Normalized defining polynomial
\( x^{8} - 2x^{7} - 8x^{6} - 14x^{4} - 40x^{3} + 48x^{2} + 144x + 76 \)
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)
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Discriminant: | \(5184000000\) \(\medspace = 2^{12}\cdot 3^{4}\cdot 5^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(16.38\) | 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: | $2^{3/2}3^{1/2}5^{3/4}\approx 16.3807251762544$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{10}a^{6}+\frac{1}{5}a^{5}+\frac{1}{10}a^{4}-\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{2}{5}$, $\frac{1}{50}a^{7}-\frac{4}{25}a^{5}+\frac{9}{50}a^{4}+\frac{2}{25}a^{3}+\frac{9}{25}a^{2}-\frac{8}{25}a+\frac{6}{25}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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)
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Fundamental units: | $\frac{3}{10}a^{7}-\frac{4}{5}a^{6}-2a^{5}+\frac{19}{10}a^{4}-\frac{28}{5}a^{3}-\frac{42}{5}a^{2}+\frac{116}{5}a+\frac{132}{5}$, $\frac{57}{50}a^{7}-\frac{37}{10}a^{6}-\frac{113}{25}a^{5}+\frac{139}{25}a^{4}-\frac{566}{25}a^{3}-\frac{417}{25}a^{2}+\frac{1894}{25}a+\frac{1747}{25}$, $\frac{81}{50}a^{7}-\frac{27}{5}a^{6}-\frac{144}{25}a^{5}+\frac{192}{25}a^{4}-\frac{823}{25}a^{3}-\frac{531}{25}a^{2}+\frac{2652}{25}a+\frac{2321}{25}$, $\frac{137}{50}a^{7}-9a^{6}-\frac{521}{50}a^{5}+\frac{683}{50}a^{4}-\frac{1401}{25}a^{3}-\frac{942}{25}a^{2}+\frac{4529}{25}a+\frac{4022}{25}$, $\frac{7}{50}a^{7}-\frac{7}{10}a^{6}+\frac{12}{25}a^{5}+\frac{14}{25}a^{4}-\frac{91}{25}a^{3}+\frac{108}{25}a^{2}+\frac{169}{25}a-\frac{53}{25}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 47.495360053 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 47.495360053 \cdot 2}{2\cdot\sqrt{5184000000}}\cr\approx \mathstrut & 0.41667592410 \end{aligned}\]
Galois group
$C_2^2:C_4$ (as 8T10):
A solvable group of order 16 |
The 10 conjugacy class representatives for $C_2^2:C_4$ |
Character table for $C_2^2:C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 4.2.18000.1, 4.2.3600.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 16.0.26873856000000000000.4 |
Degree 8 sibling: | 8.0.324000000.5 |
Minimal sibling: | 8.0.324000000.5 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.15 | $x^{8} + 4 x^{7} + 4 x^{6} + 12$ | $4$ | $2$ | $12$ | $C_2^2:C_4$ | $[2, 2]^{4}$ |
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(5\) | 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |