Normalized defining polynomial
\( x^{8} - 2x^{7} - 5x^{6} + 3x^{5} + 34x^{4} - 33x^{3} - 45x^{2} + 27x + 81 \)
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: | \(268140625\) \(\medspace = 5^{6}\cdot 131^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.31\) | 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: | $5^{3/4}131^{1/2}\approx 38.2704131838469$ | ||
Ramified primes: | \(5\), \(131\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{198}a^{6}+\frac{13}{198}a^{5}-\frac{53}{198}a^{4}+\frac{1}{22}a^{3}-\frac{10}{99}a^{2}+\frac{3}{11}a+\frac{9}{22}$, $\frac{1}{594}a^{7}+\frac{1}{594}a^{6}-\frac{1}{54}a^{5}-\frac{49}{198}a^{4}+\frac{35}{297}a^{3}+\frac{16}{99}a^{2}-\frac{19}{66}a+\frac{4}{11}$
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)
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Torsion generator: | \( -\frac{1}{66} a^{7} + \frac{1}{33} a^{5} + \frac{1}{11} a^{4} - \frac{17}{66} a^{3} + \frac{8}{33} a^{2} + \frac{5}{66} a + \frac{21}{22} \) (order $10$) | 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: | $\frac{7}{594}a^{7}-\frac{17}{594}a^{6}+\frac{7}{594}a^{5}+\frac{5}{66}a^{4}+\frac{38}{297}a^{3}-\frac{35}{33}a^{2}+\frac{31}{66}a+\frac{14}{11}$, $\frac{16}{297}a^{7}-\frac{8}{297}a^{6}-\frac{92}{297}a^{5}-\frac{10}{33}a^{4}+\frac{409}{297}a^{3}+\frac{26}{33}a^{2}-\frac{41}{33}a-\frac{21}{11}$, $\frac{155}{594}a^{7}-\frac{8}{297}a^{6}-\frac{380}{297}a^{5}-\frac{175}{99}a^{4}+\frac{2975}{594}a^{3}+\frac{80}{99}a^{2}-\frac{47}{6}a-\frac{197}{22}$ | 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: | \( 40.8873841353 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 40.8873841353 \cdot 1}{10\cdot\sqrt{268140625}}\cr\approx \mathstrut & 0.389159369612 \end{aligned}\]
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
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.2.3275.1, 4.2.16375.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 16 |
Degree 8 sibling: | 8.4.4601561265625.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.1.0.1}{1} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
\(131\) | 131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
131.2.1.1 | $x^{2} + 262$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
131.2.1.1 | $x^{2} + 262$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |