Normalized defining polynomial
\( x^{28} - x^{27} + 4 x^{26} - 20 x^{25} + 110 x^{24} + 544 x^{23} - 1294 x^{22} + 5613 x^{21} + \cdots + 493619587 \)
Invariants
| Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(89245865645132062133059821789304344522517366724620268695771613\)
\(\medspace = 197^{27}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(163.13\) | 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: | $197^{27/28}\approx 163.12517996059472$ | ||
| Ramified primes: |
\(197\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{197}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $28$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(197\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{197}(128,·)$, $\chi_{197}(1,·)$, $\chi_{197}(68,·)$, $\chi_{197}(69,·)$, $\chi_{197}(6,·)$, $\chi_{197}(129,·)$, $\chi_{197}(77,·)$, $\chi_{197}(14,·)$, $\chi_{197}(19,·)$, $\chi_{197}(20,·)$, $\chi_{197}(87,·)$, $\chi_{197}(196,·)$, $\chi_{197}(93,·)$, $\chi_{197}(161,·)$, $\chi_{197}(164,·)$, $\chi_{197}(33,·)$, $\chi_{197}(113,·)$, $\chi_{197}(104,·)$, $\chi_{197}(114,·)$, $\chi_{197}(110,·)$, $\chi_{197}(177,·)$, $\chi_{197}(178,·)$, $\chi_{197}(83,·)$, $\chi_{197}(183,·)$, $\chi_{197}(120,·)$, $\chi_{197}(84,·)$, $\chi_{197}(36,·)$, $\chi_{197}(191,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{8192}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{464323}a^{26}-\frac{41024}{464323}a^{25}+\frac{54910}{464323}a^{24}+\frac{222477}{464323}a^{23}+\frac{69853}{464323}a^{22}-\frac{119371}{464323}a^{21}+\frac{79500}{464323}a^{20}-\frac{167684}{464323}a^{19}-\frac{54301}{464323}a^{18}+\frac{60272}{464323}a^{17}-\frac{85412}{464323}a^{16}+\frac{46632}{464323}a^{15}-\frac{183925}{464323}a^{14}+\frac{188463}{464323}a^{13}+\frac{201646}{464323}a^{12}+\frac{208160}{464323}a^{11}-\frac{37055}{464323}a^{10}-\frac{33280}{464323}a^{9}+\frac{48223}{464323}a^{8}+\frac{138404}{464323}a^{7}+\frac{153793}{464323}a^{6}-\frac{67586}{464323}a^{5}+\frac{230516}{464323}a^{4}+\frac{16036}{464323}a^{3}-\frac{186530}{464323}a^{2}+\frac{221005}{464323}a+\frac{54491}{464323}$, $\frac{1}{15\!\cdots\!33}a^{27}+\frac{95\!\cdots\!47}{15\!\cdots\!33}a^{26}-\frac{44\!\cdots\!47}{15\!\cdots\!33}a^{25}+\frac{17\!\cdots\!56}{15\!\cdots\!33}a^{24}-\frac{69\!\cdots\!07}{15\!\cdots\!33}a^{23}+\frac{31\!\cdots\!35}{15\!\cdots\!33}a^{22}+\frac{49\!\cdots\!78}{15\!\cdots\!33}a^{21}+\frac{17\!\cdots\!19}{15\!\cdots\!33}a^{20}+\frac{71\!\cdots\!33}{15\!\cdots\!33}a^{19}-\frac{26\!\cdots\!35}{15\!\cdots\!33}a^{18}+\frac{44\!\cdots\!19}{15\!\cdots\!33}a^{17}+\frac{74\!\cdots\!07}{15\!\cdots\!33}a^{16}-\frac{65\!\cdots\!84}{15\!\cdots\!33}a^{15}+\frac{25\!\cdots\!96}{15\!\cdots\!33}a^{14}+\frac{19\!\cdots\!23}{80\!\cdots\!07}a^{13}+\frac{72\!\cdots\!85}{15\!\cdots\!33}a^{12}-\frac{38\!\cdots\!16}{80\!\cdots\!07}a^{11}+\frac{18\!\cdots\!59}{15\!\cdots\!33}a^{10}-\frac{63\!\cdots\!48}{15\!\cdots\!33}a^{9}+\frac{47\!\cdots\!37}{15\!\cdots\!33}a^{8}-\frac{97\!\cdots\!52}{15\!\cdots\!33}a^{7}+\frac{71\!\cdots\!40}{15\!\cdots\!33}a^{6}-\frac{69\!\cdots\!50}{15\!\cdots\!33}a^{5}-\frac{47\!\cdots\!48}{15\!\cdots\!33}a^{4}-\frac{17\!\cdots\!28}{15\!\cdots\!33}a^{3}+\frac{64\!\cdots\!72}{15\!\cdots\!33}a^{2}-\frac{30\!\cdots\!90}{15\!\cdots\!33}a+\frac{76\!\cdots\!78}{15\!\cdots\!33}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $13$ | 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 28 |
| The 28 conjugacy class representatives for $C_{28}$ |
| Character table for $C_{28}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{197}) \), 4.0.7645373.1, 7.7.58451728309129.1, 14.14.673071094837873811440793512277.1 |
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 | $28$ | $28$ | $28$ | ${\href{/padicField/7.14.0.1}{14} }^{2}$ | $28$ | $28$ | $28$ | ${\href{/padicField/19.2.0.1}{2} }^{14}$ | ${\href{/padicField/23.7.0.1}{7} }^{4}$ | ${\href{/padicField/29.7.0.1}{7} }^{4}$ | $28$ | ${\href{/padicField/37.7.0.1}{7} }^{4}$ | ${\href{/padicField/41.14.0.1}{14} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }^{4}$ | ${\href{/padicField/59.7.0.1}{7} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(197\)
| Deg $28$ | $28$ | $1$ | $27$ |