Normalized defining polynomial
\( x^{8} - 4x^{6} - 8x^{5} + 9x^{4} + 16x^{3} + 6x^{2} - 20x - 52 \)
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: | \(-754507653376\) \(\medspace = -\,2^{8}\cdot 233^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(30.53\) | 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: | $2^{43/24}233^{1/2}\approx 52.84736882095583$ | ||
Ramified primes: | \(2\), \(233\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{566}a^{7}-\frac{117}{566}a^{6}+\frac{101}{566}a^{5}+\frac{61}{566}a^{4}+\frac{115}{283}a^{3}+\frac{137}{283}a^{2}+\frac{105}{283}a-\frac{126}{283}$
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: | $\frac{7}{283}a^{7}+\frac{30}{283}a^{6}-\frac{142}{283}a^{5}+\frac{5}{566}a^{4}+\frac{195}{283}a^{3}+\frac{157}{566}a^{2}+\frac{55}{283}a-\frac{632}{283}$, $\frac{53}{566}a^{7}+\frac{25}{566}a^{6}-\frac{307}{566}a^{5}-\frac{223}{283}a^{4}+\frac{435}{283}a^{3}+\frac{1221}{566}a^{2}-\frac{661}{283}a-\frac{735}{283}$, $\frac{7}{566}a^{7}+\frac{15}{283}a^{6}-\frac{71}{283}a^{5}-\frac{139}{566}a^{4}+\frac{195}{566}a^{3}+\frac{1069}{566}a^{2}-\frac{397}{283}a-\frac{33}{283}$, $\frac{31}{566}a^{7}+\frac{26}{283}a^{6}-\frac{274}{283}a^{5}+\frac{193}{566}a^{4}+\frac{621}{566}a^{3}+\frac{853}{566}a^{2}+\frac{425}{283}a-\frac{1076}{283}$ | 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: | \( 1448.8995475 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 1448.8995475 \cdot 1}{2\cdot\sqrt{754507653376}}\cr\approx \mathstrut & 0.82751533924 \end{aligned}\]
Galois group
$S_4\wr C_2$ (as 8T47):
A solvable group of order 1152 |
The 20 conjugacy class representatives for $S_4\wr C_2$ |
Character table for $S_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{233}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 36 siblings: | data not computed |
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 | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{5}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.4 | $x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
2.4.4.5 | $x^{4} + 2 x + 2$ | $4$ | $1$ | $4$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
\(233\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $2$ | $3$ | $3$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.932.2t1.a.a | $1$ | $ 2^{2} \cdot 233 $ | \(\Q(\sqrt{-233}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.233.2t1.a.a | $1$ | $ 233 $ | \(\Q(\sqrt{233}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.932.4t3.b.a | $2$ | $ 2^{2} \cdot 233 $ | 4.0.3728.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.809557568.12t34.a.a | $4$ | $ 2^{6} \cdot 233^{3}$ | 6.0.59648.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.3474496.12t34.a.a | $4$ | $ 2^{6} \cdot 233^{2}$ | 6.0.59648.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.14912.6t13.a.a | $4$ | $ 2^{6} \cdot 233 $ | 6.0.59648.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.868624.6t13.a.a | $4$ | $ 2^{4} \cdot 233^{2}$ | 6.0.59648.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.12952921088.12t201.a.a | $6$ | $ 2^{10} \cdot 233^{3}$ | 8.2.754507653376.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.51811684352.12t202.a.a | $6$ | $ 2^{12} \cdot 233^{3}$ | 8.2.754507653376.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.3238230272.8t47.b.a | $6$ | $ 2^{8} \cdot 233^{3}$ | 8.2.754507653376.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.12952921088.12t200.b.a | $6$ | $ 2^{10} \cdot 233^{3}$ | 8.2.754507653376.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.828986949632.16t1294.a.a | $9$ | $ 2^{16} \cdot 233^{3}$ | 8.2.754507653376.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.331...528.18t272.a.a | $9$ | $ 2^{18} \cdot 233^{3}$ | 8.2.754507653376.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.419...936.18t273.a.a | $9$ | $ 2^{18} \cdot 233^{6}$ | 8.2.754507653376.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.104...984.18t274.a.a | $9$ | $ 2^{16} \cdot 233^{6}$ | 8.2.754507653376.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.671...976.36t1763.a.a | $12$ | $ 2^{22} \cdot 233^{6}$ | 8.2.754507653376.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.167...744.24t2821.a.a | $12$ | $ 2^{20} \cdot 233^{6}$ | 8.2.754507653376.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.347...552.36t1758.a.a | $18$ | $ 2^{34} \cdot 233^{9}$ | 8.2.754507653376.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |