Normalized defining polynomial
\( x^{8} - 4x^{7} - 120x^{6} + 244x^{5} + 4901x^{4} - 1850x^{3} - 73103x^{2} - 58581x + 176255 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(4239150758955121\) \(\medspace = 8069^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(89.83\) | 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: | $8069^{1/2}\approx 89.8276126811795$ | ||
Ramified primes: | \(8069\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{775}a^{6}+\frac{59}{775}a^{5}-\frac{14}{775}a^{4}+\frac{213}{775}a^{3}-\frac{101}{775}a^{2}-\frac{1}{25}a-\frac{39}{155}$, $\frac{1}{26090375}a^{7}+\frac{2197}{5218075}a^{6}-\frac{432641}{5218075}a^{5}+\frac{238624}{26090375}a^{4}+\frac{11523287}{26090375}a^{3}+\frac{8600993}{26090375}a^{2}+\frac{8484474}{26090375}a-\frac{2234809}{5218075}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{6}$, which has order $6$
Unit group
Rank: | $7$ | 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: | $\frac{64654}{26090375}a^{7}-\frac{102258}{5218075}a^{6}-\frac{1190508}{5218075}a^{5}+\frac{39320616}{26090375}a^{4}+\frac{183052633}{26090375}a^{3}-\frac{826881473}{26090375}a^{2}-\frac{2007042099}{26090375}a+\frac{667338584}{5218075}$, $\frac{64654}{26090375}a^{7}-\frac{102258}{5218075}a^{6}-\frac{1190508}{5218075}a^{5}+\frac{39320616}{26090375}a^{4}+\frac{183052633}{26090375}a^{3}-\frac{826881473}{26090375}a^{2}-\frac{2007042099}{26090375}a+\frac{672556659}{5218075}$, $\frac{85646}{26090375}a^{7}-\frac{130816}{5218075}a^{6}-\frac{1592053}{5218075}a^{5}+\frac{49970089}{26090375}a^{4}+\frac{244436982}{26090375}a^{3}-\frac{1049018757}{26090375}a^{2}-\frac{2621076256}{26090375}a+\frac{876340496}{5218075}$, $\frac{84138}{26090375}a^{7}-\frac{137989}{5218075}a^{6}-\frac{1536528}{5218075}a^{5}+\frac{53572437}{26090375}a^{4}+\frac{236877931}{26090375}a^{3}-\frac{1130367966}{26090375}a^{2}-\frac{2637640988}{26090375}a+\frac{888382383}{5218075}$, $\frac{95179}{26090375}a^{7}-\frac{133178}{5218075}a^{6}-\frac{1787483}{5218075}a^{5}+\frac{49658316}{26090375}a^{4}+\frac{270501458}{26090375}a^{3}-\frac{1020399298}{26090375}a^{2}-\frac{2742938299}{26090375}a+\frac{876094584}{5218075}$, $\frac{7396}{208723}a^{7}-\frac{1391142}{5218075}a^{6}-\frac{17289513}{5218075}a^{5}+\frac{106111758}{5218075}a^{4}+\frac{532873349}{5218075}a^{3}-\frac{71692093}{168325}a^{2}-\frac{5703087938}{5218075}a+\frac{59736388}{33665}$, $\frac{103989}{26090375}a^{7}-\frac{209046}{5218075}a^{6}-\frac{318934}{1043615}a^{5}+\frac{83861141}{26090375}a^{4}+\frac{179489433}{26090375}a^{3}-\frac{2000639028}{26090375}a^{2}-\frac{1250233669}{26090375}a+\frac{2749321154}{5218075}$ | 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: | \( 31689.4416637 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 31689.4416637 \cdot 6}{2\cdot\sqrt{4239150758955121}}\cr\approx \mathstrut & 0.373797486297 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
\(\Q(\sqrt{8069}) \), 4.4.8069.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 4 sibling: | 4.4.8069.1 |
Degree 6 siblings: | 6.6.525362592509.2, 6.6.65108761.1 |
Degree 12 siblings: | deg 12, deg 12 |
Minimal sibling: | 4.4.8069.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(8069\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |