Normalized defining polynomial
\( x^{8} - 4x^{7} + 16x^{6} - 31x^{5} + 64x^{4} - 71x^{3} + 99x^{2} - 54x + 61 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(675710893\) \(\medspace = 11\cdot 13\cdot 59\cdot 283^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.70\) | 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))
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Galois root discriminant: | $11^{1/2}13^{1/2}59^{1/2}283^{1/2}\approx 1545.2090473460216$ | ||
Ramified primes: | \(11\), \(13\), \(59\), \(283\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{8437}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{9}a^{6}+\frac{1}{9}a^{5}-\frac{1}{9}a^{4}-\frac{4}{9}a^{3}+\frac{1}{3}a^{2}-\frac{4}{9}a+\frac{4}{9}$, $\frac{1}{27}a^{7}-\frac{2}{27}a^{5}-\frac{1}{9}a^{4}-\frac{11}{27}a^{3}+\frac{11}{27}a^{2}+\frac{8}{27}a-\frac{4}{27}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | 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: | $\frac{2}{27}a^{7}-\frac{1}{3}a^{6}+\frac{32}{27}a^{5}-\frac{23}{9}a^{4}+\frac{113}{27}a^{3}-\frac{140}{27}a^{2}+\frac{106}{27}a-\frac{89}{27}$, $\frac{1}{9}a^{7}-\frac{1}{3}a^{6}+\frac{10}{9}a^{5}-\frac{5}{3}a^{4}+\frac{25}{9}a^{3}-\frac{25}{9}a^{2}+\frac{20}{9}a-\frac{13}{9}$, $\frac{4}{27}a^{7}-\frac{1}{3}a^{6}+\frac{28}{27}a^{5}-\frac{7}{9}a^{4}+\frac{37}{27}a^{3}+\frac{17}{27}a^{2}+\frac{14}{27}a+\frac{38}{27}$ | 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: | \( 5.94498452296 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 5.94498452296 \cdot 2}{2\cdot\sqrt{675710893}}\cr\approx \mathstrut & 0.356442713915 \end{aligned}\]
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 | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | 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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(13\) | 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(59\) | 59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
59.6.0.1 | $x^{6} + 2 x^{4} + 18 x^{3} + 38 x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(283\) | $\Q_{283}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{283}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{283}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{283}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $4$ | $2$ | $2$ | $2$ |