Normalized defining polynomial
\( x^{9} - 57x^{7} - 38x^{6} + 855x^{5} + 228x^{4} - 4902x^{3} + 1710x^{2} + 9063x - 7201 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[9, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(9025761726072081\) \(\medspace = 3^{12}\cdot 19^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(59.27\) | 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: | $3^{4/3}19^{8/9}\approx 59.269537795580746$ | ||
Ramified primes: | \(3\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $9$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(171=3^{2}\cdot 19\) | ||
Dirichlet character group: | $\lbrace$$\chi_{171}(64,·)$, $\chi_{171}(1,·)$, $\chi_{171}(130,·)$, $\chi_{171}(163,·)$, $\chi_{171}(142,·)$, $\chi_{171}(157,·)$, $\chi_{171}(112,·)$, $\chi_{171}(25,·)$, $\chi_{171}(61,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{369127429}a^{8}+\frac{99144680}{369127429}a^{7}+\frac{144697142}{369127429}a^{6}-\frac{16483691}{369127429}a^{5}+\frac{80212427}{369127429}a^{4}+\frac{150424695}{369127429}a^{3}-\frac{15708390}{369127429}a^{2}-\frac{46052998}{369127429}a+\frac{451705}{973951}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $8$ | 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: | $\frac{3829680}{33557039}a^{8}+\frac{5171796}{33557039}a^{7}-\frac{211409706}{33557039}a^{6}-\frac{431122860}{33557039}a^{5}+\frac{2698468602}{33557039}a^{4}+\frac{4522535327}{33557039}a^{3}-\frac{12752190252}{33557039}a^{2}-\frac{10669868817}{33557039}a+\frac{54509317}{88541}$, $\frac{3829680}{33557039}a^{8}+\frac{5171796}{33557039}a^{7}-\frac{211409706}{33557039}a^{6}-\frac{431122860}{33557039}a^{5}+\frac{2698468602}{33557039}a^{4}+\frac{4522535327}{33557039}a^{3}-\frac{12752190252}{33557039}a^{2}-\frac{10669868817}{33557039}a+\frac{54420776}{88541}$, $\frac{10094846}{369127429}a^{8}+\frac{10965825}{369127429}a^{7}-\frac{562684079}{369127429}a^{6}-\frac{999310247}{369127429}a^{5}+\frac{7525303120}{369127429}a^{4}+\frac{10632661785}{369127429}a^{3}-\frac{37976731588}{369127429}a^{2}-\frac{26590241717}{369127429}a+\frac{162572897}{973951}$, $\frac{59163018}{369127429}a^{8}+\frac{83649934}{369127429}a^{7}-\frac{3246893922}{369127429}a^{6}-\frac{6850655743}{369127429}a^{5}+\frac{40592263466}{369127429}a^{4}+\frac{70920004832}{369127429}a^{3}-\frac{187088659933}{369127429}a^{2}-\frac{163118312746}{369127429}a+\frac{793066265}{973951}$, $\frac{245908505}{369127429}a^{8}+\frac{331836660}{369127429}a^{7}-\frac{13570869989}{369127429}a^{6}-\frac{27653797738}{369127429}a^{5}+\frac{173032107575}{369127429}a^{4}+\frac{289417244786}{369127429}a^{3}-\frac{815840719861}{369127429}a^{2}-\frac{679098608785}{369127429}a+\frac{3467927723}{973951}$, $\frac{5156561}{33557039}a^{8}+\frac{7121722}{33557039}a^{7}-\frac{283990363}{33557039}a^{6}-\frac{587755289}{33557039}a^{5}+\frac{3592618127}{33557039}a^{4}+\frac{6112208327}{33557039}a^{3}-\frac{16827417647}{33557039}a^{2}-\frac{14243847813}{33557039}a+\frac{71559355}{88541}$, $\frac{24540674}{369127429}a^{8}+\frac{30508279}{369127429}a^{7}-\frac{1348918818}{369127429}a^{6}-\frac{2627637501}{369127429}a^{5}+\frac{17120725779}{369127429}a^{4}+\frac{27265959601}{369127429}a^{3}-\frac{79670377664}{369127429}a^{2}-\frac{63624346838}{369127429}a+\frac{333190037}{973951}$, $\frac{104382963}{369127429}a^{8}+\frac{139848370}{369127429}a^{7}-\frac{5760506786}{369127429}a^{6}-\frac{11682017029}{369127429}a^{5}+\frac{73461859553}{369127429}a^{4}+\frac{122193573621}{369127429}a^{3}-\frac{346131659658}{369127429}a^{2}-\frac{286475653833}{369127429}a+\frac{1470849251}{973951}$ | 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: | \( 58633.710835 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 58633.710835 \cdot 3}{2\cdot\sqrt{9025761726072081}}\cr\approx \mathstrut & 0.47398726007 \end{aligned}\]
Galois group
A cyclic group of order 9 |
The 9 conjugacy class representatives for $C_9$ |
Character table for $C_9$ |
Intermediate fields
3.3.361.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.9.0.1}{9} }$ | R | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.1.0.1}{1} }^{9}$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.9.0.1}{9} }$ | R | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.1.0.1}{1} }^{9}$ | ${\href{/padicField/37.1.0.1}{1} }^{9}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.12.3 | $x^{9} + 18 x^{8} + 108 x^{7} + 153 x^{6} - 756 x^{5} - 2268 x^{4} + 891 x^{3} + 5346 x^{2} + 9099$ | $3$ | $3$ | $12$ | $C_9$ | $[2]^{3}$ |
\(19\) | 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.171.9t1.b.a | $1$ | $ 3^{2} \cdot 19 $ | 9.9.9025761726072081.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.171.9t1.b.b | $1$ | $ 3^{2} \cdot 19 $ | 9.9.9025761726072081.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.171.9t1.b.c | $1$ | $ 3^{2} \cdot 19 $ | 9.9.9025761726072081.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.171.9t1.b.d | $1$ | $ 3^{2} \cdot 19 $ | 9.9.9025761726072081.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.171.9t1.b.e | $1$ | $ 3^{2} \cdot 19 $ | 9.9.9025761726072081.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.171.9t1.b.f | $1$ | $ 3^{2} \cdot 19 $ | 9.9.9025761726072081.1 | $C_9$ (as 9T1) | $0$ | $1$ |