Normalized defining polynomial
\( x^{8} - 2x^{7} + 9x^{6} - 28x^{5} + 28x^{4} - 72x^{3} + 68x^{2} + 80x - 67 \)
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: | \(12149330176\) \(\medspace = 2^{8}\cdot 83^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.22\) | 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: | $2\cdot 83^{1/2}\approx 18.2208671582886$ | ||
Ramified primes: | \(2\), \(83\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{254623}a^{7}+\frac{70763}{254623}a^{6}-\frac{126837}{254623}a^{5}+\frac{95040}{254623}a^{4}-\frac{106294}{254623}a^{3}-\frac{76939}{254623}a^{2}+\frac{15342}{254623}a-\frac{35762}{254623}$
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: | $\frac{3573}{254623}a^{7}-\frac{4440}{254623}a^{6}+\frac{40339}{254623}a^{5}-\frac{89162}{254623}a^{4}+\frac{109054}{254623}a^{3}-\frac{419453}{254623}a^{2}+\frac{73021}{254623}a+\frac{43120}{254623}$, $\frac{313}{254623}a^{7}-\frac{3382}{254623}a^{6}+\frac{21207}{254623}a^{5}-\frac{43371}{254623}a^{4}+\frac{85591}{254623}a^{3}-\frac{147345}{254623}a^{2}-\frac{35791}{254623}a+\frac{264529}{254623}$, $\frac{1594}{254623}a^{7}-\frac{1767}{254623}a^{6}-\frac{7516}{254623}a^{5}-\frac{6925}{254623}a^{4}-\frac{108341}{254623}a^{3}+\frac{87520}{254623}a^{2}+\frac{11340}{254623}a+\frac{30924}{254623}$, $\frac{867}{254623}a^{7}-\frac{12622}{254623}a^{6}+\frac{29457}{254623}a^{5}-\frac{98172}{254623}a^{4}+\frac{271251}{254623}a^{3}-\frac{249510}{254623}a^{2}+\frac{570364}{254623}a-\frac{705517}{254623}$, $\frac{3886}{254623}a^{7}-\frac{7822}{254623}a^{6}+\frac{61546}{254623}a^{5}-\frac{132533}{254623}a^{4}+\frac{194645}{254623}a^{3}-\frac{566798}{254623}a^{2}+\frac{291853}{254623}a+\frac{307649}{254623}$ | 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: | \( 87.4535393799 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 87.4535393799 \cdot 1}{2\cdot\sqrt{12149330176}}\cr\approx \mathstrut & 0.250582620745 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 8T24):
A solvable group of order 48 |
The 10 conjugacy class representatives for $S_4\times C_2$ |
Character table for $S_4\times C_2$ |
Intermediate fields
\(\Q(\sqrt{83}) \), 4.2.1328.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.2287148.1, 6.0.27556.1 |
Degree 8 sibling: | 8.0.1763584.1 |
Degree 12 siblings: | 12.2.1008394404608.1, 12.0.194389282816.1, 12.0.5231045973904.1, 12.4.1339147769319424.1, 12.0.1339147769319424.1, 12.0.1339147769319424.3 |
Degree 16 sibling: | deg 16 |
Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
Minimal sibling: | 6.0.27556.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
\(83\) | 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |