Normalized defining polynomial
\( x^{8} - x^{7} + 6x^{6} + 9x^{4} + 5x^{3} + 29x^{2} + 23x + 19 \)
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: | \(94139253\) \(\medspace = 3^{4}\cdot 13^{3}\cdot 23^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(9.92\) | 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: | $3^{1/2}13^{3/4}23^{1/2}\approx 56.86984696736222$ | ||
Ramified primes: | \(3\), \(13\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1643}a^{7}+\frac{462}{1643}a^{6}+\frac{322}{1643}a^{5}-\frac{427}{1643}a^{4}-\frac{532}{1643}a^{3}+\frac{139}{1643}a^{2}+\frac{309}{1643}a+\frac{149}{1643}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{9}{53} a^{7} - \frac{29}{53} a^{6} + \frac{89}{53} a^{5} - \frac{133}{53} a^{4} + \frac{141}{53} a^{3} - \frac{74}{53} a^{2} + \frac{131}{53} a - \frac{143}{53} \) (order $6$) | 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: | $\frac{415}{1643}a^{7}-\frac{501}{1643}a^{6}+\frac{2190}{1643}a^{5}+\frac{239}{1643}a^{4}+\frac{1025}{1643}a^{3}+\frac{3466}{1643}a^{2}+\frac{8296}{1643}a+\frac{7616}{1643}$, $\frac{411}{1643}a^{7}-\frac{706}{1643}a^{6}+\frac{2545}{1643}a^{5}-\frac{1339}{1643}a^{4}+\frac{1510}{1643}a^{3}+\frac{1267}{1643}a^{2}+\frac{5417}{1643}a+\frac{2091}{1643}$, $\frac{226}{1643}a^{7}-\frac{740}{1643}a^{6}+\frac{2123}{1643}a^{5}-\frac{2851}{1643}a^{4}+\frac{2993}{1643}a^{3}-\frac{1446}{1643}a^{2}+\frac{4114}{1643}a-\frac{2472}{1643}$ | 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: | \( 11.5614305416 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 11.5614305416 \cdot 1}{6\cdot\sqrt{94139253}}\cr\approx \mathstrut & 0.309524067244 \end{aligned}\]
Galois group
A solvable group of order 64 |
The 16 conjugacy class representatives for $(C_4^2 : C_2):C_2$ |
Character table for $(C_4^2 : C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.117.1 |
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 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 | ${\href{/padicField/2.8.0.1}{8} }$ | R | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | R | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(13\) | 13.4.3.3 | $x^{4} + 26$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(23\) | 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
23.4.2.2 | $x^{4} - 483 x^{2} + 2645$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |