Normalized defining polynomial
\( x^{9} - 57x^{7} - 38x^{6} + 855x^{5} + 1254x^{4} - 3192x^{3} - 7524x^{2} - 4275x - 703 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
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$$\chi_{171}(64,·)$, $\chi_{171}(1,·)$, $\chi_{171}(43,·)$, $\chi_{171}(4,·)$, $\chi_{171}(169,·)$, $\chi_{171}(139,·)$, $\chi_{171}(16,·)$, $\chi_{171}(163,·)$, $\chi_{171}(85,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{7}a^{5}+\frac{1}{7}a^{4}+\frac{1}{7}a^{3}+\frac{1}{7}a^{2}+\frac{1}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{6}-\frac{1}{7}$, $\frac{1}{49}a^{7}-\frac{3}{49}a^{6}-\frac{1}{49}a^{5}+\frac{20}{49}a^{4}+\frac{13}{49}a^{3}-\frac{1}{49}a^{2}+\frac{19}{49}a+\frac{16}{49}$, $\frac{1}{7627291}a^{8}+\frac{59856}{7627291}a^{7}+\frac{35542}{1089613}a^{6}-\frac{394222}{7627291}a^{5}+\frac{1041905}{7627291}a^{4}-\frac{2068006}{7627291}a^{3}+\frac{1669265}{7627291}a^{2}-\frac{3239056}{7627291}a+\frac{23200}{206143}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $7$ |
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: | $\frac{36072}{7627291}a^{8}-\frac{176016}{7627291}a^{7}-\frac{225581}{1089613}a^{6}+\frac{7479252}{7627291}a^{5}+\frac{13429602}{7627291}a^{4}-\frac{63536098}{7627291}a^{3}-\frac{38914299}{7627291}a^{2}+\frac{143180238}{7627291}a+\frac{1949180}{206143}$, $\frac{85644}{7627291}a^{8}-\frac{166242}{7627291}a^{7}-\frac{642044}{1089613}a^{6}+\frac{5291856}{7627291}a^{5}+\frac{59917854}{7627291}a^{4}-\frac{2406051}{7627291}a^{3}-\frac{237833298}{7627291}a^{2}-\frac{216103473}{7627291}a-\frac{1358919}{206143}$, $\frac{6025}{7627291}a^{8}-\frac{29503}{7627291}a^{7}-\frac{46234}{1089613}a^{6}+\frac{1258403}{7627291}a^{5}+\frac{4575584}{7627291}a^{4}-\frac{7638786}{7627291}a^{3}-\frac{21598625}{7627291}a^{2}-\frac{5791635}{7627291}a+\frac{44495}{206143}$, $\frac{33786}{7627291}a^{8}-\frac{26912}{7627291}a^{7}-\frac{242832}{1089613}a^{6}+\frac{244820}{7627291}a^{5}+\frac{19288173}{7627291}a^{4}+\frac{13768652}{7627291}a^{3}-\frac{49614175}{7627291}a^{2}-\frac{63751528}{7627291}a-\frac{421119}{206143}$, $\frac{169401}{7627291}a^{8}-\frac{427981}{7627291}a^{7}-\frac{1218339}{1089613}a^{6}+\frac{15100376}{7627291}a^{5}+\frac{104635220}{7627291}a^{4}-\frac{51864928}{7627291}a^{3}-\frac{391131290}{7627291}a^{2}-\frac{277178204}{7627291}a-\frac{1435061}{206143}$, $\frac{36062}{7627291}a^{8}+\frac{159378}{7627291}a^{7}-\frac{7319}{22237}a^{6}-\frac{9125516}{7627291}a^{5}+\frac{47840344}{7627291}a^{4}+\frac{132727314}{7627291}a^{3}-\frac{205817884}{7627291}a^{2}-\frac{581398919}{7627291}a-\frac{6154117}{206143}$, $\frac{302997}{7627291}a^{8}-\frac{731271}{7627291}a^{7}-\frac{2194345}{1089613}a^{6}+\frac{25024595}{7627291}a^{5}+\frac{191508388}{7627291}a^{4}-\frac{60931534}{7627291}a^{3}-\frac{731631818}{7627291}a^{2}-\frac{646622701}{7627291}a-\frac{4115037}{206143}$, $\frac{12788}{7627291}a^{8}+\frac{63225}{7627291}a^{7}-\frac{17767}{155659}a^{6}-\frac{3563060}{7627291}a^{5}+\frac{15814935}{7627291}a^{4}+\frac{48746344}{7627291}a^{3}-\frac{59509201}{7627291}a^{2}-\frac{168455607}{7627291}a-\frac{1220277}{206143}$ | 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: | \( 86043.1062668 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 86043.1062668 \cdot 3}{2\cdot\sqrt{9025761726072081}}\cr\approx \mathstrut & 0.695561232715 \end{aligned}\]
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 | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.1.0.1}{1} }^{9}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.9.0.1}{9} }$ | R | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.1.0.1}{1} }^{9}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.12.2 | $x^{9} + 18 x^{8} + 108 x^{7} + 243 x^{6} + 324 x^{5} + 972 x^{4} + 1269 x^{3} + 7614 x^{2} + 15984$ | $3$ | $3$ | $12$ | $C_9$ | $[2]^{3}$ |
\(19\) | 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.a.a | $1$ | $ 3^{2} \cdot 19 $ | 9.9.9025761726072081.2 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.171.9t1.a.b | $1$ | $ 3^{2} \cdot 19 $ | 9.9.9025761726072081.2 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.171.9t1.a.c | $1$ | $ 3^{2} \cdot 19 $ | 9.9.9025761726072081.2 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.171.9t1.a.d | $1$ | $ 3^{2} \cdot 19 $ | 9.9.9025761726072081.2 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.171.9t1.a.e | $1$ | $ 3^{2} \cdot 19 $ | 9.9.9025761726072081.2 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.171.9t1.a.f | $1$ | $ 3^{2} \cdot 19 $ | 9.9.9025761726072081.2 | $C_9$ (as 9T1) | $0$ | $1$ |