Normalized defining polynomial
\( x^{8} - 4x^{7} + 22x^{6} - 52x^{5} + 111x^{4} - 140x^{3} + 130x^{2} - 68x + 22 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(12745506816\) \(\medspace = 2^{16}\cdot 3^{4}\cdot 7^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.33\) | 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: | $2^{2}3^{1/2}7^{1/2}\approx 18.33030277982336$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(168=2^{3}\cdot 3\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{168}(1,·)$, $\chi_{168}(43,·)$, $\chi_{168}(97,·)$, $\chi_{168}(139,·)$, $\chi_{168}(125,·)$, $\chi_{168}(29,·)$, $\chi_{168}(71,·)$, $\chi_{168}(167,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-21}) \), 8.0.12745506816.2$^{4}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{235}a^{7}+\frac{4}{47}a^{6}+\frac{32}{235}a^{5}+\frac{11}{235}a^{4}-\frac{19}{47}a^{3}-\frac{14}{47}a^{2}+\frac{19}{47}a+\frac{97}{235}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$
Unit group
Rank: | $3$ | 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: | $\frac{108}{235}a^{7}-\frac{378}{235}a^{6}+\frac{428}{47}a^{5}-\frac{881}{47}a^{4}+\frac{8916}{235}a^{3}-\frac{9158}{235}a^{2}+\frac{6359}{235}a-\frac{1791}{235}$, $\frac{6}{235}a^{7}-\frac{21}{235}a^{6}+\frac{29}{47}a^{5}-\frac{62}{47}a^{4}+\frac{887}{235}a^{3}-\frac{1031}{235}a^{2}+\frac{758}{235}a-\frac{217}{235}$, $\frac{228}{235}a^{7}-\frac{798}{235}a^{6}+\frac{914}{47}a^{5}-\frac{1886}{47}a^{4}+\frac{19606}{235}a^{3}-\frac{20378}{235}a^{2}+\frac{14704}{235}a-\frac{4251}{235}$ | 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: | \( 58.329447273 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 58.329447273 \cdot 4}{2\cdot\sqrt{12745506816}}\cr\approx \mathstrut & 1.6104927549 \end{aligned}\]
Galois group
An abelian group of order 8 |
The 8 conjugacy class representatives for $C_2^3$ |
Character table for $C_2^3$ |
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 | R | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/11.1.0.1}{1} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.8.4 | $x^{4} + 6 x^{2} + 4 x + 6$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
2.4.8.4 | $x^{4} + 6 x^{2} + 4 x + 6$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(7\) | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |