Normalized defining polynomial
\( x^{8} + x^{6} - 2x^{5} - x^{4} - 2x^{3} + x^{2} + 1 \)
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: | \(-6668032\) \(\medspace = -\,2^{8}\cdot 7\cdot 61^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(7.13\) | 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: | $2^{3/2}7^{1/2}61^{1/2}\approx 58.44655678480983$ | ||
Ramified primes: | \(2\), \(7\), \(61\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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: | $a$, $a^{6}+a^{4}-a^{3}-a^{2}-a$, $a^{6}-a^{5}+a^{4}-2a^{3}$, $a^{7}+a^{5}-2a^{4}-a^{3}-a^{2}$ | 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: | \( 1.1374982737 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 1.1374982737 \cdot 1}{2\cdot\sqrt{6668032}}\cr\approx \mathstrut & 0.21853525509 \end{aligned}\]
Galois group
$C_2\wr S_4$ (as 8T44):
A solvable group of order 384 |
The 20 conjugacy class representatives for $C_2 \wr S_4$ |
Character table for $C_2 \wr S_4$ |
Intermediate fields
4.2.976.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 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 | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.2 | $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(61\) | 61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
61.4.0.1 | $x^{4} + 3 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.244.2t1.a.a | $1$ | $ 2^{2} \cdot 61 $ | \(\Q(\sqrt{-61}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.1708.2t1.a.a | $1$ | $ 2^{2} \cdot 7 \cdot 61 $ | \(\Q(\sqrt{427}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.244.3t2.a.a | $2$ | $ 2^{2} \cdot 61 $ | 3.1.244.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.11956.6t3.d.a | $2$ | $ 2^{2} \cdot 7^{2} \cdot 61 $ | 6.2.4982686912.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 3.976.4t5.b.a | $3$ | $ 2^{4} \cdot 61 $ | 4.2.976.1 | $S_4$ (as 4T5) | $1$ | $1$ |
3.238144.6t8.b.a | $3$ | $ 2^{6} \cdot 61^{2}$ | 4.2.976.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.81683392.6t11.a.a | $3$ | $ 2^{6} \cdot 7^{3} \cdot 61^{2}$ | 6.0.81683392.2 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
3.334768.6t11.a.a | $3$ | $ 2^{4} \cdot 7^{3} \cdot 61 $ | 6.0.81683392.2 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
* | 4.6832.8t44.b.a | $4$ | $ 2^{4} \cdot 7 \cdot 61 $ | 8.2.6668032.1 | $C_2 \wr S_4$ (as 8T44) | $1$ | $0$ |
4.334768.8t44.b.a | $4$ | $ 2^{4} \cdot 7^{3} \cdot 61 $ | 8.2.6668032.1 | $C_2 \wr S_4$ (as 8T44) | $1$ | $0$ | |
4.406749952.8t44.b.a | $4$ | $ 2^{8} \cdot 7 \cdot 61^{3}$ | 8.2.6668032.1 | $C_2 \wr S_4$ (as 8T44) | $1$ | $0$ | |
4.19930747648.8t44.b.a | $4$ | $ 2^{8} \cdot 7^{3} \cdot 61^{3}$ | 8.2.6668032.1 | $C_2 \wr S_4$ (as 8T44) | $1$ | $0$ | |
6.81683392.8t41.a.a | $6$ | $ 2^{6} \cdot 7^{3} \cdot 61^{2}$ | 8.0.571783744.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $0$ | |
6.486...112.12t108.a.a | $6$ | $ 2^{10} \cdot 7^{3} \cdot 61^{4}$ | 8.0.571783744.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $0$ | |
6.79722990592.8t41.a.a | $6$ | $ 2^{10} \cdot 7^{3} \cdot 61^{3}$ | 8.0.571783744.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $2$ | |
6.79722990592.12t108.a.a | $6$ | $ 2^{10} \cdot 7^{3} \cdot 61^{3}$ | 8.0.571783744.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $-2$ | |
8.667...664.24t708.b.a | $8$ | $ 2^{12} \cdot 7^{6} \cdot 61^{4}$ | 8.2.6668032.1 | $C_2 \wr S_4$ (as 8T44) | $1$ | $0$ | |
8.277...064.24t708.b.a | $8$ | $ 2^{12} \cdot 7^{2} \cdot 61^{4}$ | 8.2.6668032.1 | $C_2 \wr S_4$ (as 8T44) | $1$ | $0$ |