Normalized defining polynomial
\( x^{8} - 8x^{6} + 51x^{4} - 136x^{2} + 289 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(5473632256\) \(\medspace = 2^{16}\cdot 17^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.49\) | 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: | $2^{2}17^{3/4}\approx 33.48857611437076$ | ||
Ramified primes: | \(2\), \(17\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{221}a^{6}+\frac{94}{221}a^{4}-\frac{5}{13}a^{2}+\frac{2}{13}$, $\frac{1}{221}a^{7}+\frac{94}{221}a^{5}-\frac{5}{13}a^{3}+\frac{2}{13}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{1}{17} a^{6} - \frac{8}{17} a^{4} + 2 a^{2} - 4 \) (order $4$) | 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: | $\frac{7}{221}a^{6}-\frac{5}{221}a^{4}+\frac{4}{13}a^{2}+\frac{27}{13}$, $\frac{4}{221}a^{7}-\frac{3}{221}a^{6}-\frac{66}{221}a^{5}-\frac{61}{221}a^{4}+\frac{19}{13}a^{3}+\frac{15}{13}a^{2}-\frac{70}{13}a-\frac{110}{13}$, $\frac{2}{13}a^{7}+\frac{2}{13}a^{6}-\frac{7}{13}a^{5}-\frac{20}{13}a^{4}+\frac{25}{13}a^{3}+\frac{64}{13}a^{2}+\frac{29}{13}a-\frac{205}{13}$ | 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: | \( 45.5644963366 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 45.5644963366 \cdot 2}{4\cdot\sqrt{5473632256}}\cr\approx \mathstrut & 0.479930381874 \end{aligned}\]
Galois group
$C_2^3:C_4$ (as 8T19):
A solvable group of order 32 |
The 11 conjugacy class representatives for $C_2^3 : C_4 $ |
Character table for $C_2^3 : C_4 $ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 4.0.272.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 8 siblings: | data not computed |
Degree 16 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 | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.4 | $x^{8} + 12 x^{7} + 58 x^{6} + 160 x^{5} + 329 x^{4} + 500 x^{3} + 408 x^{2} + 68 x + 61$ | $4$ | $2$ | $16$ | $D_4$ | $[2, 3]^{2}$ |
\(17\) | $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.4.3.1 | $x^{4} + 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.68.2t1.a.a | $1$ | $ 2^{2} \cdot 17 $ | \(\Q(\sqrt{-17}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.17.4t1.a.a | $1$ | $ 17 $ | 4.4.4913.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.68.4t1.a.a | $1$ | $ 2^{2} \cdot 17 $ | 4.0.78608.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.68.4t1.a.b | $1$ | $ 2^{2} \cdot 17 $ | 4.0.78608.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.17.4t1.a.b | $1$ | $ 17 $ | 4.4.4913.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
* | 2.68.4t3.b.a | $2$ | $ 2^{2} \cdot 17 $ | 4.2.1156.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
2.1156.4t3.a.a | $2$ | $ 2^{2} \cdot 17^{2}$ | 4.2.19652.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 4.20123648.8t19.a.a | $4$ | $ 2^{12} \cdot 17^{3}$ | 8.0.5473632256.9 | $C_2^3 : C_4 $ (as 8T19) | $1$ | $0$ |