Normalized defining polynomial
\( x^{8} - 2x^{7} - 2x^{6} + 6x^{5} + 66x^{3} + 68x^{2} + 68x - 79 \)
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)
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Discriminant: | \(-393660000000\) \(\medspace = -\,2^{8}\cdot 3^{9}\cdot 5^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(28.14\) | 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: | $2^{31/28}3^{37/18}5^{71/60}\approx 138.4034641689339$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-15}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}+\frac{1}{5}$, $\frac{1}{15}a^{6}+\frac{2}{5}a+\frac{1}{3}$, $\frac{1}{31305}a^{7}+\frac{119}{31305}a^{6}+\frac{125}{2087}a^{5}+\frac{99}{2087}a^{4}-\frac{543}{2087}a^{3}-\frac{5008}{10435}a^{2}-\frac{2146}{31305}a+\frac{1925}{6261}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $\frac{5243}{31305}a^{7}-\frac{2119}{10435}a^{6}-\frac{5976}{10435}a^{5}+\frac{1481}{2087}a^{4}+\frac{1806}{2087}a^{3}+\frac{112301}{10435}a^{2}+\frac{619378}{31305}a+\frac{247729}{10435}$, $\frac{1066}{31305}a^{7}-\frac{508}{6261}a^{6}-\frac{318}{2087}a^{5}+\frac{1184}{2087}a^{4}-\frac{739}{2087}a^{3}+\frac{14627}{10435}a^{2}+\frac{19561}{6261}a-\frac{43298}{6261}$, $\frac{968}{6261}a^{7}-\frac{6313}{31305}a^{6}-\frac{5324}{10435}a^{5}+\frac{1237}{2087}a^{4}+\frac{1500}{2087}a^{3}+\frac{21227}{2087}a^{2}+\frac{488717}{31305}a+\frac{585433}{31305}$, $\frac{8789}{31305}a^{7}-\frac{2861}{6261}a^{6}-\frac{1224}{2087}a^{5}+\frac{1919}{2087}a^{4}+\frac{2629}{2087}a^{3}+\frac{187348}{10435}a^{2}+\frac{183461}{6261}a+\frac{212390}{6261}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 3136.20698725 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 3136.20698725 \cdot 1}{2\cdot\sqrt{393660000000}}\cr\approx \mathstrut & 2.47978297905 \end{aligned}\]
Galois group
A non-solvable group of order 40320 |
The 22 conjugacy class representatives for $S_8$ |
Character table for $S_8$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 16 sibling: | deg 16 |
Degree 28 sibling: | deg 28 |
Degree 30 sibling: | deg 30 |
Degree 35 sibling: | deg 35 |
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 | R | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.13 | $x^{8} + 2 x + 2$ | $8$ | $1$ | $8$ | $C_2^3:(C_7: C_3)$ | $[8/7, 8/7, 8/7]_{7}^{3}$ |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.3.3.2 | $x^{3} + 3 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
3.3.5.1 | $x^{3} + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
\(5\) | 5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
5.5.5.2 | $x^{5} + 5 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ |