Normalized defining polynomial
\( x^{8} - 2x^{7} + x^{6} - 12x^{5} - 17x^{4} + 29x^{3} + 36x^{2} + 174x + 61 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(127027375281\) \(\medspace = 3^{4}\cdot 199^{4}\) | 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: | \(24.43\) | 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: | $3^{1/2}199^{1/2}\approx 24.43358344574123$ | ||
Ramified primes: | \(3\), \(199\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}+\frac{2}{7}a^{5}+\frac{3}{7}a^{4}+\frac{2}{7}a^{3}+\frac{1}{7}a^{2}+\frac{2}{7}$, $\frac{1}{72331}a^{7}+\frac{4125}{72331}a^{6}-\frac{35907}{72331}a^{5}-\frac{33647}{72331}a^{4}-\frac{26998}{72331}a^{3}-\frac{1473}{10333}a^{2}-\frac{12500}{72331}a-\frac{2189}{10333}$
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)
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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: | $\frac{44}{10333}a^{7}-\frac{459}{72331}a^{6}-\frac{3046}{72331}a^{5}+\frac{723}{72331}a^{4}-\frac{7652}{72331}a^{3}+\frac{37775}{72331}a^{2}+\frac{7982}{10333}a+\frac{44029}{72331}$, $\frac{33}{72331}a^{7}+\frac{1796}{72331}a^{6}-\frac{6969}{72331}a^{5}+\frac{5613}{72331}a^{4}-\frac{328}{10333}a^{3}-\frac{40606}{72331}a^{2}+\frac{93817}{72331}a+\frac{21324}{72331}$, $\frac{891}{72331}a^{7}-\frac{3173}{72331}a^{6}-\frac{2169}{72331}a^{5}-\frac{492}{10333}a^{4}-\frac{20660}{72331}a^{3}+\frac{81600}{72331}a^{2}+\frac{73805}{72331}a+\frac{38432}{72331}$, $\frac{583}{72331}a^{7}-\frac{2714}{72331}a^{6}+\frac{877}{72331}a^{5}-\frac{4167}{72331}a^{4}-\frac{13008}{72331}a^{3}+\frac{43825}{72331}a^{2}+\frac{17931}{72331}a-\frac{5597}{72331}$, $\frac{907}{10333}a^{7}-\frac{25196}{72331}a^{6}+\frac{24102}{72331}a^{5}-\frac{52026}{72331}a^{4}-\frac{47830}{72331}a^{3}+\frac{470334}{72331}a^{2}-\frac{12532}{10333}a+\frac{226794}{72331}$ | 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: | \( 367.02107817 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 367.02107817 \cdot 1}{2\cdot\sqrt{127027375281}}\cr\approx \mathstrut & 0.32523110008 \end{aligned}\]
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
\(\Q(\sqrt{597}) \), 4.2.1791.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.23641797.1, 6.0.118803.1 |
Degree 8 sibling: | 8.0.3207681.1 |
Degree 12 siblings: | deg 12, deg 12, deg 12, deg 12, deg 12, deg 12 |
Degree 16 sibling: | deg 16 |
Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
Minimal sibling: | 6.0.118803.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(199\) | 199.4.2.1 | $x^{4} + 386 x^{3} + 37653 x^{2} + 77972 x + 7450967$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
199.4.2.1 | $x^{4} + 386 x^{3} + 37653 x^{2} + 77972 x + 7450967$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |