Normalized defining polynomial
\( x^{8} - 4x^{7} + 4x^{6} + 2x^{5} + 37x^{4} - 82x^{3} - 76x^{2} + 118x + 401 \)
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: | \(8999178496\) \(\medspace = 2^{8}\cdot 7^{4}\cdot 11^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(17.55\) | 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: | $2\cdot 7^{1/2}11^{1/2}\approx 17.549928774784245$ | ||
Ramified primes: | \(2\), \(7\), \(11\) | 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: | \(308=2^{2}\cdot 7\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{308}(1,·)$, $\chi_{308}(197,·)$, $\chi_{308}(265,·)$, $\chi_{308}(43,·)$, $\chi_{308}(111,·)$, $\chi_{308}(307,·)$, $\chi_{308}(153,·)$, $\chi_{308}(155,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-77}) \), 8.0.8999178496.1$^{4}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{6}a^{4}+\frac{1}{6}a^{2}+\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{18}a^{5}-\frac{1}{18}a^{4}-\frac{1}{6}a^{3}-\frac{5}{18}a^{2}-\frac{1}{2}a+\frac{7}{18}$, $\frac{1}{54}a^{6}+\frac{1}{27}a^{4}+\frac{2}{27}a^{3}+\frac{5}{27}a^{2}+\frac{11}{27}a-\frac{23}{54}$, $\frac{1}{1998}a^{7}+\frac{5}{666}a^{6}-\frac{7}{1998}a^{5}-\frac{10}{999}a^{4}-\frac{47}{1998}a^{3}-\frac{247}{999}a^{2}+\frac{5}{999}a+\frac{263}{666}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{1}{333} a^{7} + \frac{7}{666} a^{6} - \frac{23}{666} a^{5} + \frac{20}{333} a^{4} - \frac{91}{666} a^{3} + \frac{50}{333} a^{2} - \frac{205}{666} a + \frac{29}{222} \) (order $4$) | 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: | $\frac{5}{999}a^{7}+\frac{1}{999}a^{6}-\frac{35}{999}a^{5}+\frac{85}{999}a^{4}+\frac{5}{37}a^{3}+\frac{40}{333}a^{2}-\frac{82}{333}a-\frac{14}{999}$, $\frac{19}{1998}a^{7}-\frac{85}{1998}a^{6}+\frac{100}{999}a^{5}-\frac{121}{1998}a^{4}+\frac{104}{333}a^{3}-\frac{53}{74}a^{2}+\frac{569}{666}a+\frac{2926}{999}$, $\frac{8}{999}a^{7}-\frac{31}{666}a^{6}+\frac{221}{1998}a^{5}-\frac{160}{999}a^{4}+\frac{913}{1998}a^{3}-\frac{955}{999}a^{2}-\frac{173}{1998}a+\frac{1211}{666}$ | 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: | \( 72.4184469969 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 72.4184469969 \cdot 2}{4\cdot\sqrt{8999178496}}\cr\approx \mathstrut & 0.594890799128 \end{aligned}\]
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 | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | R | R | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.1.0.1}{1} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.1.0.1}{1} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
\(7\) | 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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(11\) | 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |