Normalized defining polynomial
\( x^{8} - 2x^{7} + 4x^{6} - 12x^{5} + 16x^{4} - 18x^{3} + 10x^{2} - 1 \)
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)
| |
Discriminant: | \(-325660672\) \(\medspace = -\,2^{12}\cdot 43^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.59\) | 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: | $2^{19/12}43^{1/2}\approx 19.65011309429094$ | ||
Ramified primes: | \(2\), \(43\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-43}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$
Monogenic: | Yes | |
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: | $83a^{7}-116a^{6}+262a^{5}-838a^{4}+823a^{3}-997a^{2}+228a+138$, $138a^{7}-193a^{6}+436a^{5}-1394a^{4}+1370a^{3}-1661a^{2}+382a+229$, $a$, $84a^{7}-117a^{6}+265a^{5}-847a^{4}+830a^{3}-1008a^{2}+228a+138$ | 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: | \( 16.7435125502 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 16.7435125502 \cdot 1}{2\cdot\sqrt{325660672}}\cr\approx \mathstrut & 0.460292093890 \end{aligned}\]
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
4.2.688.1 |
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: | 8.2.325660672.3 |
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 | R | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | R | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.29 | $x^{8} + 2 x^{5} + 2 x^{2} + 6$ | $8$ | $1$ | $12$ | $\textrm{GL(2,3)}$ | $[4/3, 4/3, 2]_{3}^{2}$ |
\(43\) | $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.43.2t1.a.a | $1$ | $ 43 $ | \(\Q(\sqrt{-43}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.172.3t2.a.a | $2$ | $ 2^{2} \cdot 43 $ | 3.1.172.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.688.24t22.a.a | $2$ | $ 2^{4} \cdot 43 $ | 8.2.325660672.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
2.688.24t22.a.b | $2$ | $ 2^{4} \cdot 43 $ | 8.2.325660672.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
3.29584.6t8.c.a | $3$ | $ 2^{4} \cdot 43^{2}$ | 4.2.688.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 3.688.4t5.b.a | $3$ | $ 2^{4} \cdot 43 $ | 4.2.688.1 | $S_4$ (as 4T5) | $1$ | $1$ |
* | 4.473344.8t23.a.a | $4$ | $ 2^{8} \cdot 43^{2}$ | 8.2.325660672.2 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ |