} + 99640 x^{18} + 442444 x^{16} + 1229759 x^{14} + \cdots + 53 \)
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: |
\(-858374143948696578292647084480714777491390439882752\)
\(\medspace = -\,2^{26}\cdot 53^{25}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(90.99\) | 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: | $2\cdot 53^{25/26}\approx 90.98872147271963$ | ||
| Ramified primes: |
\(2\), \(53\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-53}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $26$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(212=2^{2}\cdot 53\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{212}(123,·)$, $\chi_{212}(1,·)$, $\chi_{212}(131,·)$, $\chi_{212}(69,·)$, $\chi_{212}(7,·)$, $\chi_{212}(201,·)$, $\chi_{212}(11,·)$, $\chi_{212}(13,·)$, $\chi_{212}(77,·)$, $\chi_{212}(143,·)$, $\chi_{212}(81,·)$, $\chi_{212}(205,·)$, $\chi_{212}(211,·)$, $\chi_{212}(89,·)$, $\chi_{212}(153,·)$, $\chi_{212}(91,·)$, $\chi_{212}(135,·)$, $\chi_{212}(199,·)$, $\chi_{212}(97,·)$, $\chi_{212}(163,·)$, $\chi_{212}(169,·)$, $\chi_{212}(43,·)$, $\chi_{212}(49,·)$, $\chi_{212}(115,·)$, $\chi_{212}(121,·)$, $\chi_{212}(59,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{4096}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23}a^{14}-\frac{11}{23}a^{12}+\frac{5}{23}a^{10}-\frac{7}{23}a^{8}+\frac{9}{23}a^{6}-\frac{11}{23}a^{4}+\frac{9}{23}a^{2}-\frac{1}{23}$, $\frac{1}{23}a^{15}-\frac{11}{23}a^{13}+\frac{5}{23}a^{11}-\frac{7}{23}a^{9}+\frac{9}{23}a^{7}-\frac{11}{23}a^{5}+\frac{9}{23}a^{3}-\frac{1}{23}a$, $\frac{1}{23}a^{16}-\frac{1}{23}a^{12}+\frac{2}{23}a^{10}+\frac{1}{23}a^{8}-\frac{4}{23}a^{6}+\frac{3}{23}a^{4}+\frac{6}{23}a^{2}-\frac{11}{23}$, $\frac{1}{23}a^{17}-\frac{1}{23}a^{13}+\frac{2}{23}a^{11}+\frac{1}{23}a^{9}-\frac{4}{23}a^{7}+\frac{3}{23}a^{5}+\frac{6}{23}a^{3}-\frac{11}{23}a$, $\frac{1}{23}a^{18}-\frac{9}{23}a^{12}+\frac{6}{23}a^{10}-\frac{11}{23}a^{8}-\frac{11}{23}a^{6}-\frac{5}{23}a^{4}-\frac{2}{23}a^{2}-\frac{1}{23}$, $\frac{1}{23}a^{19}-\frac{9}{23}a^{13}+\frac{6}{23}a^{11}-\frac{11}{23}a^{9}-\frac{11}{23}a^{7}-\frac{5}{23}a^{5}-\frac{2}{23}a^{3}-\frac{1}{23}a$, $\frac{1}{23}a^{20}-\frac{1}{23}a^{12}+\frac{11}{23}a^{10}-\frac{5}{23}a^{8}+\frac{7}{23}a^{6}-\frac{9}{23}a^{4}+\frac{11}{23}a^{2}-\frac{9}{23}$, $\frac{1}{23}a^{21}-\frac{1}{23}a^{13}+\frac{11}{23}a^{11}-\frac{5}{23}a^{9}+\frac{7}{23}a^{7}-\frac{9}{23}a^{5}+\frac{11}{23}a^{3}-\frac{9}{23}a$, $\frac{1}{23}a^{22}-\frac{1}{23}$, $\frac{1}{23}a^{23}-\frac{1}{23}a$, $\frac{1}{95\!\cdots\!23}a^{24}-\frac{59\!\cdots\!39}{95\!\cdots\!23}a^{22}+\frac{34\!\cdots\!90}{95\!\cdots\!23}a^{20}+\frac{96\!\cdots\!29}{95\!\cdots\!23}a^{18}-\frac{38\!\cdots\!81}{95\!\cdots\!23}a^{16}+\frac{60\!\cdots\!93}{95\!\cdots\!23}a^{14}-\frac{31\!\cdots\!60}{95\!\cdots\!23}a^{12}-\frac{34\!\cdots\!90}{95\!\cdots\!23}a^{10}-\frac{12\!\cdots\!89}{95\!\cdots\!23}a^{8}-\frac{13\!\cdots\!95}{95\!\cdots\!23}a^{6}-\frac{48\!\cdots\!78}{95\!\cdots\!23}a^{4}-\frac{22\!\cdots\!08}{95\!\cdots\!23}a^{2}+\frac{11\!\cdots\!85}{95\!\cdots\!23}$, $\frac{1}{95\!\cdots\!23}a^{25}-\frac{59\!\cdots\!39}{95\!\cdots\!23}a^{23}+\frac{34\!\cdots\!90}{95\!\cdots\!23}a^{21}+\frac{96\!\cdots\!29}{95\!\cdots\!23}a^{19}-\frac{38\!\cdots\!81}{95\!\cdots\!23}a^{17}+\frac{60\!\cdots\!93}{95\!\cdots\!23}a^{15}-\frac{31\!\cdots\!60}{95\!\cdots\!23}a^{13}-\frac{34\!\cdots\!90}{95\!\cdots\!23}a^{11}-\frac{12\!\cdots\!89}{95\!\cdots\!23}a^{9}-\frac{13\!\cdots\!95}{95\!\cdots\!23}a^{7}-\frac{48\!\cdots\!78}{95\!\cdots\!23}a^{5}-\frac{22\!\cdots\!08}{95\!\cdots\!23}a^{3}+\frac{11\!\cdots\!85}{95\!\cdots\!23}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $23$ |
Class group and class number
$C_{195474}$, which has order $195474$ (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: |
$\frac{49\!\cdots\!06}{95\!\cdots\!23}a^{24}+\frac{26\!\cdots\!80}{95\!\cdots\!23}a^{22}+\frac{56\!\cdots\!68}{95\!\cdots\!23}a^{20}+\frac{67\!\cdots\!52}{95\!\cdots\!23}a^{18}+\frac{47\!\cdots\!59}{95\!\cdots\!23}a^{16}+\frac{20\!\cdots\!94}{95\!\cdots\!23}a^{14}+\frac{54\!\cdots\!62}{95\!\cdots\!23}a^{12}+\frac{88\!\cdots\!30}{95\!\cdots\!23}a^{10}+\frac{79\!\cdots\!60}{95\!\cdots\!23}a^{8}+\frac{36\!\cdots\!78}{95\!\cdots\!23}a^{6}+\frac{75\!\cdots\!50}{95\!\cdots\!23}a^{4}+\frac{54\!\cdots\!44}{95\!\cdots\!23}a^{2}+\frac{10\!\cdots\!60}{95\!\cdots\!23}$, $\frac{79\!\cdots\!39}{95\!\cdots\!23}a^{24}+\frac{41\!\cdots\!37}{95\!\cdots\!23}a^{22}+\frac{90\!\cdots\!35}{95\!\cdots\!23}a^{20}+\frac{10\!\cdots\!79}{95\!\cdots\!23}a^{18}+\frac{74\!\cdots\!05}{95\!\cdots\!23}a^{16}+\frac{31\!\cdots\!73}{95\!\cdots\!23}a^{14}+\frac{83\!\cdots\!82}{95\!\cdots\!23}a^{12}+\frac{13\!\cdots\!20}{95\!\cdots\!23}a^{10}+\frac{11\!\cdots\!25}{95\!\cdots\!23}a^{8}+\frac{43\!\cdots\!05}{95\!\cdots\!23}a^{6}+\frac{54\!\cdots\!31}{95\!\cdots\!23}a^{4}-\frac{39\!\cdots\!39}{95\!\cdots\!23}a^{2}-\frac{42\!\cdots\!58}{95\!\cdots\!23}$, $\frac{58\!\cdots\!01}{95\!\cdots\!23}a^{24}+\frac{30\!\cdots\!13}{95\!\cdots\!23}a^{22}+\frac{66\!\cdots\!53}{95\!\cdots\!23}a^{20}+\frac{78\!\cdots\!83}{95\!\cdots\!23}a^{18}+\frac{55\!\cdots\!38}{95\!\cdots\!23}a^{16}+\frac{23\!\cdots\!31}{95\!\cdots\!23}a^{14}+\frac{63\!\cdots\!62}{95\!\cdots\!23}a^{12}+\frac{10\!\cdots\!60}{95\!\cdots\!23}a^{10}+\frac{92\!\cdots\!03}{95\!\cdots\!23}a^{8}+\frac{42\!\cdots\!62}{95\!\cdots\!23}a^{6}+\frac{88\!\cdots\!39}{95\!\cdots\!23}a^{4}+\frac{71\!\cdots\!24}{95\!\cdots\!23}a^{2}+\frac{16\!\cdots\!94}{95\!\cdots\!23}$, $\frac{10\!\cdots\!72}{95\!\cdots\!23}a^{24}+\frac{56\!\cdots\!36}{95\!\cdots\!23}a^{22}+\frac{12\!\cdots\!16}{95\!\cdots\!23}a^{20}+\frac{14\!\cdots\!36}{95\!\cdots\!23}a^{18}+\frac{98\!\cdots\!81}{95\!\cdots\!23}a^{16}+\frac{41\!\cdots\!64}{95\!\cdots\!23}a^{14}+\frac{10\!\cdots\!99}{95\!\cdots\!23}a^{12}+\frac{15\!\cdots\!52}{95\!\cdots\!23}a^{10}+\frac{11\!\cdots\!08}{95\!\cdots\!23}a^{8}+\frac{28\!\cdots\!75}{95\!\cdots\!23}a^{6}-\frac{46\!\cdots\!29}{95\!\cdots\!23}a^{4}-\frac{22\!\cdots\!77}{95\!\cdots\!23}a^{2}-\frac{10\!\cdots\!46}{95\!\cdots\!23}$, $\frac{45\!\cdots\!84}{95\!\cdots\!23}a^{24}+\frac{24\!\cdots\!51}{95\!\cdots\!23}a^{22}+\frac{53\!\cdots\!81}{95\!\cdots\!23}a^{20}+\frac{63\!\cdots\!21}{95\!\cdots\!23}a^{18}+\frac{45\!\cdots\!89}{95\!\cdots\!23}a^{16}+\frac{20\!\cdots\!13}{95\!\cdots\!23}a^{14}+\frac{57\!\cdots\!57}{95\!\cdots\!23}a^{12}+\frac{98\!\cdots\!52}{95\!\cdots\!23}a^{10}+\frac{10\!\cdots\!34}{95\!\cdots\!23}a^{8}+\frac{57\!\cdots\!73}{95\!\cdots\!23}a^{6}+\frac{17\!\cdots\!54}{95\!\cdots\!23}a^{4}+\frac{21\!\cdots\!97}{95\!\cdots\!23}a^{2}+\frac{43\!\cdots\!50}{95\!\cdots\!23}$, $\frac{40\!\cdots\!04}{95\!\cdots\!23}a^{24}+\frac{21\!\cdots\!18}{95\!\cdots\!23}a^{22}+\frac{46\!\cdots\!75}{95\!\cdots\!23}a^{20}+\frac{54\!\cdots\!29}{95\!\cdots\!23}a^{18}+\frac{38\!\cdots\!47}{95\!\cdots\!23}a^{16}+\frac{16\!\cdots\!81}{95\!\cdots\!23}a^{14}+\frac{43\!\cdots\!84}{95\!\cdots\!23}a^{12}+\frac{67\!\cdots\!35}{95\!\cdots\!23}a^{10}+\frac{58\!\cdots\!75}{95\!\cdots\!23}a^{8}+\frac{23\!\cdots\!84}{95\!\cdots\!23}a^{6}+\frac{37\!\cdots\!51}{95\!\cdots\!23}a^{4}+\frac{83\!\cdots\!86}{95\!\cdots\!23}a^{2}-\frac{17\!\cdots\!94}{95\!\cdots\!23}$, $\frac{27\!\cdots\!85}{95\!\cdots\!23}a^{24}+\frac{14\!\cdots\!76}{95\!\cdots\!23}a^{22}+\frac{31\!\cdots\!16}{95\!\cdots\!23}a^{20}+\frac{37\!\cdots\!56}{95\!\cdots\!23}a^{18}+\frac{26\!\cdots\!96}{95\!\cdots\!23}a^{16}+\frac{11\!\cdots\!13}{95\!\cdots\!23}a^{14}+\frac{29\!\cdots\!29}{95\!\cdots\!23}a^{12}+\frac{46\!\cdots\!31}{95\!\cdots\!23}a^{10}+\frac{40\!\cdots\!52}{95\!\cdots\!23}a^{8}+\frac{16\!\cdots\!27}{95\!\cdots\!23}a^{6}+\frac{26\!\cdots\!47}{95\!\cdots\!23}a^{4}+\frac{41\!\cdots\!10}{95\!\cdots\!23}a^{2}-\frac{24\!\cdots\!04}{95\!\cdots\!23}$, $\frac{30\!\cdots\!72}{41\!\cdots\!01}a^{24}+\frac{16\!\cdots\!50}{41\!\cdots\!01}a^{22}+\frac{35\!\cdots\!12}{41\!\cdots\!01}a^{20}+\frac{41\!\cdots\!56}{41\!\cdots\!01}a^{18}+\frac{29\!\cdots\!75}{41\!\cdots\!01}a^{16}+\frac{12\!\cdots\!25}{41\!\cdots\!01}a^{14}+\frac{33\!\cdots\!43}{41\!\cdots\!01}a^{12}+\frac{52\!\cdots\!38}{41\!\cdots\!01}a^{10}+\frac{46\!\cdots\!38}{41\!\cdots\!01}a^{8}+\frac{20\!\cdots\!95}{41\!\cdots\!01}a^{6}+\frac{37\!\cdots\!52}{41\!\cdots\!01}a^{4}+\frac{19\!\cdots\!63}{41\!\cdots\!01}a^{2}+\frac{30\!\cdots\!05}{41\!\cdots\!01}$, $\frac{32\!\cdots\!16}{95\!\cdots\!23}a^{24}+\frac{16\!\cdots\!08}{95\!\cdots\!23}a^{22}+\frac{36\!\cdots\!48}{95\!\cdots\!23}a^{20}+\frac{42\!\cdots\!08}{95\!\cdots\!23}a^{18}+\frac{29\!\cdots\!43}{95\!\cdots\!23}a^{16}+\frac{12\!\cdots\!92}{95\!\cdots\!23}a^{14}+\frac{31\!\cdots\!97}{95\!\cdots\!23}a^{12}+\frac{45\!\cdots\!56}{95\!\cdots\!23}a^{10}+\frac{34\!\cdots\!24}{95\!\cdots\!23}a^{8}+\frac{84\!\cdots\!25}{95\!\cdots\!23}a^{6}-\frac{14\!\cdots\!87}{95\!\cdots\!23}a^{4}-\frac{68\!\cdots\!08}{95\!\cdots\!23}a^{2}-\frac{29\!\cdots\!69}{95\!\cdots\!23}$, $\frac{40\!\cdots\!72}{95\!\cdots\!23}a^{24}+\frac{21\!\cdots\!97}{95\!\cdots\!23}a^{22}+\frac{47\!\cdots\!31}{95\!\cdots\!23}a^{20}+\frac{55\!\cdots\!59}{95\!\cdots\!23}a^{18}+\frac{39\!\cdots\!83}{95\!\cdots\!23}a^{16}+\frac{17\!\cdots\!35}{95\!\cdots\!23}a^{14}+\frac{46\!\cdots\!15}{95\!\cdots\!23}a^{12}+\frac{75\!\cdots\!08}{95\!\cdots\!23}a^{10}+\frac{71\!\cdots\!34}{95\!\cdots\!23}a^{8}+\frac{34\!\cdots\!31}{95\!\cdots\!23}a^{6}+\frac{83\!\cdots\!31}{95\!\cdots\!23}a^{4}+\frac{81\!\cdots\!59}{95\!\cdots\!23}a^{2}+\frac{18\!\cdots\!83}{95\!\cdots\!23}$, $\frac{16\!\cdots\!76}{95\!\cdots\!23}a^{24}+\frac{87\!\cdots\!26}{95\!\cdots\!23}a^{22}+\frac{18\!\cdots\!25}{95\!\cdots\!23}a^{20}+\frac{22\!\cdots\!71}{95\!\cdots\!23}a^{18}+\frac{15\!\cdots\!01}{95\!\cdots\!23}a^{16}+\frac{64\!\cdots\!11}{95\!\cdots\!23}a^{14}+\frac{16\!\cdots\!84}{95\!\cdots\!23}a^{12}+\frac{24\!\cdots\!29}{95\!\cdots\!23}a^{10}+\frac{18\!\cdots\!40}{95\!\cdots\!23}a^{8}+\frac{52\!\cdots\!40}{95\!\cdots\!23}a^{6}-\frac{36\!\cdots\!31}{95\!\cdots\!23}a^{4}-\frac{29\!\cdots\!06}{95\!\cdots\!23}a^{2}-\frac{12\!\cdots\!51}{95\!\cdots\!23}$, $\frac{24\!\cdots\!28}{95\!\cdots\!23}a^{24}+\frac{12\!\cdots\!92}{95\!\cdots\!23}a^{22}+\frac{27\!\cdots\!50}{95\!\cdots\!23}a^{20}+\frac{32\!\cdots\!58}{95\!\cdots\!23}a^{18}+\frac{23\!\cdots\!46}{95\!\cdots\!23}a^{16}+\frac{10\!\cdots\!70}{95\!\cdots\!23}a^{14}+\frac{26\!\cdots\!00}{95\!\cdots\!23}a^{12}+\frac{43\!\cdots\!06}{95\!\cdots\!23}a^{10}+\frac{39\!\cdots\!35}{95\!\cdots\!23}a^{8}+\frac{18\!\cdots\!44}{95\!\cdots\!23}a^{6}+\frac{41\!\cdots\!82}{95\!\cdots\!23}a^{4}+\frac{38\!\cdots\!92}{95\!\cdots\!23}a^{2}+\frac{11\!\cdots\!80}{95\!\cdots\!23}$
| 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: | \( 5382739421.971964 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 5382739421.971964 \cdot 195474}{2\cdot\sqrt{858374143948696578292647084480714777491390439882752}}\cr\approx \mathstrut & 0.427132222587746 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 26 |
| The 26 conjugacy class representatives for $C_{26}$ |
| Character table for $C_{26}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-53}) \), 13.13.491258904256726154641.1 |
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 | ${\href{/padicField/3.13.0.1}{13} }^{2}$ | $26$ | $26$ | $26$ | ${\href{/padicField/13.13.0.1}{13} }^{2}$ | ${\href{/padicField/17.13.0.1}{13} }^{2}$ | ${\href{/padicField/19.13.0.1}{13} }^{2}$ | ${\href{/padicField/23.1.0.1}{1} }^{26}$ | ${\href{/padicField/29.13.0.1}{13} }^{2}$ | ${\href{/padicField/31.13.0.1}{13} }^{2}$ | ${\href{/padicField/37.13.0.1}{13} }^{2}$ | $26$ | $26$ | $26$ | R | $26$ |
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 $26$ | $2$ | $13$ | $26$ | |||
|
\(53\)
| Deg $26$ | $26$ | $1$ | $25$ |