Normalized defining polynomial
\( x^{8} - 3x^{7} + x^{6} + 7x^{5} - 16x^{4} + 7x^{3} + x^{2} - 3x + 1 \)
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: | \(312050000\) \(\medspace = 2^{4}\cdot 5^{5}\cdot 79^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.53\) | 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: | $2\cdot 5^{3/4}79^{1/2}\approx 59.438938453253584$ | ||
Ramified primes: | \(2\), \(5\), \(79\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{8}a^{6}+\frac{1}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{8}a+\frac{1}{8}$, $\frac{1}{8}a^{7}+\frac{3}{8}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{3}{8}a^{2}-\frac{1}{8}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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)
| |
Fundamental units: | $\frac{7}{8}a^{7}-\frac{17}{8}a^{6}-\frac{1}{2}a^{5}+\frac{25}{4}a^{4}-\frac{21}{2}a^{3}-\frac{9}{8}a^{2}+\frac{23}{8}a-1$, $\frac{1}{8}a^{7}-\frac{5}{8}a^{5}+\frac{1}{4}a^{4}+\frac{3}{4}a^{3}-\frac{19}{8}a^{2}-2a+\frac{7}{8}$, $a$, $\frac{15}{8}a^{7}-\frac{41}{8}a^{6}+\frac{1}{2}a^{5}+\frac{53}{4}a^{4}-\frac{53}{2}a^{3}+\frac{47}{8}a^{2}+\frac{31}{8}a-5$, $\frac{7}{8}a^{7}-\frac{17}{8}a^{6}-\frac{1}{2}a^{5}+\frac{25}{4}a^{4}-\frac{21}{2}a^{3}-\frac{9}{8}a^{2}+\frac{15}{8}a-2$ | 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: | \( 17.6162768168 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 17.6162768168 \cdot 1}{2\cdot\sqrt{312050000}}\cr\approx \mathstrut & 0.314957356228 \end{aligned}\]
Galois group
$C_2\wr C_4$ (as 8T28):
A solvable group of order 64 |
The 13 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Character table for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.2.1975.1 |
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: | 8.6.316000000.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(79\) | 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |