Normalized defining polynomial
\( x^{8} - 4x^{7} + 6x^{6} - 6x^{4} + 4x^{3} + 2x^{2} + 9 \)
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: | \(321978368\) \(\medspace = 2^{16}\cdot 17^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.57\) | 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: | $2^{2}17^{3/4}\approx 33.48857611437076$ | ||
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: | \(\Q(\sqrt{17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{1}{2}a^{3}+\frac{1}{6}a^{2}+\frac{1}{3}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
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: | \( -1 \) (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: | $\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{1}{2}a^{5}+a^{4}+\frac{1}{2}a^{3}-\frac{5}{6}a^{2}-\frac{1}{6}a$, $\frac{1}{2}a^{4}-a^{3}+2a-\frac{1}{2}$, $\frac{1}{3}a^{7}-\frac{5}{6}a^{6}+\frac{1}{2}a^{5}+\frac{1}{2}a^{4}-\frac{1}{6}a^{2}+\frac{1}{6}a-\frac{3}{2}$ | 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: | \( 12.7171485653 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 12.7171485653 \cdot 2}{2\cdot\sqrt{321978368}}\cr\approx \mathstrut & 1.10457668211 \end{aligned}\]
Galois group
$C_4\wr C_2$ (as 8T17):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_4\wr C_2$ |
Character table for $C_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{-2}) \), 4.0.1088.1 |
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 sibling: | 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 | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.1 | $x^{8} + 8 x^{7} + 24 x^{6} + 40 x^{5} + 60 x^{4} + 32 x^{3} + 88 x^{2} + 108$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ |
\(17\) | 17.4.3.3 | $x^{4} + 51$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.136.2t1.b.a | $1$ | $ 2^{3} \cdot 17 $ | \(\Q(\sqrt{-34}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.136.4t1.b.a | $1$ | $ 2^{3} \cdot 17 $ | 4.0.314432.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.136.4t1.b.b | $1$ | $ 2^{3} \cdot 17 $ | 4.0.314432.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.17.4t1.a.a | $1$ | $ 17 $ | 4.4.4913.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.17.4t1.a.b | $1$ | $ 17 $ | 4.4.4913.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
2.2312.4t3.a.a | $2$ | $ 2^{3} \cdot 17^{2}$ | 4.0.314432.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 2.136.4t3.b.a | $2$ | $ 2^{3} \cdot 17 $ | 4.0.1088.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
* | 2.544.8t17.a.a | $2$ | $ 2^{5} \cdot 17 $ | 8.0.321978368.6 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
2.9248.8t17.a.a | $2$ | $ 2^{5} \cdot 17^{2}$ | 8.0.321978368.6 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ | |
* | 2.544.8t17.a.b | $2$ | $ 2^{5} \cdot 17 $ | 8.0.321978368.6 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
2.9248.8t17.a.b | $2$ | $ 2^{5} \cdot 17^{2}$ | 8.0.321978368.6 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |