Normalized defining polynomial
\( x^{8} + 10x^{6} - 12x^{5} + 36x^{4} - 68x^{3} + 92x^{2} - 96x + 90 \)
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: | \(4297326592\) \(\medspace = 2^{18}\cdot 13^{2}\cdot 97\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.00\) | 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: | $2^{9/4}13^{1/2}97^{1/2}\approx 168.91765104464235$ | ||
Ramified primes: | \(2\), \(13\), \(97\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{97}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{6}+\frac{1}{9}a^{5}+\frac{1}{9}a^{4}+\frac{1}{3}a^{3}-\frac{1}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{27}a^{7}-\frac{1}{27}a^{6}+\frac{2}{27}a^{5}+\frac{4}{27}a^{4}-\frac{4}{27}a^{3}-\frac{10}{27}a^{2}+\frac{1}{9}a+\frac{1}{3}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
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: | \( -\frac{1}{9} a^{6} + \frac{2}{9} a^{5} - \frac{7}{9} a^{4} + 2 a^{3} - \frac{26}{9} a^{2} + 4 a - 3 \) (order $4$) | 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: | $\frac{1}{27}a^{7}-\frac{1}{27}a^{6}+\frac{2}{27}a^{5}-\frac{5}{27}a^{4}-\frac{4}{27}a^{3}+\frac{35}{27}a^{2}-\frac{14}{9}a+\frac{1}{3}$, $\frac{1}{27}a^{7}-\frac{1}{27}a^{6}+\frac{2}{27}a^{5}-\frac{32}{27}a^{4}-\frac{31}{27}a^{3}-\frac{127}{27}a^{2}-\frac{14}{9}a-\frac{11}{3}$, $\frac{20}{27}a^{7}-\frac{8}{27}a^{6}+\frac{151}{27}a^{5}-\frac{313}{27}a^{4}+\frac{460}{27}a^{3}-\frac{1040}{27}a^{2}+\frac{494}{9}a-\frac{73}{3}$ | 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: | \( 199.942087266 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 199.942087266 \cdot 1}{4\cdot\sqrt{4297326592}}\cr\approx \mathstrut & 1.18840514472 \end{aligned}\]
Galois group
$C_2\wr D_4$ (as 8T35):
A solvable group of order 128 |
The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ |
Character table for $C_2 \wr C_2\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 4.0.832.1 |
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 | R | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.18.61 | $x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{3} + 2$ | $8$ | $1$ | $18$ | $C_2^3 : C_4 $ | $[2, 2, 3]^{4}$ |
\(13\) | 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(97\) | $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
97.2.1.1 | $x^{2} + 97$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |