Normalized defining polynomial
\( x^{9} + 9x^{7} - 6x^{6} + 9x^{5} - 27x^{4} - 36x^{3} + 81x^{2} - 36x + 3 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(9448798306221\) \(\medspace = 3^{18}\cdot 29^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(27.65\) | 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: | $3^{58/27}29^{1/2}\approx 57.03298977688336$ | ||
Ramified primes: | \(3\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{29}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{3182}a^{8}+\frac{81}{3182}a^{7}-\frac{1385}{3182}a^{6}+\frac{385}{1591}a^{5}+\frac{165}{1591}a^{4}+\frac{29}{74}a^{3}-\frac{853}{3182}a^{2}-\frac{599}{3182}a+\frac{383}{1591}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{3355}{3182}a^{8}+\frac{1438}{1591}a^{7}+\frac{16228}{1591}a^{6}+\frac{7521}{3182}a^{5}+\frac{17408}{1591}a^{4}-\frac{1421}{74}a^{3}-\frac{87308}{1591}a^{2}+\frac{63534}{1591}a-\frac{5899}{3182}$, $\frac{1259}{1591}a^{8}+\frac{1901}{3182}a^{7}+\frac{23907}{3182}a^{6}+\frac{2613}{3182}a^{5}+\frac{11356}{1591}a^{4}-\frac{600}{37}a^{3}-\frac{132057}{3182}a^{2}+\frac{112947}{3182}a+\frac{2087}{3182}$, $\frac{2835}{1591}a^{8}+\frac{2122}{1591}a^{7}+\frac{27160}{1591}a^{6}+\frac{3280}{1591}a^{5}+\frac{28680}{1591}a^{4}-\frac{1294}{37}a^{3}-\frac{141534}{1591}a^{2}+\frac{120348}{1591}a-\frac{11242}{1591}$, $\frac{3779}{1591}a^{8}+\frac{2218}{1591}a^{7}+\frac{35477}{1591}a^{6}-\frac{1700}{1591}a^{5}+\frac{34728}{1591}a^{4}-\frac{1890}{37}a^{3}-\frac{181495}{1591}a^{2}+\frac{196065}{1591}a-\frac{31135}{1591}$, $\frac{397}{3182}a^{8}+\frac{337}{3182}a^{7}+\frac{3823}{3182}a^{6}+\frac{109}{1591}a^{5}+\frac{1865}{1591}a^{4}-\frac{253}{74}a^{3}-\frac{20441}{3182}a^{2}+\frac{23121}{3182}a-\frac{685}{1591}$, $\frac{2012}{1591}a^{8}+\frac{2971}{3182}a^{7}+\frac{38217}{3182}a^{6}+\frac{3985}{3182}a^{5}+\frac{19605}{1591}a^{4}-\frac{926}{37}a^{3}-\frac{194787}{3182}a^{2}+\frac{184545}{3182}a-\frac{15293}{3182}$ | 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: | \( 5642.53122375 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 5642.53122375 \cdot 1}{2\cdot\sqrt{9448798306221}}\cr\approx \mathstrut & 1.15948571122 \end{aligned}\]
Galois group
$C_3^3:S_4$ (as 9T29):
A solvable group of order 648 |
The 17 conjugacy class representatives for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
Character table for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
Intermediate fields
3.3.2349.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 siblings: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.9.0.1}{9} }$ | R | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.18.46 | $x^{9} + 3 x^{6} + 18 x^{2} + 18 x + 21$ | $9$ | $1$ | $18$ | $C_3 \wr S_3 $ | $[2, 2, 7/3]^{6}$ |
\(29\) | $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 2.2349.3t2.a.a | $2$ | $ 3^{4} \cdot 29 $ | 3.3.2349.1 | $S_3$ (as 3T2) | $1$ | $2$ |
3.2349.4t5.a.a | $3$ | $ 3^{4} \cdot 29 $ | 4.0.2349.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.68121.6t8.a.a | $3$ | $ 3^{4} \cdot 29^{2}$ | 4.0.2349.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
4.53338743.24t1527.a.a | $4$ | $ 3^{7} \cdot 29^{3}$ | 9.5.9448798306221.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $0$ | |
4.63423.12t175.a.a | $4$ | $ 3^{7} \cdot 29 $ | 9.5.9448798306221.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $0$ | |
4.53338743.24t1527.a.b | $4$ | $ 3^{7} \cdot 29^{3}$ | 9.5.9448798306221.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $0$ | |
4.63423.12t175.a.b | $4$ | $ 3^{7} \cdot 29 $ | 9.5.9448798306221.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $0$ | |
6.338...289.18t220.a.a | $6$ | $ 3^{14} \cdot 29^{4}$ | 9.5.9448798306221.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $1$ | $2$ | |
* | 6.4022476929.9t29.a.a | $6$ | $ 3^{14} \cdot 29^{2}$ | 9.5.9448798306221.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $1$ | $2$ |
6.116651830941.36t1131.a.a | $6$ | $ 3^{14} \cdot 29^{3}$ | 9.5.9448798306221.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $-2$ | |
6.116651830941.36t1131.a.b | $6$ | $ 3^{14} \cdot 29^{3}$ | 9.5.9448798306221.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $-2$ | |
8.274...409.24t1540.a.a | $8$ | $ 3^{18} \cdot 29^{4}$ | 9.5.9448798306221.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $0$ | |
8.274...409.24t1540.a.b | $8$ | $ 3^{18} \cdot 29^{4}$ | 9.5.9448798306221.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $0$ | |
12.521...021.18t219.a.a | $12$ | $ 3^{26} \cdot 29^{5}$ | 9.5.9448798306221.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $1$ | $0$ | |
12.438...661.36t1126.a.a | $12$ | $ 3^{26} \cdot 29^{7}$ | 9.5.9448798306221.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $1$ | $0$ |