Normalized defining polynomial
\( x^{8} - 4x^{7} - x^{6} + 17x^{5} - 5x^{4} - 23x^{3} + 6x^{2} + 9x - 1 \)
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: | \(309593125\) \(\medspace = 5^{4}\cdot 19\cdot 29^{2}\cdot 31\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.52\) | 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: | $5^{1/2}19^{1/2}29^{1/2}31^{1/2}\approx 292.24133862272123$ | ||
Ramified primes: | \(5\), \(19\), \(29\), \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{589}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$
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: | $a^{6}-3a^{5}-2a^{4}+9a^{3}+a^{2}-6a-1$, $a^{2}-a-2$, $a^{5}-3a^{4}-a^{3}+7a^{2}-a-3$, $a^{3}-2a^{2}-2a+3$, $a^{6}-3a^{5}-2a^{4}+10a^{3}-a^{2}-8a+2$, $a^{7}-3a^{6}-2a^{5}+9a^{4}+a^{3}-7a^{2}+1$, $a^{5}-2a^{4}-3a^{3}+5a^{2}+2a-2$ | 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: | \( 23.6967890277 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 23.6967890277 \cdot 1}{2\cdot\sqrt{309593125}}\cr\approx \mathstrut & 0.172386724486 \end{aligned}\]
Galois group
$C_2\wr D_4$ (as 8T35):
A solvable group of order 128 |
The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ |
Character table for $C_2 \wr C_2\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.4.725.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | data not computed |
Degree 16 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 | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }$ | R | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | R | R | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(19\) | 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(29\) | 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.145.2t1.a.a | $1$ | $ 5 \cdot 29 $ | \(\Q(\sqrt{145}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.85405.2t1.a.a | $1$ | $ 5 \cdot 19 \cdot 29 \cdot 31 $ | \(\Q(\sqrt{85405}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.589.2t1.a.a | $1$ | $ 19 \cdot 31 $ | \(\Q(\sqrt{589}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.2945.2t1.a.a | $1$ | $ 5 \cdot 19 \cdot 31 $ | \(\Q(\sqrt{2945}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.17081.2t1.a.a | $1$ | $ 19 \cdot 29 \cdot 31 $ | \(\Q(\sqrt{17081}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.85405.4t3.b.a | $2$ | $ 5 \cdot 19 \cdot 29 \cdot 31 $ | 4.4.1458802805.2 | $D_{4}$ (as 4T3) | $1$ | $2$ | |
2.85405.4t3.a.a | $2$ | $ 5 \cdot 19 \cdot 29 \cdot 31 $ | 4.4.1458802805.1 | $D_{4}$ (as 4T3) | $1$ | $2$ | |
2.2945.4t3.f.a | $2$ | $ 5 \cdot 19 \cdot 31 $ | 4.4.1734605.1 | $D_{4}$ (as 4T3) | $1$ | $2$ | |
2.2476745.4t3.a.a | $2$ | $ 5 \cdot 19 \cdot 29^{2} \cdot 31 $ | 4.4.1458802805.3 | $D_{4}$ (as 4T3) | $1$ | $2$ | |
2.50303545.4t3.a.a | $2$ | $ 5 \cdot 19^{2} \cdot 29 \cdot 31^{2}$ | 4.4.1458802805.4 | $D_{4}$ (as 4T3) | $1$ | $2$ | |
* | 2.145.4t3.a.a | $2$ | $ 5 \cdot 29 $ | 4.4.4205.1 | $D_{4}$ (as 4T3) | $1$ | $2$ |
4.148143940025.8t35.a.a | $4$ | $ 5^{2} \cdot 19^{3} \cdot 29 \cdot 31^{3}$ | 8.8.309593125.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $4$ | |
4.36470070125.8t29.b.a | $4$ | $ 5^{3} \cdot 19^{2} \cdot 29^{2} \cdot 31^{2}$ | 8.8.6134265795025.1 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $4$ | |
4.359128025.8t35.a.a | $4$ | $ 5^{2} \cdot 19 \cdot 29^{3} \cdot 31 $ | 8.8.309593125.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $4$ | |
4.1458802805.8t29.b.a | $4$ | $ 5 \cdot 19^{2} \cdot 29^{2} \cdot 31^{2}$ | 8.8.6134265795025.1 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $4$ | |
4.124...025.8t35.a.a | $4$ | $ 5^{2} \cdot 19^{3} \cdot 29^{3} \cdot 31^{3}$ | 8.8.309593125.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $4$ | |
* | 4.427025.8t35.a.a | $4$ | $ 5^{2} \cdot 19 \cdot 29 \cdot 31 $ | 8.8.309593125.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $4$ |