Normalized defining polynomial
\( x^{8} + 12x^{6} - 6x^{4} - 48x^{2} - 51 \)
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: | \(-44010418176\) \(\medspace = -\,2^{12}\cdot 3^{7}\cdot 17^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.40\) | 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: | $2^{19/12}3^{7/8}17^{1/2}\approx 32.309957372245684$ | ||
Ramified primes: | \(2\), \(3\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-51}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{388}a^{6}-\frac{49}{388}a^{4}-\frac{1}{2}a^{3}-\frac{121}{388}a^{2}-\frac{1}{2}a-\frac{39}{388}$, $\frac{1}{388}a^{7}-\frac{49}{388}a^{5}-\frac{121}{388}a^{3}-\frac{1}{2}a^{2}-\frac{39}{388}a-\frac{1}{2}$
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: | $\frac{8}{97}a^{6}+\frac{93}{97}a^{4}-\frac{95}{97}a^{2}-\frac{409}{97}$, $\frac{11}{194}a^{6}+\frac{70}{97}a^{4}+\frac{27}{194}a^{2}-\frac{263}{97}$, $\frac{17}{97}a^{7}-\frac{7}{194}a^{6}+\frac{371}{194}a^{5}-\frac{45}{194}a^{4}-\frac{311}{97}a^{3}+\frac{459}{194}a^{2}-\frac{1035}{194}a-\frac{503}{194}$, $\frac{15}{388}a^{7}+\frac{27}{388}a^{6}+\frac{235}{388}a^{5}+\frac{423}{388}a^{4}+\frac{707}{388}a^{3}+\frac{1195}{388}a^{2}+\frac{967}{388}a+\frac{887}{388}$ | 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: | \( 418.372577599 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 418.372577599 \cdot 1}{2\cdot\sqrt{44010418176}}\cr\approx \mathstrut & 0.989361631501 \end{aligned}\]
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
4.2.7344.1 |
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: | 8.2.44010418176.2 |
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 | R | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.29 | $x^{8} + 2 x^{5} + 2 x^{2} + 6$ | $8$ | $1$ | $12$ | $\textrm{GL(2,3)}$ | $[4/3, 4/3, 2]_{3}^{2}$ |
\(3\) | 3.8.7.1 | $x^{8} + 3$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |
\(17\) | 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.51.2t1.a.a | $1$ | $ 3 \cdot 17 $ | \(\Q(\sqrt{-51}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.204.3t2.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 17 $ | 3.1.204.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.2448.24t22.a.a | $2$ | $ 2^{4} \cdot 3^{2} \cdot 17 $ | 8.2.44010418176.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
2.2448.24t22.a.b | $2$ | $ 2^{4} \cdot 3^{2} \cdot 17 $ | 8.2.44010418176.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
3.41616.6t8.a.a | $3$ | $ 2^{4} \cdot 3^{2} \cdot 17^{2}$ | 4.2.7344.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 3.7344.4t5.a.a | $3$ | $ 2^{4} \cdot 3^{3} \cdot 17 $ | 4.2.7344.1 | $S_4$ (as 4T5) | $1$ | $1$ |
* | 4.5992704.8t23.a.a | $4$ | $ 2^{8} \cdot 3^{4} \cdot 17^{2}$ | 8.2.44010418176.1 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ |