Normalized defining polynomial
\( x^{8} - 5824x - 5096 \)
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: | \(-1865133024596601830654345216\) \(\medspace = -\,2^{31}\cdot 7^{12}\cdot 13^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(2563.53\) | 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: | $2^{67/16}7^{12/7}13^{7/8}\approx 4830.621803680782$ | ||
Ramified primes: | \(2\), \(7\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-26}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{756}a^{7}-\frac{1}{27}a^{6}+\frac{1}{27}a^{5}-\frac{1}{27}a^{4}+\frac{1}{27}a^{3}-\frac{1}{27}a^{2}+\frac{1}{27}a+\frac{7}{27}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
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: | $\frac{78\!\cdots\!30}{63}a^{7}-\frac{51\!\cdots\!36}{9}a^{6}-\frac{16\!\cdots\!84}{9}a^{5}-\frac{67\!\cdots\!78}{9}a^{4}-\frac{10\!\cdots\!84}{9}a^{3}+\frac{37\!\cdots\!64}{9}a^{2}+\frac{30\!\cdots\!60}{9}a+\frac{23\!\cdots\!31}{9}$, $\frac{90\!\cdots\!24}{189}a^{7}-\frac{40\!\cdots\!05}{27}a^{6}+\frac{18\!\cdots\!85}{54}a^{5}-\frac{23\!\cdots\!87}{54}a^{4}-\frac{35\!\cdots\!25}{27}a^{3}+\frac{34\!\cdots\!42}{27}a^{2}-\frac{16\!\cdots\!86}{27}a-\frac{16\!\cdots\!77}{27}$, $\frac{10\!\cdots\!32}{189}a^{7}+\frac{10\!\cdots\!01}{54}a^{6}+\frac{63\!\cdots\!41}{27}a^{5}-\frac{78\!\cdots\!07}{54}a^{4}-\frac{25\!\cdots\!52}{27}a^{3}-\frac{67\!\cdots\!02}{27}a^{2}+\frac{14\!\cdots\!68}{27}a+\frac{49\!\cdots\!55}{27}$, $\frac{42\!\cdots\!77}{27}a^{7}-\frac{28\!\cdots\!07}{27}a^{6}+\frac{24\!\cdots\!63}{54}a^{5}-\frac{83\!\cdots\!05}{54}a^{4}+\frac{10\!\cdots\!41}{27}a^{3}-\frac{18\!\cdots\!08}{27}a^{2}-\frac{97\!\cdots\!76}{27}a+\frac{16\!\cdots\!65}{27}$ (assuming GRH) | 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: | \( 16760064929.9 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 16760064929.9 \cdot 4}{2\cdot\sqrt{1865133024596601830654345216}}\cr\approx \mathstrut & 0.770105779861 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 40320 |
The 22 conjugacy class representatives for $S_8$ |
Character table for $S_8$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 16 sibling: | data not computed |
Degree 28 sibling: | data not computed |
Degree 30 sibling: | data not computed |
Degree 35 sibling: | 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 | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.31.261 | $x^{8} + 16 x^{7} + 24 x^{6} + 12 x^{4} + 8 x^{2} + 16 x + 26$ | $8$ | $1$ | $31$ | $(C_4^2 : C_2):C_2$ | $[2, 3, 7/2, 4, 5]^{2}$ |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ | |
\(13\) | 13.8.7.4 | $x^{8} + 91$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |