Normalized defining polynomial
\( x^{8} - 3x^{7} + 6x^{6} - 12x^{5} + 6x^{4} - 18x^{3} + 18x^{2} - 3x - 3 \)
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)
| |
Discriminant: | \(-2832228423\) \(\medspace = -\,3^{7}\cdot 109^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.19\) | 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: | $3^{7/8}109^{1/2}\approx 27.301992740901024$ | ||
Ramified primes: | \(3\), \(109\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-327}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2594}a^{7}+\frac{57}{1297}a^{6}+\frac{187}{1297}a^{5}-\frac{176}{1297}a^{4}+\frac{163}{1297}a^{3}-\frac{393}{1297}a^{2}-\frac{577}{1297}a-\frac{133}{2594}$
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: | $\frac{43}{1297}a^{7}-\frac{286}{1297}a^{6}+\frac{518}{1297}a^{5}-\frac{869}{1297}a^{4}+\frac{1048}{1297}a^{3}-\frac{76}{1297}a^{2}+\frac{3555}{1297}a+\frac{766}{1297}$, $\frac{278}{1297}a^{7}-\frac{733}{1297}a^{6}+\frac{1509}{1297}a^{5}-\frac{3175}{1297}a^{4}+\frac{1135}{1297}a^{3}-\frac{5800}{1297}a^{2}+\frac{3438}{1297}a-\frac{1955}{1297}$, $\frac{56}{1297}a^{7}-\frac{101}{1297}a^{6}+\frac{192}{1297}a^{5}-\frac{257}{1297}a^{4}+\frac{98}{1297}a^{3}-\frac{1215}{1297}a^{2}+\frac{226}{1297}a+\frac{334}{1297}$, $\frac{321}{1297}a^{7}-\frac{1019}{1297}a^{6}+\frac{2027}{1297}a^{5}-\frac{4044}{1297}a^{4}+\frac{2183}{1297}a^{3}-\frac{4579}{1297}a^{2}+\frac{5696}{1297}a+\frac{2702}{1297}$ | 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: | \( 143.077001818 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 143.077001818 \cdot 1}{2\cdot\sqrt{2832228423}}\cr\approx \mathstrut & 1.33375264357 \end{aligned}\]
Galois group
$\GL(2,3)$ (as 8T23):
A solvable group of order 48 |
The 8 conjugacy class representatives for $\textrm{GL(2,3)}$ |
Character table for $\textrm{GL(2,3)}$ |
Intermediate fields
4.2.2943.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 sibling: | deg 16 |
Degree 24 sibling: | deg 24 |
Arithmetically equvalently sibling: | 8.2.2832228423.1 |
Minimal sibling: | 8.2.2832228423.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.7.2 | $x^{8} + 6$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |
\(109\) | $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
109.2.1.1 | $x^{2} + 109$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.1 | $x^{2} + 109$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.1 | $x^{2} + 109$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.327.2t1.a.a | $1$ | $ 3 \cdot 109 $ | \(\Q(\sqrt{-327}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.327.3t2.a.a | $2$ | $ 3 \cdot 109 $ | 3.1.327.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.981.24t22.b.a | $2$ | $ 3^{2} \cdot 109 $ | 8.2.2832228423.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
2.981.24t22.b.b | $2$ | $ 3^{2} \cdot 109 $ | 8.2.2832228423.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
3.106929.6t8.a.a | $3$ | $ 3^{2} \cdot 109^{2}$ | 4.2.2943.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 3.2943.4t5.a.a | $3$ | $ 3^{3} \cdot 109 $ | 4.2.2943.1 | $S_4$ (as 4T5) | $1$ | $1$ |
* | 4.962361.8t23.b.a | $4$ | $ 3^{4} \cdot 109^{2}$ | 8.2.2832228423.2 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ |