Normalized defining polynomial
\( x^{28} - 174 x^{26} + 13572 x^{24} - 626400 x^{22} + 19017504 x^{20} - 399367584 x^{18} + \cdots + 2272560758784 \)
Invariants
Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[28, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(64224883807413352704377353768048295264353596334281414672384\) \(\medspace = 2^{42}\cdot 3^{14}\cdot 29^{27}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(125.97\) | 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: | $2^{3/2}3^{1/2}29^{27/28}\approx 125.9723174208094$ | ||
Ramified primes: | \(2\), \(3\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{29}) \) | ||
$\card{ \Gal(K/\Q) }$: | $28$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(696=2^{3}\cdot 3\cdot 29\) | ||
Dirichlet character group: | $\lbrace$$\chi_{696}(1,·)$, $\chi_{696}(361,·)$, $\chi_{696}(389,·)$, $\chi_{696}(673,·)$, $\chi_{696}(265,·)$, $\chi_{696}(241,·)$, $\chi_{696}(269,·)$, $\chi_{696}(77,·)$, $\chi_{696}(461,·)$, $\chi_{696}(529,·)$, $\chi_{696}(533,·)$, $\chi_{696}(121,·)$, $\chi_{696}(25,·)$, $\chi_{696}(101,·)$, $\chi_{696}(221,·)$, $\chi_{696}(485,·)$, $\chi_{696}(653,·)$, $\chi_{696}(289,·)$, $\chi_{696}(677,·)$, $\chi_{696}(49,·)$, $\chi_{696}(169,·)$, $\chi_{696}(365,·)$, $\chi_{696}(625,·)$, $\chi_{696}(437,·)$, $\chi_{696}(457,·)$, $\chi_{696}(313,·)$, $\chi_{696}(293,·)$, $\chi_{696}(317,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{6}a^{2}$, $\frac{1}{6}a^{3}$, $\frac{1}{36}a^{4}$, $\frac{1}{36}a^{5}$, $\frac{1}{216}a^{6}$, $\frac{1}{216}a^{7}$, $\frac{1}{1296}a^{8}$, $\frac{1}{1296}a^{9}$, $\frac{1}{7776}a^{10}$, $\frac{1}{7776}a^{11}$, $\frac{1}{46656}a^{12}$, $\frac{1}{46656}a^{13}$, $\frac{1}{279936}a^{14}$, $\frac{1}{279936}a^{15}$, $\frac{1}{1679616}a^{16}$, $\frac{1}{1679616}a^{17}$, $\frac{1}{10077696}a^{18}$, $\frac{1}{10077696}a^{19}$, $\frac{1}{60466176}a^{20}$, $\frac{1}{60466176}a^{21}$, $\frac{1}{362797056}a^{22}$, $\frac{1}{362797056}a^{23}$, $\frac{1}{2176782336}a^{24}$, $\frac{1}{2176782336}a^{25}$, $\frac{1}{13060694016}a^{26}$, $\frac{1}{13060694016}a^{27}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $27$ | 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
\(\Q(\sqrt{29}) \), 4.4.14048064.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\) |
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 | R | ${\href{/padicField/5.14.0.1}{14} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }^{4}$ | $28$ | ${\href{/padicField/13.7.0.1}{7} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{7}$ | $28$ | ${\href{/padicField/23.14.0.1}{14} }^{2}$ | R | $28$ | $28$ | ${\href{/padicField/41.4.0.1}{4} }^{7}$ | $28$ | $28$ | ${\href{/padicField/53.7.0.1}{7} }^{4}$ | ${\href{/padicField/59.1.0.1}{1} }^{28}$ |
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 $28$ | $2$ | $14$ | $42$ | |||
\(3\) | Deg $28$ | $2$ | $14$ | $14$ | |||
\(29\) | Deg $28$ | $28$ | $1$ | $27$ |