Normalized defining polynomial
\( x^{9} - x^{8} + 6x^{7} - 6x^{6} + 11x^{5} + 5x^{4} - 15x^{3} + 50x^{2} - 100x + 125 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(9294114390625\) \(\medspace = 5^{6}\cdot 29^{6}\) | 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: | \(27.60\) | 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: | $5^{3/4}29^{2/3}\approx 31.561635639558343$ | ||
Ramified primes: | \(5\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}$, $\frac{1}{5}a^{7}+\frac{1}{5}a^{2}$, $\frac{1}{24775}a^{8}+\frac{1594}{24775}a^{7}+\frac{521}{24775}a^{6}+\frac{3504}{24775}a^{5}-\frac{349}{24775}a^{4}+\frac{653}{4955}a^{3}-\frac{1991}{4955}a^{2}+\frac{104}{991}a+\frac{379}{991}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $4$ | 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: | $\frac{314}{24775}a^{8}+\frac{61}{24775}a^{7}+\frac{79}{24775}a^{6}+\frac{246}{24775}a^{5}-\frac{576}{24775}a^{4}-\frac{19}{991}a^{3}+\frac{147}{4955}a^{2}-\frac{47}{991}a+\frac{86}{991}$, $\frac{87}{24775}a^{8}-\frac{62}{24775}a^{7}+\frac{732}{24775}a^{6}+\frac{2593}{24775}a^{5}-\frac{633}{24775}a^{4}+\frac{263}{991}a^{3}-\frac{1774}{4955}a^{2}-\frac{862}{991}a+\frac{1261}{991}$, $\frac{84}{991}a^{8}-\frac{436}{4955}a^{7}+\frac{160}{991}a^{6}+\frac{9}{991}a^{5}-\frac{577}{991}a^{4}+\frac{1735}{991}a^{3}-\frac{71}{4955}a^{2}-\frac{3584}{991}a+\frac{4091}{991}$, $\frac{6236}{24775}a^{8}+\frac{5409}{24775}a^{7}+\frac{28206}{24775}a^{6}-\frac{606}{24775}a^{5}-\frac{20939}{24775}a^{4}-\frac{902}{4955}a^{3}-\frac{8556}{4955}a^{2}+\frac{10340}{991}a-\frac{91}{991}$ | 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: | \( 971.958599203 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 971.958599203 \cdot 3}{2\cdot\sqrt{9294114390625}}\cr\approx \mathstrut & 1.49068021689 \end{aligned}\]
Galois group
$C_3^2:C_4$ (as 9T9):
A solvable group of order 36 |
The 6 conjugacy class representatives for $C_3^2:C_4$ |
Character table for $C_3^2:C_4$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Galois closure: | data not computed |
Degree 6 siblings: | 6.2.525625.1, 6.2.442050625.2 |
Degree 12 siblings: | 12.0.1381408203125.1, deg 12 |
Degree 18 sibling: | deg 18 |
Minimal sibling: | 6.2.525625.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | R | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(29\) | 29.3.2.1 | $x^{3} + 29$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
29.6.4.1 | $x^{6} + 72 x^{5} + 1734 x^{4} + 14170 x^{3} + 5556 x^{2} + 50052 x + 397569$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |