Normalized defining polynomial
\( x^{16} + 175x^{8} + 625 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(68719476736000000000000\) \(\medspace = 2^{48}\cdot 5^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(26.75\) | 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: | $2^{3}5^{3/4}\approx 26.74961219905688$ | ||
Ramified primes: | \(2\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(80=2^{4}\cdot 5\) | ||
Dirichlet character group: | $\lbrace$$\chi_{80}(1,·)$, $\chi_{80}(3,·)$, $\chi_{80}(71,·)$, $\chi_{80}(9,·)$, $\chi_{80}(79,·)$, $\chi_{80}(13,·)$, $\chi_{80}(77,·)$, $\chi_{80}(67,·)$, $\chi_{80}(27,·)$, $\chi_{80}(31,·)$, $\chi_{80}(37,·)$, $\chi_{80}(39,·)$, $\chi_{80}(41,·)$, $\chi_{80}(43,·)$, $\chi_{80}(49,·)$, $\chi_{80}(53,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{75}a^{8}-\frac{1}{3}$, $\frac{1}{75}a^{9}-\frac{1}{3}a$, $\frac{1}{375}a^{10}-\frac{4}{15}a^{2}$, $\frac{1}{375}a^{11}-\frac{4}{15}a^{3}$, $\frac{1}{375}a^{12}-\frac{1}{15}a^{4}$, $\frac{1}{375}a^{13}-\frac{1}{15}a^{5}$, $\frac{1}{1875}a^{14}-\frac{4}{75}a^{6}$, $\frac{1}{1875}a^{15}-\frac{4}{75}a^{7}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{10}$, which has order $10$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{4}{1875} a^{14} + \frac{29}{75} a^{6} \) (order $8$) | 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: | $\frac{2}{375}a^{10}+\frac{7}{15}a^{2}$, $\frac{4}{1875}a^{14}+\frac{1}{375}a^{12}+\frac{29}{75}a^{6}+\frac{8}{15}a^{4}-1$, $\frac{1}{1875}a^{14}-\frac{1}{375}a^{12}+\frac{1}{75}a^{8}+\frac{11}{75}a^{6}-\frac{1}{3}a^{4}+\frac{2}{3}$, $\frac{4}{1875}a^{15}-\frac{1}{375}a^{12}+\frac{1}{375}a^{10}+\frac{29}{75}a^{7}-\frac{8}{15}a^{4}+\frac{11}{15}a^{2}$, $\frac{1}{375}a^{12}-\frac{1}{125}a^{11}+\frac{1}{375}a^{10}+\frac{8}{15}a^{4}-\frac{6}{5}a^{3}+\frac{11}{15}a^{2}$, $\frac{1}{625}a^{15}-\frac{2}{1875}a^{14}-\frac{1}{375}a^{12}+\frac{2}{375}a^{11}-\frac{2}{375}a^{10}+\frac{1}{75}a^{8}+\frac{6}{25}a^{7}-\frac{7}{75}a^{6}-\frac{1}{5}a^{5}-\frac{2}{15}a^{4}+\frac{7}{15}a^{3}-\frac{7}{15}a^{2}+\frac{2}{3}$, $\frac{7}{1875}a^{15}-\frac{1}{375}a^{12}+\frac{1}{375}a^{10}+\frac{47}{75}a^{7}-\frac{8}{15}a^{4}+\frac{11}{15}a^{2}$ (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: | \( 42649.7127656 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 42649.7127656 \cdot 10}{8\cdot\sqrt{68719476736000000000000}}\cr\approx \mathstrut & 0.493997952152 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\times C_4$ (as 16T2):
An abelian group of order 16 |
The 16 conjugacy class representatives for $C_4\times C_2^2$ |
Character table for $C_4\times C_2^2$ |
Intermediate fields
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 | ${\href{/padicField/3.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 $16$ | $8$ | $2$ | $48$ | |||
\(5\) | 5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |