Normalized defining polynomial
\( x^{9} - 111x^{7} - 74x^{6} + 3663x^{5} + 3774x^{4} - 40848x^{3} - 30636x^{2} + 137529x - 42661 \)
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: | \(1866675593471230161\) \(\medspace = 3^{12}\cdot 37^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(107.18\) | 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: | $3^{4/3}37^{8/9}\approx 107.18123601368417$ | ||
Ramified primes: | \(3\), \(37\) | 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: | \(333=3^{2}\cdot 37\) | ||
Dirichlet character group: | $\lbrace$$\chi_{333}(1,·)$, $\chi_{333}(34,·)$, $\chi_{333}(100,·)$, $\chi_{333}(70,·)$, $\chi_{333}(7,·)$, $\chi_{333}(10,·)$, $\chi_{333}(238,·)$, $\chi_{333}(49,·)$, $\chi_{333}(157,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{607623885394441}a^{8}+\frac{255189087937095}{607623885394441}a^{7}-\frac{68036324506765}{607623885394441}a^{6}+\frac{52086422263566}{607623885394441}a^{5}-\frac{247923452819436}{607623885394441}a^{4}-\frac{205581437849824}{607623885394441}a^{3}-\frac{155162761137803}{607623885394441}a^{2}+\frac{3248115088898}{607623885394441}a-\frac{274073517077616}{607623885394441}$
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: | $\frac{3837178440}{6264163766953}a^{8}-\frac{14196930924}{6264163766953}a^{7}-\frac{371427121145}{6264163766953}a^{6}+\frac{1091310705828}{6264163766953}a^{5}+\frac{9819080989302}{6264163766953}a^{4}-\frac{21923771201496}{6264163766953}a^{3}-\frac{69996790036773}{6264163766953}a^{2}+\frac{141571347106500}{6264163766953}a-\frac{41136543787446}{6264163766953}$, $\frac{280731636}{6264163766953}a^{8}-\frac{3475958766}{6264163766953}a^{7}-\frac{22305185284}{6264163766953}a^{6}+\frac{318702859488}{6264163766953}a^{5}+\frac{437626041462}{6264163766953}a^{4}-\frac{7992925753871}{6264163766953}a^{3}-\frac{1517412641094}{6264163766953}a^{2}+\frac{57659536109193}{6264163766953}a-\frac{20808483860405}{6264163766953}$, $\frac{97620548043}{607623885394441}a^{8}+\frac{833402012752}{607623885394441}a^{7}-\frac{10253555186754}{607623885394441}a^{6}-\frac{88756508845205}{607623885394441}a^{5}+\frac{231340524155459}{607623885394441}a^{4}+\frac{23\!\cdots\!94}{607623885394441}a^{3}+\frac{329178038207373}{607623885394441}a^{2}-\frac{92\!\cdots\!59}{607623885394441}a+\frac{44\!\cdots\!27}{607623885394441}$, $\frac{222472064778}{607623885394441}a^{8}+\frac{1329636025202}{607623885394441}a^{7}-\frac{22670713346056}{607623885394441}a^{6}-\frac{146598840320074}{607623885394441}a^{5}+\frac{505130774332732}{607623885394441}a^{4}+\frac{38\!\cdots\!01}{607623885394441}a^{3}+\frac{511167050228628}{607623885394441}a^{2}-\frac{13\!\cdots\!38}{607623885394441}a+\frac{45\!\cdots\!05}{607623885394441}$, $\frac{2911678193667}{607623885394441}a^{8}-\frac{12388113236891}{607623885394441}a^{7}-\frac{269478253374233}{607623885394441}a^{6}+\frac{927464563231951}{607623885394441}a^{5}+\frac{66\!\cdots\!53}{607623885394441}a^{4}-\frac{16\!\cdots\!71}{607623885394441}a^{3}-\frac{44\!\cdots\!94}{607623885394441}a^{2}+\frac{96\!\cdots\!93}{607623885394441}a-\frac{25\!\cdots\!85}{607623885394441}$, $\frac{93407850276}{607623885394441}a^{8}-\frac{349151175552}{607623885394441}a^{7}-\frac{9501218380269}{607623885394441}a^{6}+\frac{24576063377665}{607623885394441}a^{5}+\frac{265714013674373}{607623885394441}a^{4}-\frac{448441401341608}{607623885394441}a^{3}-\frac{20\!\cdots\!57}{607623885394441}a^{2}+\frac{28\!\cdots\!18}{607623885394441}a-\frac{404951428409543}{607623885394441}$, $\frac{165202444766}{607623885394441}a^{8}-\frac{990942818041}{607623885394441}a^{7}-\frac{15874801376977}{607623885394441}a^{6}+\frac{79014062164413}{607623885394441}a^{5}+\frac{428877921154557}{607623885394441}a^{4}-\frac{15\!\cdots\!56}{607623885394441}a^{3}-\frac{29\!\cdots\!11}{607623885394441}a^{2}+\frac{91\!\cdots\!04}{607623885394441}a-\frac{32\!\cdots\!57}{607623885394441}$, $\frac{42277645169}{607623885394441}a^{8}-\frac{49293520295}{607623885394441}a^{7}-\frac{4960598593657}{607623885394441}a^{6}+\frac{6604906743694}{607623885394441}a^{5}+\frac{155661442967877}{607623885394441}a^{4}-\frac{230238985127943}{607623885394441}a^{3}-\frac{12\!\cdots\!84}{607623885394441}a^{2}+\frac{24\!\cdots\!27}{607623885394441}a-\frac{755652945694009}{607623885394441}$ | 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: | \( 380116.876345 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 380116.876345 \cdot 3}{2\cdot\sqrt{1866675593471230161}}\cr\approx \mathstrut & 0.213670195408 \end{aligned}\]
Galois group
A cyclic group of order 9 |
The 9 conjugacy class representatives for $C_9$ |
Character table for $C_9$ |
Intermediate fields
3.3.1369.1 |
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 | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.9.0.1}{9} }$ | ${\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.1.0.1}{1} }^{9}$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 | $x^{9} + 18 x^{8} + 108 x^{7} + 153 x^{6} - 756 x^{5} - 2268 x^{4} + 891 x^{3} + 5346 x^{2} + 9099$ | $3$ | $3$ | $12$ | $C_9$ | $[2]^{3}$ |
\(37\) | 37.9.8.1 | $x^{9} + 37$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.333.9t1.a.a | $1$ | $ 3^{2} \cdot 37 $ | 9.9.1866675593471230161.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.333.9t1.a.b | $1$ | $ 3^{2} \cdot 37 $ | 9.9.1866675593471230161.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.37.3t1.a.a | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.333.9t1.a.c | $1$ | $ 3^{2} \cdot 37 $ | 9.9.1866675593471230161.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.333.9t1.a.d | $1$ | $ 3^{2} \cdot 37 $ | 9.9.1866675593471230161.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.37.3t1.a.b | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.333.9t1.a.e | $1$ | $ 3^{2} \cdot 37 $ | 9.9.1866675593471230161.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.333.9t1.a.f | $1$ | $ 3^{2} \cdot 37 $ | 9.9.1866675593471230161.1 | $C_9$ (as 9T1) | $0$ | $1$ |