Normalized defining polynomial
\( x^{8} - 2x^{7} + 11x^{6} - 39x^{5} + 87x^{4} - 178x^{3} + 108x^{2} + 56x + 5 \)
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: | \(6529513515605\) \(\medspace = 5\cdot 1069^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(39.98\) | 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: | $5^{1/2}1069^{1/2}\approx 73.10950690573696$ | ||
Ramified primes: | \(5\), \(1069\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{9}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{2}-\frac{4}{9}a+\frac{4}{9}$, $\frac{1}{5985}a^{7}+\frac{307}{5985}a^{6}+\frac{148}{1995}a^{5}-\frac{277}{665}a^{4}+\frac{121}{399}a^{3}-\frac{1933}{5985}a^{2}-\frac{47}{105}a-\frac{232}{1197}$
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: | $\frac{484}{1995}a^{7}-\frac{1037}{1995}a^{6}+\frac{1807}{665}a^{5}-\frac{6529}{665}a^{4}+\frac{2970}{133}a^{3}-\frac{91687}{1995}a^{2}+\frac{1122}{35}a+\frac{4220}{399}$, $\frac{484}{1995}a^{7}-\frac{1037}{1995}a^{6}+\frac{1807}{665}a^{5}-\frac{6529}{665}a^{4}+\frac{2970}{133}a^{3}-\frac{91687}{1995}a^{2}+\frac{1122}{35}a+\frac{3821}{399}$, $\frac{73}{5985}a^{7}-\frac{199}{5985}a^{6}+\frac{164}{1995}a^{5}-\frac{271}{665}a^{4}+\frac{454}{399}a^{3}-\frac{1459}{5985}a^{2}-\frac{493}{315}a-\frac{311}{1197}$, $\frac{2671}{5985}a^{7}-\frac{5933}{5985}a^{6}+\frac{10273}{1995}a^{5}-\frac{12357}{665}a^{4}+\frac{17158}{399}a^{3}-\frac{536638}{5985}a^{2}+\frac{7183}{105}a+\frac{12344}{1197}$, $\frac{520}{171}a^{7}-\frac{1138}{171}a^{6}+\frac{1853}{57}a^{5}-\frac{2319}{19}a^{4}+\frac{15749}{57}a^{3}-\frac{89512}{171}a^{2}+\frac{3071}{9}a+\frac{19772}{171}$ | 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: | \( 5071.00887402 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 5071.00887402 \cdot 1}{2\cdot\sqrt{6529513515605}}\cr\approx \mathstrut & 0.626763477729 \end{aligned}\]
Galois group
$S_4\wr C_2$ (as 8T47):
A solvable group of order 1152 |
The 20 conjugacy class representatives for $S_4\wr C_2$ |
Character table for $S_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{1069}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 36 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 | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(1069\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $2$ | $3$ | $3$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.5345.2t1.a.a | $1$ | $ 5 \cdot 1069 $ | \(\Q(\sqrt{5345}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.1069.2t1.a.a | $1$ | $ 1069 $ | \(\Q(\sqrt{1069}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.5345.4t3.a.a | $2$ | $ 5 \cdot 1069 $ | 4.0.26725.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
4.26725.6t13.b.a | $4$ | $ 5^{2} \cdot 1069 $ | 6.2.133625.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.142845125.12t34.b.a | $4$ | $ 5^{3} \cdot 1069^{2}$ | 6.2.133625.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.30540287725.12t34.b.a | $4$ | $ 5^{2} \cdot 1069^{3}$ | 6.2.133625.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.5713805.6t13.b.a | $4$ | $ 5 \cdot 1069^{2}$ | 6.2.133625.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
6.763507193125.12t201.a.a | $6$ | $ 5^{4} \cdot 1069^{3}$ | 8.4.6529513515605.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
6.381...625.12t202.a.a | $6$ | $ 5^{5} \cdot 1069^{3}$ | 8.4.6529513515605.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
* | 6.6108057545.8t47.a.a | $6$ | $ 5 \cdot 1069^{3}$ | 8.4.6529513515605.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ |
6.30540287725.12t200.a.a | $6$ | $ 5^{2} \cdot 1069^{3}$ | 8.4.6529513515605.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.152701438625.16t1294.a.a | $9$ | $ 5^{3} \cdot 1069^{3}$ | 8.4.6529513515605.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.190...125.18t272.a.a | $9$ | $ 5^{6} \cdot 1069^{3}$ | 8.4.6529513515605.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.233...625.18t273.a.a | $9$ | $ 5^{6} \cdot 1069^{6}$ | 8.4.6529513515605.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.186...125.18t274.a.a | $9$ | $ 5^{3} \cdot 1069^{6}$ | 8.4.6529513515605.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
12.116...125.36t1763.a.a | $12$ | $ 5^{7} \cdot 1069^{6}$ | 8.4.6529513515605.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
12.466...125.24t2821.a.a | $12$ | $ 5^{5} \cdot 1069^{6}$ | 8.4.6529513515605.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
18.356...625.36t1758.a.a | $18$ | $ 5^{9} \cdot 1069^{9}$ | 8.4.6529513515605.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ |