Normalized defining polynomial
\( x^{8} - 10x^{6} - 4x^{5} + 27x^{4} + 20x^{3} - 10x^{2} - 8x - 1 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(132705746944\) \(\medspace = 2^{16}\cdot 1423^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24.57\) | 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: | $2^{55/24}1423^{1/2}\approx 184.69828241304324$ | ||
Ramified primes: | \(2\), \(1423\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $a$, $a+1$, $a^{7}-10a^{5}-4a^{4}+28a^{3}+19a^{2}-13a-7$, $2a^{7}-a^{6}-19a^{5}+a^{4}+49a^{3}+17a^{2}-18a-3$, $a^{7}-10a^{5}-4a^{4}+27a^{3}+19a^{2}-9a-5$, $a^{7}-11a^{5}-3a^{4}+34a^{3}+15a^{2}-19a-3$, $2a^{7}-2a^{6}-17a^{5}+8a^{4}+38a^{3}+6a^{2}-11a-2$ | 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: | \( 1356.97140968 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 1356.97140968 \cdot 1}{2\cdot\sqrt{132705746944}}\cr\approx \mathstrut & 0.476799511483 \end{aligned}\]
Galois group
$C_2^3:\GL(3,2)$ (as 8T48):
A non-solvable group of order 1344 |
The 11 conjugacy class representatives for $C_2^3:\GL(3,2)$ |
Character table for $C_2^3:\GL(3,2)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 8 sibling: | deg 8 |
Degree 14 siblings: | deg 14, 14.14.275168988624452583424.1 |
Degree 28 siblings: | deg 28, deg 28, deg 28 |
Degree 42 siblings: | deg 42, deg 42, deg 42, deg 42 |
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 | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.20 | $x^{8} - 4 x^{6} + 8 x^{5} + 20 x^{4} - 16 x^{3} + 8 x^{2} + 16 x + 4$ | $4$ | $2$ | $16$ | $V_4^2:S_3$ | $[4/3, 4/3, 8/3, 8/3]_{3}^{2}$ |
\(1423\) | Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
3.518381824.42t37.b.a | $3$ | $ 2^{8} \cdot 1423^{2}$ | 7.7.2073527296.2 | $\GL(3,2)$ (as 7T5) | $0$ | $3$ | |
3.518381824.42t37.b.b | $3$ | $ 2^{8} \cdot 1423^{2}$ | 7.7.2073527296.2 | $\GL(3,2)$ (as 7T5) | $0$ | $3$ | |
6.2073527296.7t5.b.a | $6$ | $ 2^{10} \cdot 1423^{2}$ | 7.7.2073527296.2 | $\GL(3,2)$ (as 7T5) | $1$ | $6$ | |
* | 7.132705746944.8t48.a.a | $7$ | $ 2^{16} \cdot 1423^{2}$ | 8.8.132705746944.1 | $C_2^3:\GL(3,2)$ (as 8T48) | $1$ | $7$ |
7.268...976.8t37.b.a | $7$ | $ 2^{16} \cdot 1423^{4}$ | 7.7.2073527296.2 | $\GL(3,2)$ (as 7T5) | $1$ | $7$ | |
7.167...936.8t48.a.a | $7$ | $ 2^{12} \cdot 1423^{4}$ | 8.8.132705746944.1 | $C_2^3:\GL(3,2)$ (as 8T48) | $1$ | $7$ | |
8.107...904.21t14.a.a | $8$ | $ 2^{18} \cdot 1423^{4}$ | 7.7.2073527296.1 | $\GL(3,2)$ (as 7T5) | $1$ | $8$ | |
14.891...336.28t159.a.a | $14$ | $ 2^{30} \cdot 1423^{6}$ | 8.8.132705746944.1 | $C_2^3:\GL(3,2)$ (as 8T48) | $1$ | $14$ | |
21.194...176.42t210.a.a | $21$ | $ 2^{48} \cdot 1423^{12}$ | 8.8.132705746944.1 | $C_2^3:\GL(3,2)$ (as 8T48) | $1$ | $21$ | |
21.153...904.42t210.a.a | $21$ | $ 2^{52} \cdot 1423^{10}$ | 8.8.132705746944.1 | $C_2^3:\GL(3,2)$ (as 8T48) | $1$ | $21$ |