Normalized defining polynomial
\( x^{8} - 4x^{7} + 10x^{6} - 16x^{5} + 14x^{4} - 6x^{3} - 3x^{2} + 4x - 1 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(32993536\) \(\medspace = 2^{8}\cdot 359^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(8.71\) | 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: | $2^{3/2}359^{1/2}\approx 53.59104402789705$ | ||
Ramified primes: | \(2\), \(359\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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: | $5$ | 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: | $a$, $a^{7}-3a^{6}+7a^{5}-9a^{4}+5a^{3}-a^{2}-4a$, $10a^{7}-35a^{6}+83a^{5}-120a^{4}+83a^{3}-22a^{2}-40a+20$, $11a^{7}-39a^{6}+92a^{5}-133a^{4}+91a^{3}-21a^{2}-45a+23$, $8a^{7}-28a^{6}+66a^{5}-95a^{4}+64a^{3}-15a^{2}-34a+17$ | 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: | \( 3.76255994435 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 3.76255994435 \cdot 1}{2\cdot\sqrt{32993536}}\cr\approx \mathstrut & 0.206880101315 \end{aligned}\]
Galois group
$C_2^3:S_4$ (as 8T39):
A solvable group of order 192 |
The 13 conjugacy class representatives for $C_2^3:S_4$ |
Character table for $C_2^3:S_4$ |
Intermediate fields
4.4.5744.1 |
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: | 8.0.8248384.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 | $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ |
\(359\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.1436.2t1.a.a | $1$ | $ 2^{2} \cdot 359 $ | \(\Q(\sqrt{359}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.1436.3t2.a.a | $2$ | $ 2^{2} \cdot 359 $ | 3.3.1436.1 | $S_3$ (as 3T2) | $1$ | $2$ | |
3.1436.4t5.a.a | $3$ | $ 2^{2} \cdot 359 $ | 4.0.1436.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.8248384.6t8.b.a | $3$ | $ 2^{6} \cdot 359^{2}$ | 4.0.5744.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 3.5744.4t5.a.a | $3$ | $ 2^{4} \cdot 359 $ | 4.4.5744.1 | $S_4$ (as 4T5) | $1$ | $3$ |
3.8248384.6t8.a.a | $3$ | $ 2^{6} \cdot 359^{2}$ | 4.4.5744.1 | $S_4$ (as 4T5) | $1$ | $3$ | |
3.5744.4t5.b.a | $3$ | $ 2^{4} \cdot 359 $ | 4.0.5744.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.2062096.6t8.a.a | $3$ | $ 2^{4} \cdot 359^{2}$ | 4.0.1436.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 4.5744.8t39.a.a | $4$ | $ 2^{4} \cdot 359 $ | 8.4.32993536.1 | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ |
4.11844679424.8t39.a.a | $4$ | $ 2^{8} \cdot 359^{3}$ | 8.4.32993536.1 | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ | |
6.47378717696.8t34.a.a | $6$ | $ 2^{10} \cdot 359^{3}$ | 8.0.68035838611456.3 | $V_4^2:S_3$ (as 8T34) | $1$ | $-2$ | |
8.680...456.24t333.b.a | $8$ | $ 2^{12} \cdot 359^{4}$ | 8.4.32993536.1 | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ |