Normalized defining polynomial
\( x^{8} - 4x^{7} + 4x^{6} + 21x^{4} - 84x^{3} + 132x^{2} - 96x + 24 \)
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: | \(-11609505792\) \(\medspace = -\,2^{16}\cdot 3^{11}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.12\) | 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: | $2^{9/4}3^{25/12}\approx 46.91590565166763$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{692}a^{7}+\frac{41}{346}a^{6}+\frac{34}{173}a^{5}-\frac{17}{173}a^{4}-\frac{291}{692}a^{3}-\frac{99}{346}a^{2}-\frac{72}{173}a+\frac{12}{173}$
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: | $\frac{35}{346}a^{7}-\frac{71}{346}a^{6}-\frac{42}{173}a^{5}+\frac{21}{173}a^{4}+\frac{887}{346}a^{3}-\frac{1221}{346}a^{2}+\frac{150}{173}a-\frac{25}{173}$, $\frac{39}{692}a^{7}-\frac{131}{346}a^{6}+\frac{115}{173}a^{5}+\frac{29}{173}a^{4}+\frac{415}{692}a^{3}-\frac{3169}{346}a^{2}+\frac{2728}{173}a-\frac{1435}{173}$, $\frac{783}{692}a^{7}-\frac{1113}{346}a^{6}+\frac{153}{173}a^{5}+\frac{183}{173}a^{4}+\frac{17115}{692}a^{3}-\frac{23195}{346}a^{2}+\frac{12651}{173}a-\frac{3925}{173}$, $\frac{163}{692}a^{7}-\frac{32}{173}a^{6}-\frac{167}{173}a^{5}-\frac{3}{173}a^{4}+\frac{3775}{692}a^{3}-\frac{543}{173}a^{2}-\frac{1183}{173}a+\frac{2129}{173}$ | 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: | \( 400.071910328 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 400.071910328 \cdot 1}{2\cdot\sqrt{11609505792}}\cr\approx \mathstrut & 1.84204737053 \end{aligned}\]
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
\(\Q(\sqrt{3}) \) |
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 | R | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.8.5 | $x^{4} + 2 x^{2} + 4 x + 6$ | $4$ | $1$ | $8$ | $D_{4}$ | $[2, 3]^{2}$ |
2.4.8.3 | $x^{4} + 6 x^{2} + 4 x + 14$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.6.10.11 | $x^{6} + 3 x^{5} + 9 x^{2} + 9 x + 24$ | $6$ | $1$ | $10$ | $C_3^2:D_4$ | $[9/4, 9/4]_{4}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.144.4t3.b.a | $2$ | $ 2^{4} \cdot 3^{2}$ | 4.0.432.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.314928.6t13.b.a | $4$ | $ 2^{4} \cdot 3^{9}$ | 6.0.944784.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.1259712.12t34.b.a | $4$ | $ 2^{6} \cdot 3^{9}$ | 6.0.944784.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.5038848.12t34.c.a | $4$ | $ 2^{8} \cdot 3^{9}$ | 6.0.944784.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.1259712.6t13.c.a | $4$ | $ 2^{6} \cdot 3^{9}$ | 6.0.944784.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.11609505792.12t201.c.a | $6$ | $ 2^{16} \cdot 3^{11}$ | 8.2.11609505792.9 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.967458816.12t202.c.a | $6$ | $ 2^{14} \cdot 3^{10}$ | 8.2.11609505792.9 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.967458816.8t47.c.a | $6$ | $ 2^{14} \cdot 3^{10}$ | 8.2.11609505792.9 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.11609505792.12t200.a.a | $6$ | $ 2^{16} \cdot 3^{11}$ | 8.2.11609505792.9 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.253...104.16t1294.b.a | $9$ | $ 2^{16} \cdot 3^{18}$ | 8.2.11609505792.9 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.761...312.18t272.a.a | $9$ | $ 2^{16} \cdot 3^{19}$ | 8.2.11609505792.9 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.162...656.18t273.a.a | $9$ | $ 2^{22} \cdot 3^{18}$ | 8.2.11609505792.9 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.487...968.18t274.a.a | $9$ | $ 2^{22} \cdot 3^{19}$ | 8.2.11609505792.9 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.818...288.36t1763.a.a | $12$ | $ 2^{30} \cdot 3^{27}$ | 8.2.11609505792.9 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.818...288.24t2821.a.a | $12$ | $ 2^{30} \cdot 3^{27}$ | 8.2.11609505792.9 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.148...064.36t1758.a.a | $18$ | $ 2^{40} \cdot 3^{38}$ | 8.2.11609505792.9 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |