Normalized defining polynomial
\( x^{8} - x^{7} - 9x^{6} + 5x^{5} + 34x^{4} - 22x^{3} - 39x^{2} + 21x + 17 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(713512901\) \(\medspace = 59\cdot 151\cdot 283^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.78\) | 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: | $59^{1/2}151^{1/2}283^{1/2}\approx 1587.843506142844$ | ||
Ramified primes: | \(59\), \(151\), \(283\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{8909}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2473}a^{7}+\frac{113}{2473}a^{6}+\frac{508}{2473}a^{5}+\frac{1038}{2473}a^{4}-\frac{338}{2473}a^{3}+\frac{1014}{2473}a^{2}-\frac{674}{2473}a-\frac{152}{2473}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{436}{2473}a^{7}-\frac{192}{2473}a^{6}-\frac{3555}{2473}a^{5}+\frac{9}{2473}a^{4}+\frac{10904}{2473}a^{3}-\frac{3036}{2473}a^{2}-\frac{6996}{2473}a+\frac{499}{2473}$, $\frac{197}{2473}a^{7}+\frac{4}{2473}a^{6}-\frac{1317}{2473}a^{5}-\frac{773}{2473}a^{4}+\frac{2658}{2473}a^{3}-\frac{555}{2473}a^{2}+\frac{764}{2473}a-\frac{2741}{2473}$, $\frac{611}{2473}a^{7}-\frac{201}{2473}a^{6}-\frac{6156}{2473}a^{5}-\frac{1343}{2473}a^{4}+\frac{23471}{2473}a^{3}+\frac{6250}{2473}a^{2}-\frac{26026}{2473}a-\frac{8790}{2473}$, $\frac{451}{2473}a^{7}-\frac{970}{2473}a^{6}-\frac{3354}{2473}a^{5}+\frac{5687}{2473}a^{4}+\frac{10780}{2473}a^{3}-\frac{19975}{2473}a^{2}+\frac{205}{2473}a+\frac{8111}{2473}$, $\frac{943}{2473}a^{7}+\frac{220}{2473}a^{6}-\frac{8137}{2473}a^{5}-\frac{5420}{2473}a^{4}+\frac{25013}{2473}a^{3}+\frac{11516}{2473}a^{2}-\frac{19805}{2473}a-\frac{9794}{2473}$ | 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: | \( 25.7875819228 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 25.7875819228 \cdot 1}{2\cdot\sqrt{713512901}}\cr\approx \mathstrut & 0.304901407640 \end{aligned}\]
Galois group
$C_2\wr S_4$ (as 8T44):
A solvable group of order 384 |
The 20 conjugacy class representatives for $C_2 \wr S_4$ |
Character table for $C_2 \wr S_4$ |
Intermediate fields
4.2.283.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 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 | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | R |
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 |
---|---|---|---|---|---|---|---|
\(59\) | 59.2.1.1 | $x^{2} + 118$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
59.6.0.1 | $x^{6} + 2 x^{4} + 18 x^{3} + 38 x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(151\) | 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
151.6.0.1 | $x^{6} + 125 x^{3} + 18 x^{2} + 15 x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(283\) | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |