Normalized defining polynomial
\( x^{28} + 54 x^{26} + 1300 x^{24} + 18400 x^{22} + 170016 x^{20} + 1076768 x^{18} + 4775232 x^{16} + \cdots + 16384 \)
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: | \(463028542684026225381227850734902390950731116969984\) \(\medspace = 2^{42}\cdot 29^{26}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(64.49\) | 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: | $2^{3/2}29^{13/14}\approx 64.48905181635536$ | ||
Ramified primes: | \(2\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $28$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(232=2^{3}\cdot 29\) | ||
Dirichlet character group: | $\lbrace$$\chi_{232}(123,·)$, $\chi_{232}(1,·)$, $\chi_{232}(67,·)$, $\chi_{232}(51,·)$, $\chi_{232}(129,·)$, $\chi_{232}(9,·)$, $\chi_{232}(139,·)$, $\chi_{232}(161,·)$, $\chi_{232}(115,·)$, $\chi_{232}(209,·)$, $\chi_{232}(83,·)$, $\chi_{232}(121,·)$, $\chi_{232}(25,·)$, $\chi_{232}(91,·)$, $\chi_{232}(81,·)$, $\chi_{232}(107,·)$, $\chi_{232}(33,·)$, $\chi_{232}(35,·)$, $\chi_{232}(65,·)$, $\chi_{232}(227,·)$, $\chi_{232}(169,·)$, $\chi_{232}(225,·)$, $\chi_{232}(49,·)$, $\chi_{232}(179,·)$, $\chi_{232}(187,·)$, $\chi_{232}(219,·)$, $\chi_{232}(57,·)$, $\chi_{232}(59,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{8192}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}\times C_{172}\times C_{172}$, which has order $118336$ (assuming GRH)
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: | $\frac{1}{512}a^{18}+\frac{9}{128}a^{16}+\frac{135}{128}a^{14}+\frac{273}{32}a^{12}+\frac{1287}{32}a^{10}+\frac{891}{8}a^{8}+\frac{693}{4}a^{6}+135a^{4}+\frac{81}{2}a^{2}+2$, $\frac{1}{4}a^{4}+2a^{2}+2$, $\frac{1}{128}a^{14}+\frac{7}{32}a^{12}+\frac{77}{32}a^{10}+\frac{105}{8}a^{8}+\frac{147}{4}a^{6}+49a^{4}+\frac{49}{2}a^{2}+1$, $\frac{1}{1024}a^{20}+\frac{5}{128}a^{18}+\frac{85}{128}a^{16}+\frac{25}{4}a^{14}+\frac{2275}{64}a^{12}+\frac{1001}{8}a^{10}+\frac{2145}{8}a^{8}+330a^{6}+\frac{825}{4}a^{4}+50a^{2}+2$, $\frac{1}{32}a^{10}+\frac{5}{8}a^{8}+\frac{35}{8}a^{6}+\frac{25}{2}a^{4}+\frac{25}{2}a^{2}+2$, $\frac{1}{256}a^{16}+\frac{1}{8}a^{14}+\frac{13}{8}a^{12}+11a^{10}+\frac{165}{4}a^{8}+84a^{6}+84a^{4}+32a^{2}+3$, $\frac{1}{16}a^{8}+a^{6}+5a^{4}+8a^{2}+2$, $\frac{1}{2048}a^{22}+\frac{11}{512}a^{20}+\frac{209}{512}a^{18}+\frac{561}{128}a^{16}+\frac{935}{32}a^{14}+\frac{1001}{8}a^{12}+\frac{11011}{32}a^{10}+\frac{4719}{8}a^{8}+\frac{4719}{8}a^{6}+\frac{605}{2}a^{4}+\frac{121}{2}a^{2}+2$, $\frac{1}{2048}a^{22}+\frac{11}{512}a^{20}+\frac{209}{512}a^{18}+\frac{561}{128}a^{16}+\frac{935}{32}a^{14}+\frac{1001}{8}a^{12}+\frac{11011}{32}a^{10}+\frac{9437}{16}a^{8}+\frac{4711}{8}a^{6}+\frac{595}{2}a^{4}+\frac{105}{2}a^{2}$, $\frac{1}{2}a^{2}+2$, $\frac{1}{16}a^{8}+a^{6}+5a^{4}+8a^{2}+3$, $\frac{1}{64}a^{12}+\frac{3}{8}a^{10}+\frac{27}{8}a^{8}+14a^{6}+\frac{105}{4}a^{4}+18a^{2}+3$, $\frac{1}{4096}a^{24}+\frac{3}{256}a^{22}+\frac{63}{256}a^{20}+\frac{95}{32}a^{18}+\frac{2907}{128}a^{16}+\frac{459}{4}a^{14}+\frac{1547}{4}a^{12}+858a^{10}+\frac{9653}{8}a^{8}+1002a^{6}+434a^{4}+80a^{2}+4$ (assuming GRH) | 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: | \( 487075979.1876791 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 487075979.1876791 \cdot 118336}{2\cdot\sqrt{463028542684026225381227850734902390950731116969984}}\cr\approx \mathstrut & 0.200169487455587 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_{14}$ (as 28T2):
An abelian group of order 28 |
The 28 conjugacy class representatives for $C_2\times C_{14}$ |
Character table for $C_2\times C_{14}$ |
Intermediate fields
\(\Q(\sqrt{29}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-58}) \), \(\Q(\sqrt{-2}, \sqrt{29})\), 7.7.594823321.1, \(\Q(\zeta_{29})^+\), 14.0.742003380228915810271232.1, 14.0.21518098026638558497865728.1 |
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 | ${\href{/padicField/3.14.0.1}{14} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }^{2}$ | ${\href{/padicField/7.14.0.1}{14} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{14}$ | ${\href{/padicField/19.14.0.1}{14} }^{2}$ | ${\href{/padicField/23.14.0.1}{14} }^{2}$ | R | ${\href{/padicField/31.14.0.1}{14} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{14}$ | ${\href{/padicField/43.14.0.1}{14} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }^{2}$ | ${\href{/padicField/59.1.0.1}{1} }^{28}$ |
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 $28$ | $2$ | $14$ | $42$ | |||
\(29\) | Deg $28$ | $14$ | $2$ | $26$ |