Normalized defining polynomial
\( x^{26} - x^{25} + 134 x^{24} - 135 x^{23} + 7272 x^{22} - 7408 x^{21} + 207314 x^{20} + \cdots + 60572712821 \)
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: |
\(-441572122027487213320469174167608817426459689061923004183\)
\(\medspace = -\,11^{13}\cdot 53^{25}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(150.89\) | 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: | $11^{1/2}53^{25/26}\approx 150.8877246395823$ | ||
| Ramified primes: |
\(11\), \(53\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-583}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $26$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(583=11\cdot 53\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{583}(1,·)$, $\chi_{583}(386,·)$, $\chi_{583}(131,·)$, $\chi_{583}(452,·)$, $\chi_{583}(197,·)$, $\chi_{583}(582,·)$, $\chi_{583}(329,·)$, $\chi_{583}(331,·)$, $\chi_{583}(461,·)$, $\chi_{583}(274,·)$, $\chi_{583}(483,·)$, $\chi_{583}(342,·)$, $\chi_{583}(89,·)$, $\chi_{583}(155,·)$, $\chi_{583}(540,·)$, $\chi_{583}(219,·)$, $\chi_{583}(100,·)$, $\chi_{583}(43,·)$, $\chi_{583}(428,·)$, $\chi_{583}(494,·)$, $\chi_{583}(241,·)$, $\chi_{583}(309,·)$, $\chi_{583}(364,·)$, $\chi_{583}(122,·)$, $\chi_{583}(252,·)$, $\chi_{583}(254,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{4096}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{83}a^{21}+\frac{35}{83}a^{20}-\frac{25}{83}a^{19}-\frac{23}{83}a^{18}-\frac{16}{83}a^{17}+\frac{34}{83}a^{16}+\frac{25}{83}a^{15}-\frac{3}{83}a^{14}+\frac{18}{83}a^{13}-\frac{40}{83}a^{12}-\frac{41}{83}a^{11}+\frac{1}{83}a^{10}-\frac{9}{83}a^{9}-\frac{27}{83}a^{8}-\frac{37}{83}a^{7}-\frac{40}{83}a^{6}-\frac{36}{83}a^{5}+\frac{23}{83}a^{4}+\frac{30}{83}a^{3}-\frac{8}{83}a^{2}-\frac{16}{83}a+\frac{25}{83}$, $\frac{1}{83}a^{22}-\frac{5}{83}a^{20}+\frac{22}{83}a^{19}-\frac{41}{83}a^{18}+\frac{13}{83}a^{17}-\frac{3}{83}a^{16}+\frac{35}{83}a^{15}+\frac{40}{83}a^{14}-\frac{6}{83}a^{13}+\frac{31}{83}a^{12}+\frac{25}{83}a^{11}+\frac{39}{83}a^{10}+\frac{39}{83}a^{9}-\frac{5}{83}a^{8}+\frac{10}{83}a^{7}+\frac{36}{83}a^{6}+\frac{38}{83}a^{5}-\frac{28}{83}a^{4}+\frac{21}{83}a^{3}+\frac{15}{83}a^{2}+\frac{4}{83}a+\frac{38}{83}$, $\frac{1}{83}a^{23}+\frac{31}{83}a^{20}-\frac{19}{83}a^{18}+\frac{39}{83}a^{16}-\frac{1}{83}a^{15}-\frac{21}{83}a^{14}+\frac{38}{83}a^{13}-\frac{9}{83}a^{12}-\frac{39}{83}a^{10}+\frac{33}{83}a^{9}+\frac{41}{83}a^{8}+\frac{17}{83}a^{7}+\frac{4}{83}a^{6}+\frac{41}{83}a^{5}-\frac{30}{83}a^{4}-\frac{1}{83}a^{3}-\frac{36}{83}a^{2}+\frac{41}{83}a-\frac{41}{83}$, $\frac{1}{83}a^{24}-\frac{6}{83}a^{20}+\frac{9}{83}a^{19}-\frac{34}{83}a^{18}+\frac{37}{83}a^{17}+\frac{24}{83}a^{16}+\frac{34}{83}a^{15}-\frac{35}{83}a^{14}+\frac{14}{83}a^{13}-\frac{5}{83}a^{12}-\frac{13}{83}a^{11}+\frac{2}{83}a^{10}-\frac{12}{83}a^{9}+\frac{24}{83}a^{8}-\frac{11}{83}a^{7}+\frac{36}{83}a^{6}+\frac{7}{83}a^{5}+\frac{33}{83}a^{4}+\frac{30}{83}a^{3}+\frac{40}{83}a^{2}+\frac{40}{83}a-\frac{28}{83}$, $\frac{1}{70\!\cdots\!57}a^{25}+\frac{34\!\cdots\!94}{70\!\cdots\!57}a^{24}+\frac{24\!\cdots\!79}{70\!\cdots\!57}a^{23}-\frac{26\!\cdots\!38}{70\!\cdots\!57}a^{22}+\frac{26\!\cdots\!59}{70\!\cdots\!57}a^{21}-\frac{13\!\cdots\!82}{70\!\cdots\!57}a^{20}+\frac{34\!\cdots\!62}{70\!\cdots\!57}a^{19}+\frac{14\!\cdots\!78}{70\!\cdots\!57}a^{18}-\frac{65\!\cdots\!18}{70\!\cdots\!57}a^{17}+\frac{26\!\cdots\!76}{70\!\cdots\!57}a^{16}+\frac{10\!\cdots\!06}{70\!\cdots\!57}a^{15}+\frac{51\!\cdots\!40}{70\!\cdots\!57}a^{14}+\frac{18\!\cdots\!69}{70\!\cdots\!57}a^{13}-\frac{23\!\cdots\!02}{70\!\cdots\!57}a^{12}+\frac{66\!\cdots\!42}{70\!\cdots\!57}a^{11}+\frac{25\!\cdots\!94}{70\!\cdots\!57}a^{10}-\frac{86\!\cdots\!38}{70\!\cdots\!57}a^{9}-\frac{11\!\cdots\!72}{70\!\cdots\!57}a^{8}-\frac{69\!\cdots\!87}{70\!\cdots\!57}a^{7}-\frac{66\!\cdots\!01}{70\!\cdots\!57}a^{6}+\frac{16\!\cdots\!30}{70\!\cdots\!57}a^{5}+\frac{33\!\cdots\!07}{70\!\cdots\!57}a^{4}-\frac{53\!\cdots\!82}{70\!\cdots\!57}a^{3}+\frac{65\!\cdots\!73}{70\!\cdots\!57}a^{2}+\frac{18\!\cdots\!90}{70\!\cdots\!57}a+\frac{88\!\cdots\!22}{70\!\cdots\!57}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
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: | not computed | 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: | not computed | 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 $
Galois group
| A cyclic group of order 26 |
| The 26 conjugacy class representatives for $C_{26}$ |
| Character table for $C_{26}$ |
Intermediate fields
| \(\Q(\sqrt{-583}) \), 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 | ${\href{/padicField/2.13.0.1}{13} }^{2}$ | $26$ | $26$ | $26$ | R | $26$ | $26$ | ${\href{/padicField/19.13.0.1}{13} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{13}$ | $26$ | $26$ | ${\href{/padicField/37.13.0.1}{13} }^{2}$ | ${\href{/padicField/41.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/47.13.0.1}{13} }^{2}$ | R | ${\href{/padicField/59.13.0.1}{13} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| Deg $26$ | $2$ | $13$ | $13$ | |||
|
\(53\)
| Deg $26$ | $26$ | $1$ | $25$ |