Normalized defining polynomial
\( x^{8} - 2x^{6} + 8x^{4} - 10x^{2} + 5 \)
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: | \(131072000\) \(\medspace = 2^{20}\cdot 5^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.34\) | 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: | $2^{23/8}5^{3/4}\approx 24.529502540692157$ | ||
Ramified primes: | \(2\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11}a^{6}-\frac{5}{11}a^{4}+\frac{1}{11}a^{2}-\frac{2}{11}$, $\frac{1}{11}a^{7}-\frac{5}{11}a^{5}+\frac{1}{11}a^{3}-\frac{2}{11}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{3}{11} a^{6} + \frac{4}{11} a^{4} - \frac{25}{11} a^{2} + \frac{17}{11} \) (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: | $\frac{6}{11}a^{6}-\frac{8}{11}a^{4}+\frac{39}{11}a^{2}-\frac{34}{11}$, $\frac{3}{11}a^{7}-\frac{3}{11}a^{6}-\frac{4}{11}a^{5}+\frac{4}{11}a^{4}+\frac{25}{11}a^{3}-\frac{25}{11}a^{2}-\frac{17}{11}a+\frac{28}{11}$, $\frac{6}{11}a^{7}-\frac{3}{11}a^{6}-\frac{8}{11}a^{5}+\frac{4}{11}a^{4}+\frac{39}{11}a^{3}-\frac{25}{11}a^{2}-\frac{34}{11}a+\frac{28}{11}$ | 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: | \( 18.510095453 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 18.510095453 \cdot 1}{4\cdot\sqrt{131072000}}\cr\approx \mathstrut & 0.62996029441 \end{aligned}\]
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
\(\Q(\sqrt{-1}) \), 4.0.320.1 |
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 | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.20.99 | $x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{2} + 8 x + 10$ | $8$ | $1$ | $20$ | $C_4\wr C_2$ | $[2, 2, 3, 7/2]^{2}$ |
\(5\) | 5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.40.4t1.a.a | $1$ | $ 2^{3} \cdot 5 $ | 4.4.8000.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.40.4t1.b.a | $1$ | $ 2^{3} \cdot 5 $ | 4.0.8000.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.40.4t1.a.b | $1$ | $ 2^{3} \cdot 5 $ | 4.4.8000.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.40.4t1.b.b | $1$ | $ 2^{3} \cdot 5 $ | 4.0.8000.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
2.1600.4t3.a.a | $2$ | $ 2^{6} \cdot 5^{2}$ | 4.2.8000.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 2.80.4t3.c.a | $2$ | $ 2^{4} \cdot 5 $ | 4.2.400.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
2.3200.8t17.a.a | $2$ | $ 2^{7} \cdot 5^{2}$ | 8.0.131072000.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ | |
2.3200.8t17.a.b | $2$ | $ 2^{7} \cdot 5^{2}$ | 8.0.131072000.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ | |
* | 2.640.8t17.a.a | $2$ | $ 2^{7} \cdot 5 $ | 8.0.131072000.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |
* | 2.640.8t17.a.b | $2$ | $ 2^{7} \cdot 5 $ | 8.0.131072000.1 | $C_4\wr C_2$ (as 8T17) | $0$ | $0$ |