Normalized defining polynomial
\( x^{9} - 219x^{7} - 584x^{6} + 9855x^{5} + 30222x^{4} - 139284x^{3} - 437562x^{2} + 561735x + 1779813 \)
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: | \(428585957696282300721\) \(\medspace = 3^{12}\cdot 73^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(196.09\) | 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: | $3^{4/3}73^{8/9}\approx 196.08693569504558$ | ||
Ramified primes: | \(3\), \(73\) | 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: | \(657=3^{2}\cdot 73\) | ||
Dirichlet character group: | $\lbrace$$\chi_{657}(64,·)$, $\chi_{657}(1,·)$, $\chi_{657}(454,·)$, $\chi_{657}(274,·)$, $\chi_{657}(178,·)$, $\chi_{657}(148,·)$, $\chi_{657}(154,·)$, $\chi_{657}(475,·)$, $\chi_{657}(223,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{21}a^{5}+\frac{3}{7}a^{4}-\frac{8}{21}a^{2}-\frac{3}{7}a$, $\frac{1}{189}a^{6}+\frac{29}{63}a^{4}-\frac{8}{189}a^{3}+\frac{2}{21}a-\frac{1}{3}$, $\frac{1}{567}a^{7}+\frac{2}{189}a^{5}+\frac{208}{567}a^{4}+\frac{1}{3}a^{3}-\frac{31}{63}a^{2}+\frac{11}{63}a$, $\frac{1}{664551185949}a^{8}+\frac{60080257}{73839020661}a^{7}+\frac{6165113}{221517061983}a^{6}+\frac{27143881}{2207811249}a^{5}-\frac{7404283457}{73839020661}a^{4}-\frac{2129413801}{8204335629}a^{3}+\frac{33956812298}{73839020661}a^{2}+\frac{1090438180}{8204335629}a-\frac{4803661}{27256929}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $7$ |
Class group and class number
$C_{3}\times C_{3}$, which has order $9$ (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)
| |
Fundamental units: | $\frac{41846120}{664551185949}a^{8}-\frac{26983748}{73839020661}a^{7}-\frac{2791586186}{221517061983}a^{6}+\frac{61272704}{2207811249}a^{5}+\frac{46853544916}{73839020661}a^{4}+\frac{32283617750}{73839020661}a^{3}-\frac{696084801374}{73839020661}a^{2}-\frac{220867958018}{8204335629}a-\frac{565338769}{27256929}$, $\frac{22333088}{664551185949}a^{8}-\frac{13118540}{73839020661}a^{7}-\frac{1493462828}{221517061983}a^{6}+\frac{28311800}{2207811249}a^{5}+\frac{24023131072}{73839020661}a^{4}+\frac{7040300042}{73839020661}a^{3}-\frac{343403860748}{73839020661}a^{2}-\frac{78396426158}{8204335629}a-\frac{155097427}{27256929}$, $\frac{90378041}{94935883707}a^{8}-\frac{112521770}{31645294569}a^{7}-\frac{6116453639}{31645294569}a^{6}+\frac{319438073}{2207811249}a^{5}+\frac{271131812416}{31645294569}a^{4}-\frac{7721384066}{10548431523}a^{3}-\frac{1309370829932}{10548431523}a^{2}-\frac{28363936030}{3516143841}a+\frac{688737479}{1297949}$, $\frac{110841266}{221517061983}a^{8}-\frac{508310356}{221517061983}a^{7}-\frac{7765305557}{73839020661}a^{6}+\frac{140810336}{735937083}a^{5}+\frac{1184528975549}{221517061983}a^{4}-\frac{150396106193}{24613006887}a^{3}-\frac{2288567193691}{24613006887}a^{2}+\frac{1462999359394}{24613006887}a+\frac{4614034214}{9085643}$, $\frac{92269879}{664551185949}a^{8}-\frac{45911630}{73839020661}a^{7}-\frac{6272832805}{221517061983}a^{6}+\frac{87285523}{2207811249}a^{5}+\frac{103456134823}{73839020661}a^{4}-\frac{73576980899}{73839020661}a^{3}-\frac{1763087738854}{73839020661}a^{2}+\frac{24718406332}{2734778543}a+\frac{3460064023}{27256929}$, $\frac{33592810}{73839020661}a^{8}-\frac{298282391}{221517061983}a^{7}-\frac{7053732772}{73839020661}a^{6}+\frac{1365590}{81770787}a^{5}+\frac{980112941776}{221517061983}a^{4}+\frac{55737854636}{73839020661}a^{3}-\frac{1609281554722}{24613006887}a^{2}-\frac{190300817218}{24613006887}a+\frac{7556009269}{27256929}$, $\frac{41649406}{31645294569}a^{8}-\frac{131610211}{31645294569}a^{7}-\frac{2932494179}{10548431523}a^{6}+\frac{81011083}{735937083}a^{5}+\frac{59890123889}{4520756367}a^{4}-\frac{5117428660}{10548431523}a^{3}-\frac{34712740222}{167435421}a^{2}+\frac{19191077296}{3516143841}a+\frac{3836626936}{3893847}$, $\frac{6643355}{24613006887}a^{8}-\frac{318596102}{221517061983}a^{7}-\frac{4365882743}{73839020661}a^{6}+\frac{36211745}{245312361}a^{5}+\frac{768653409808}{221517061983}a^{4}-\frac{289963157864}{73839020661}a^{3}-\frac{1700476710595}{24613006887}a^{2}+\frac{376702862642}{24613006887}a+\frac{9420130940}{27256929}$ (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: | \( 17216132.8683 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 17216132.8683 \cdot 9}{2\cdot\sqrt{428585957696282300721}}\cr\approx \mathstrut & 1.91601591302 \end{aligned}\] (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
3.3.5329.1 |
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 | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.1.0.1}{1} }^{9}$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.1.0.1}{1} }^{9}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.4.3 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
3.3.4.3 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.4.3 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
\(73\) | 73.9.8.1 | $x^{9} + 73$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |