Normalized defining polynomial
\( x^{9} - x^{8} - 120x^{7} + 543x^{6} + 858x^{5} - 6780x^{4} + 7217x^{3} + 2818x^{2} - 4068x + 261 \)
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)
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Discriminant: | \(29090710405024191361\) \(\medspace = 271^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(145.43\) | 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: | $271^{8/9}\approx 145.42509951704676$ | ||
Ramified primes: | \(271\) | 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)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(271\) | ||
Dirichlet character group: | $\lbrace$$\chi_{271}(1,·)$, $\chi_{271}(258,·)$, $\chi_{271}(169,·)$, $\chi_{271}(106,·)$, $\chi_{271}(178,·)$, $\chi_{271}(242,·)$, $\chi_{271}(248,·)$, $\chi_{271}(28,·)$, $\chi_{271}(125,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{9}a^{6}+\frac{1}{9}a^{4}-\frac{2}{9}a^{2}$, $\frac{1}{513}a^{7}+\frac{4}{513}a^{6}+\frac{31}{513}a^{5}+\frac{25}{513}a^{4}+\frac{85}{513}a^{3}+\frac{169}{513}a^{2}+\frac{5}{57}a+\frac{5}{57}$, $\frac{1}{67689837}a^{8}-\frac{2005}{22563279}a^{7}-\frac{477125}{22563279}a^{6}+\frac{1386467}{22563279}a^{5}-\frac{28817}{278559}a^{4}+\frac{871649}{22563279}a^{3}-\frac{14720764}{67689837}a^{2}+\frac{274552}{2507031}a+\frac{2932651}{7521093}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $8$ | 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: | $\frac{302450}{22563279}a^{8}-\frac{91934}{7521093}a^{7}-\frac{12469081}{7521093}a^{6}+\frac{51769975}{7521093}a^{5}+\frac{513389}{30951}a^{4}-\frac{653256284}{7521093}a^{3}+\frac{963929833}{22563279}a^{2}+\frac{60584345}{835677}a-\frac{12990640}{2507031}$, $\frac{3127784}{22563279}a^{8}+\frac{4492774}{7521093}a^{7}-\frac{101530714}{7521093}a^{6}+\frac{25990699}{7521093}a^{5}+\frac{13076194}{92853}a^{4}-\frac{1461879986}{7521093}a^{3}-\frac{1461090017}{22563279}a^{2}+\frac{92318984}{835677}a-\frac{17130901}{2507031}$, $\frac{14765963}{67689837}a^{8}+\frac{4523428}{22563279}a^{7}-\frac{581900548}{22563279}a^{6}+\frac{1556187925}{22563279}a^{5}+\frac{88935539}{278559}a^{4}-\frac{19543940276}{22563279}a^{3}-\frac{5779783370}{67689837}a^{2}+\frac{1121941052}{2507031}a-\frac{228405775}{7521093}$, $\frac{2530532}{67689837}a^{8}-\frac{2256845}{22563279}a^{7}-\frac{100301794}{22563279}a^{6}+\frac{625577065}{22563279}a^{5}+\frac{176672}{278559}a^{4}-\frac{6899544515}{22563279}a^{3}+\frac{42924515158}{67689837}a^{2}-\frac{857903980}{2507031}a+\frac{153426413}{7521093}$, $\frac{4438570}{22563279}a^{8}-\frac{1352035}{7521093}a^{7}-\frac{177673469}{7521093}a^{6}+\frac{788081675}{7521093}a^{5}+\frac{16530463}{92853}a^{4}-\frac{9923300176}{7521093}a^{3}+\frac{29447405030}{22563279}a^{2}+\frac{566527438}{835677}a-\frac{100601636}{131949}$, $\frac{311728}{7521093}a^{8}-\frac{470701}{2507031}a^{7}-\frac{10791815}{2507031}a^{6}+\frac{94468928}{2507031}a^{5}-\frac{337538}{3439}a^{4}+\frac{175711190}{2507031}a^{3}+\frac{339433166}{7521093}a^{2}-\frac{12961655}{278559}a+\frac{2468047}{835677}$, $\frac{146039}{67689837}a^{8}-\frac{13364}{22563279}a^{7}-\frac{5563507}{22563279}a^{6}+\frac{24153349}{22563279}a^{5}+\frac{457727}{278559}a^{4}-\frac{307851629}{22563279}a^{3}+\frac{1042083655}{67689837}a^{2}+\frac{11084597}{2507031}a-\frac{52743901}{7521093}$, $\frac{232346}{67689837}a^{8}-\frac{29777}{22563279}a^{7}-\frac{9137806}{22563279}a^{6}+\frac{34255654}{22563279}a^{5}+\frac{1141961}{278559}a^{4}-\frac{443575784}{22563279}a^{3}+\frac{571168468}{67689837}a^{2}+\frac{42362690}{2507031}a+\frac{1530623}{7521093}$ (assuming GRH) | 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: | \( 190161211.995 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 190161211.995 \cdot 1}{2\cdot\sqrt{29090710405024191361}}\cr\approx \mathstrut & 9.02577989560 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 9 |
The 9 conjugacy class representatives for $C_9$ |
Character table for $C_9$ |
Intermediate fields
3.3.73441.1 |
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 | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.1.0.1}{1} }^{9}$ | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.9.0.1}{9} }$ | ${\href{/padicField/19.1.0.1}{1} }^{9}$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.1.0.1}{1} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(271\) | Deg $9$ | $9$ | $1$ | $8$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.271.9t1.a.a | $1$ | $ 271 $ | 9.9.29090710405024191361.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.271.9t1.a.b | $1$ | $ 271 $ | 9.9.29090710405024191361.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.271.3t1.a.a | $1$ | $ 271 $ | 3.3.73441.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.271.9t1.a.c | $1$ | $ 271 $ | 9.9.29090710405024191361.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.271.9t1.a.d | $1$ | $ 271 $ | 9.9.29090710405024191361.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.271.3t1.a.b | $1$ | $ 271 $ | 3.3.73441.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.271.9t1.a.e | $1$ | $ 271 $ | 9.9.29090710405024191361.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.271.9t1.a.f | $1$ | $ 271 $ | 9.9.29090710405024191361.1 | $C_9$ (as 9T1) | $0$ | $1$ |