Normalized defining polynomial
\( x^{8} - 2x^{7} + x^{6} + x^{5} + 7x^{4} - 14x^{3} + 7x^{2} + 7x + 1 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-3157114563\) \(\medspace = -\,3^{6}\cdot 163^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.40\) | 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: | $3^{6/7}163^{1/2}\approx 32.7382060432246$ | ||
Ramified primes: | \(3\), \(163\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-163}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{15}a^{7}+\frac{2}{15}a^{6}-\frac{2}{5}a^{5}+\frac{7}{15}a^{4}+\frac{1}{3}a^{3}+\frac{2}{5}a^{2}+\frac{1}{15}a-\frac{4}{15}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $4$ | 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: | $\frac{14}{15}a^{7}-\frac{32}{15}a^{6}+\frac{7}{5}a^{5}+\frac{8}{15}a^{4}+\frac{20}{3}a^{3}-\frac{77}{5}a^{2}+\frac{149}{15}a+\frac{64}{15}$, $\frac{17}{15}a^{7}-\frac{41}{15}a^{6}+\frac{11}{5}a^{5}+\frac{14}{15}a^{4}+\frac{23}{3}a^{3}-\frac{96}{5}a^{2}+\frac{227}{15}a+\frac{97}{15}$, $a$, $\frac{3}{5}a^{7}-\frac{9}{5}a^{6}+\frac{7}{5}a^{5}+\frac{1}{5}a^{4}+4a^{3}-\frac{62}{5}a^{2}+\frac{48}{5}a+\frac{8}{5}$ | 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: | \( 26.3306012182 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 26.3306012182 \cdot 3}{2\cdot\sqrt{3157114563}}\cr\approx \mathstrut & 0.697439371567 \end{aligned}\]
Galois group
$\PGL(2,7)$ (as 8T43):
A non-solvable group of order 336 |
The 9 conjugacy class representatives for $\PGL(2,7)$ |
Character table for $\PGL(2,7)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 14 sibling: | deg 14 |
Degree 16 sibling: | deg 16 |
Degree 21 sibling: | deg 21 |
Degree 24 sibling: | deg 24 |
Degree 28 siblings: | deg 28, deg 28 |
Degree 42 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 | ${\href{/padicField/2.8.0.1}{8} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.7.6.1 | $x^{7} + 3$ | $7$ | $1$ | $6$ | $F_7$ | $[\ ]_{7}^{6}$ | |
\(163\) | $\Q_{163}$ | $x + 161$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{163}$ | $x + 161$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
163.6.3.2 | $x^{6} + 503 x^{4} + 322 x^{3} + 79756 x^{2} - 155204 x + 3992689$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |