Normalized defining polynomial
\( x^{8} - x^{7} - x^{6} + 8x^{5} - 2x^{4} - 11x^{3} + 11x^{2} + 13x + 1 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-606436875\) \(\medspace = -\,3^{6}\cdot 5^{4}\cdot 11^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(12.53\) | 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: | $3^{3/4}5^{1/2}11^{1/2}\approx 16.90527678711191$ | ||
Ramified primes: | \(3\), \(5\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11}a^{6}-\frac{4}{11}a^{4}+\frac{4}{11}a^{3}+\frac{3}{11}a^{2}+\frac{2}{11}a+\frac{4}{11}$, $\frac{1}{11}a^{7}-\frac{4}{11}a^{5}+\frac{4}{11}a^{4}+\frac{3}{11}a^{3}+\frac{2}{11}a^{2}+\frac{4}{11}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $\frac{2}{11}a^{7}-\frac{4}{11}a^{6}+\frac{3}{11}a^{5}+\frac{13}{11}a^{4}-\frac{21}{11}a^{3}+\frac{3}{11}a^{2}+2a-\frac{16}{11}$, $\frac{1}{11}a^{6}-\frac{4}{11}a^{4}+\frac{15}{11}a^{3}-\frac{8}{11}a^{2}-\frac{9}{11}a+\frac{15}{11}$, $\frac{2}{11}a^{7}-\frac{5}{11}a^{6}+\frac{3}{11}a^{5}+\frac{17}{11}a^{4}-\frac{25}{11}a^{3}+\frac{42}{11}a+\frac{2}{11}$, $\frac{1}{11}a^{7}+\frac{1}{11}a^{6}-\frac{4}{11}a^{5}+a^{4}+\frac{7}{11}a^{3}-\frac{17}{11}a^{2}+\frac{6}{11}a+\frac{15}{11}$ | 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: | \( 16.4184768178 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 16.4184768178 \cdot 1}{2\cdot\sqrt{606436875}}\cr\approx \mathstrut & 0.330757472253 \end{aligned}\]
Galois group
A solvable group of order 16 |
The 7 conjugacy class representatives for $D_{8}$ |
Character table for $D_{8}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.2.2475.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 16.0.44499647686531640625.1 |
Degree 8 sibling: | 8.0.1334161125.1 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }$ | R | R | ${\href{/padicField/7.8.0.1}{8} }$ | R | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.55.2t1.a.a | $1$ | $ 5 \cdot 11 $ | \(\Q(\sqrt{-55}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.495.4t3.a.a | $2$ | $ 3^{2} \cdot 5 \cdot 11 $ | 4.0.5445.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
* | 2.495.8t6.b.a | $2$ | $ 3^{2} \cdot 5 \cdot 11 $ | 8.2.606436875.2 | $D_{8}$ (as 8T6) | $1$ | $0$ |
* | 2.495.8t6.b.b | $2$ | $ 3^{2} \cdot 5 \cdot 11 $ | 8.2.606436875.2 | $D_{8}$ (as 8T6) | $1$ | $0$ |