Normalized defining polynomial
\( x^{8} - x^{7} - 4x^{6} + 7x^{5} - 6x^{3} + 8x^{2} - 9x + 2 \)
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: | \(6689113369\) \(\medspace = 17^{4}\cdot 283^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.91\) | 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: | $17^{1/2}283^{1/2}\approx 69.36137253543934$ | ||
Ramified primes: | \(17\), \(283\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{11}a^{7}+\frac{4}{11}a^{6}+\frac{5}{11}a^{5}-\frac{1}{11}a^{4}-\frac{5}{11}a^{3}+\frac{2}{11}a^{2}-\frac{4}{11}a+\frac{4}{11}$
Monogenic: | Not computed | |
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: | $\frac{2}{11}a^{7}-\frac{3}{11}a^{6}-\frac{12}{11}a^{5}+\frac{20}{11}a^{4}+\frac{12}{11}a^{3}-\frac{29}{11}a^{2}+\frac{14}{11}a-\frac{3}{11}$, $\frac{2}{11}a^{7}-\frac{3}{11}a^{6}-\frac{12}{11}a^{5}+\frac{20}{11}a^{4}+\frac{12}{11}a^{3}-\frac{18}{11}a^{2}+\frac{25}{11}a-\frac{25}{11}$, $\frac{12}{11}a^{7}-\frac{7}{11}a^{6}-\frac{50}{11}a^{5}+\frac{65}{11}a^{4}+\frac{28}{11}a^{3}-\frac{64}{11}a^{2}+\frac{62}{11}a-\frac{73}{11}$, $\frac{4}{11}a^{7}-\frac{6}{11}a^{6}-\frac{13}{11}a^{5}+\frac{29}{11}a^{4}-\frac{9}{11}a^{3}-\frac{3}{11}a^{2}+\frac{6}{11}a-\frac{17}{11}$, $a^{7}-a^{6}-4a^{5}+7a^{4}-6a^{2}+9a-9$ | 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: | \( 275.246438617 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 275.246438617 \cdot 1}{2\cdot\sqrt{6689113369}}\cr\approx \mathstrut & 1.06288714322 \end{aligned}\]
Galois group
$C_2^3:S_4$ (as 8T41):
A solvable group of order 192 |
The 14 conjugacy class representatives for $V_4^2:(S_3\times C_2)$ |
Character table for $V_4^2:(S_3\times C_2)$ |
Intermediate fields
\(\Q(\sqrt{17}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 sibling: | data not computed |
Degree 12 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 | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | 17.8.4.1 | $x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(283\) | 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.4811.2t1.a.a | $1$ | $ 17 \cdot 283 $ | \(\Q(\sqrt{-4811}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.283.2t1.a.a | $1$ | $ 283 $ | \(\Q(\sqrt{-283}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.81787.6t3.b.a | $2$ | $ 17^{2} \cdot 283 $ | 6.0.111354063731.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
2.283.3t2.a.a | $2$ | $ 283 $ | 3.1.283.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
3.80089.6t8.d.a | $3$ | $ 283^{2}$ | 4.2.283.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.393477257.6t11.b.a | $3$ | $ 17^{3} \cdot 283^{2}$ | 6.0.111354063731.2 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.283.4t5.b.a | $3$ | $ 283 $ | 4.2.283.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.1390379.6t11.b.a | $3$ | $ 17^{3} \cdot 283 $ | 6.0.111354063731.2 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
6.315...873.12t108.a.a | $6$ | $ 17^{3} \cdot 283^{4}$ | 8.4.6689113369.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $-2$ | |
6.111354063731.8t41.a.a | $6$ | $ 17^{3} \cdot 283^{3}$ | 8.4.6689113369.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $0$ | |
6.111354063731.12t108.a.a | $6$ | $ 17^{3} \cdot 283^{3}$ | 8.4.6689113369.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $0$ | |
* | 6.393477257.8t41.a.a | $6$ | $ 17^{3} \cdot 283^{2}$ | 8.4.6689113369.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $2$ |