Normalized defining polynomial
\( x^{8} - 4x^{5} - 8x^{4} + 4x^{3} + 12x^{2} + 4x + 2 \)
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)
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Discriminant: | \(6585647104\) \(\medspace = 2^{16}\cdot 317^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.88\) | 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: | $2^{59/28}317^{1/2}\approx 76.70838446092021$ | ||
Ramified primes: | \(2\), \(317\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{43}a^{7}-\frac{12}{43}a^{6}+\frac{15}{43}a^{5}-\frac{12}{43}a^{4}+\frac{7}{43}a^{3}+\frac{6}{43}a^{2}-\frac{17}{43}a-\frac{7}{43}$
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: | \( -1 \) (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: | $\frac{5}{43}a^{7}-\frac{17}{43}a^{6}-\frac{11}{43}a^{5}-\frac{60}{43}a^{4}-\frac{8}{43}a^{3}+\frac{116}{43}a^{2}+\frac{130}{43}a+\frac{51}{43}$, $\frac{13}{43}a^{7}+\frac{16}{43}a^{6}+\frac{23}{43}a^{5}-\frac{27}{43}a^{4}-\frac{124}{43}a^{3}-\frac{94}{43}a^{2}-\frac{6}{43}a-\frac{5}{43}$, $\frac{2}{43}a^{7}+\frac{19}{43}a^{6}-\frac{13}{43}a^{5}-\frac{24}{43}a^{4}-\frac{115}{43}a^{3}-\frac{117}{43}a^{2}+\frac{224}{43}a+\frac{287}{43}$ | 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: | \( 105.615024891 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 105.615024891 \cdot 1}{2\cdot\sqrt{6585647104}}\cr\approx \mathstrut & 1.01418213468 \end{aligned}\]
Galois group
$C_2^3:\GL(3,2)$ (as 8T48):
A non-solvable group of order 1344 |
The 11 conjugacy class representatives for $C_2^3:\GL(3,2)$ |
Character table for $C_2^3:\GL(3,2)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 8 sibling: | 8.4.661785091833856.1 |
Degree 14 siblings: | deg 14, deg 14 |
Degree 28 siblings: | deg 28, deg 28, deg 28 |
Degree 42 siblings: | deg 42, deg 42, deg 42, deg 42 |
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 | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.62 | $x^{8} + 4 x^{2} + 4 x + 2$ | $8$ | $1$ | $16$ | $C_2^3:(C_7: C_3)$ | $[16/7, 16/7, 16/7]_{7}^{3}$ |
\(317\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |