Normalized defining polynomial
\( x^{8} - x^{7} + x^{6} + x^{2} + x + 1 \)
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: | \(13690000\) \(\medspace = 2^{4}\cdot 5^{4}\cdot 37^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(7.80\) | 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: | $2^{2/3}5^{1/2}37^{1/2}\approx 21.590988593881082$ | ||
Ramified primes: | \(2\), \(5\), \(37\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | \( -1 \) (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: | $a$, $\frac{1}{2}a^{7}-a^{6}+\frac{3}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+\frac{1}{2}$, $\frac{1}{2}a^{7}-a^{6}+\frac{3}{2}a^{5}-\frac{3}{2}a^{4}+\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$ | 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: | \( 1.81112849726 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 1.81112849726 \cdot 1}{2\cdot\sqrt{13690000}}\cr\approx \mathstrut & 0.381449471706 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 8T24):
A solvable group of order 48 |
The 10 conjugacy class representatives for $S_4\times C_2$ |
Character table for $S_4\times C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.0.3700.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.4052240.1, 6.2.109520.1 |
Degree 8 sibling: | 8.0.18741610000.2 |
Degree 12 siblings: | 12.0.11095033120000.1, 12.4.7496644000000.1, 12.4.16420649017600.1, 12.4.10262905636000000.3, 12.4.10262905636000000.4, 12.4.10262905636000000.2 |
Degree 16 sibling: | 16.0.351247945392100000000.1 |
Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
Minimal sibling: | 6.2.109520.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(5\) | 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(37\) | 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.37.2t1.a.a | $1$ | $ 37 $ | \(\Q(\sqrt{37}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.185.2t1.a.a | $1$ | $ 5 \cdot 37 $ | \(\Q(\sqrt{185}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.3700.6t3.c.a | $2$ | $ 2^{2} \cdot 5^{2} \cdot 37 $ | 6.6.101306000.1 | $D_{6}$ (as 6T3) | $1$ | $2$ | |
2.148.3t2.a.a | $2$ | $ 2^{2} \cdot 37 $ | 3.3.148.1 | $S_3$ (as 3T2) | $1$ | $2$ | |
* | 3.3700.4t5.a.a | $3$ | $ 2^{2} \cdot 5^{2} \cdot 37 $ | 4.0.3700.1 | $S_4$ (as 4T5) | $1$ | $-1$ |
3.27380.6t11.b.a | $3$ | $ 2^{2} \cdot 5 \cdot 37^{2}$ | 8.0.13690000.1 | $S_4\times C_2$ (as 8T24) | $1$ | $-1$ | |
* | 3.740.6t11.b.a | $3$ | $ 2^{2} \cdot 5 \cdot 37 $ | 8.0.13690000.1 | $S_4\times C_2$ (as 8T24) | $1$ | $-1$ |
3.136900.6t8.a.a | $3$ | $ 2^{2} \cdot 5^{2} \cdot 37^{2}$ | 4.0.3700.1 | $S_4$ (as 4T5) | $1$ | $-1$ |