Normalized defining polynomial
\( x^{9} - 3x^{8} + 7x^{7} - 18x^{6} + 47x^{5} + x^{4} - 10x^{3} - 220x^{2} - 200x - 145 \)
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: | \(659070838140625\) \(\medspace = 5^{6}\cdot 59^{6}\) | 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: | \(44.31\) | 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: | $5^{3/4}59^{2/3}\approx 50.67520462224516$ | ||
Ramified primes: | \(5\), \(59\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{3302302}a^{8}-\frac{36911}{1651151}a^{7}-\frac{491613}{1651151}a^{6}-\frac{267791}{1651151}a^{5}+\frac{968161}{3302302}a^{4}-\frac{128487}{1651151}a^{3}-\frac{510143}{3302302}a^{2}-\frac{1206111}{3302302}a-\frac{553832}{1651151}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{9}$, which has order $9$
Unit group
Rank: | $4$ | 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: | $\frac{22345}{3302302}a^{8}-\frac{52741}{3302302}a^{7}+\frac{15118}{1651151}a^{6}-\frac{18671}{1651151}a^{5}+\frac{177143}{3302302}a^{4}+\frac{2270299}{3302302}a^{3}-\frac{2901133}{3302302}a^{2}-\frac{2708563}{1651151}a-\frac{4952043}{3302302}$, $\frac{35855}{1651151}a^{8}-\frac{92757}{1651151}a^{7}+\frac{156771}{1651151}a^{6}-\frac{406480}{1651151}a^{5}+\frac{1265182}{1651151}a^{4}+\frac{1270961}{1651151}a^{3}-\frac{3028789}{1651151}a^{2}-\frac{6418668}{1651151}a-\frac{8413472}{1651151}$, $\frac{45519}{3302302}a^{8}-\frac{211333}{3302302}a^{7}+\frac{317356}{1651151}a^{6}-\frac{781847}{1651151}a^{5}+\frac{3802671}{3302302}a^{4}-\frac{5399275}{3302302}a^{3}+\frac{588447}{3302302}a^{2}-\frac{4225807}{1651151}a+\frac{4742773}{3302302}$, $\frac{534137}{3302302}a^{8}-\frac{2439018}{1651151}a^{7}+\frac{7059757}{1651151}a^{6}-\frac{20986351}{1651151}a^{5}+\frac{99094823}{3302302}a^{4}-\frac{87080407}{1651151}a^{3}+\frac{158406133}{3302302}a^{2}-\frac{61669275}{3302302}a+\frac{201141749}{1651151}$ | 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: | \( 3003.75272723 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 3003.75272723 \cdot 9}{2\cdot\sqrt{659070838140625}}\cr\approx \mathstrut & 1.64119472742 \end{aligned}\]
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: | 6.2.2175625.1, 6.2.7573350625.1 |
Degree 12 siblings: | deg 12, deg 12 |
Degree 18 sibling: | deg 18 |
Minimal sibling: | 6.2.2175625.1 |
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 | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | R |
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 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(59\) | 59.3.2.1 | $x^{3} + 59$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
59.6.4.1 | $x^{6} + 174 x^{5} + 10098 x^{4} + 195926 x^{3} + 30462 x^{2} + 595416 x + 11494565$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |