Normalized defining polynomial
\( x^{9} - x^{8} - 198x^{7} - 107x^{6} + 10319x^{5} + 24533x^{4} - 118922x^{3} - 460626x^{2} - 367949x + 57721 \)
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: | \(15072974715383053921\) \(\medspace = 19^{8}\cdot 31^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(135.18\) | 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: | $19^{8/9}31^{2/3}\approx 135.17955010958042$ | ||
Ramified primes: | \(19\), \(31\) | 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: | \(589=19\cdot 31\) | ||
Dirichlet character group: | $\lbrace$$\chi_{589}(1,·)$, $\chi_{589}(397,·)$, $\chi_{589}(366,·)$, $\chi_{589}(125,·)$, $\chi_{589}(149,·)$, $\chi_{589}(311,·)$, $\chi_{589}(408,·)$, $\chi_{589}(346,·)$, $\chi_{589}(253,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{25\!\cdots\!29}a^{8}-\frac{34\!\cdots\!91}{25\!\cdots\!29}a^{7}-\frac{46\!\cdots\!88}{25\!\cdots\!29}a^{6}+\frac{10\!\cdots\!34}{25\!\cdots\!29}a^{5}+\frac{78\!\cdots\!26}{25\!\cdots\!29}a^{4}+\frac{24\!\cdots\!26}{25\!\cdots\!29}a^{3}+\frac{68\!\cdots\!71}{25\!\cdots\!29}a^{2}-\frac{34\!\cdots\!73}{25\!\cdots\!29}a-\frac{23\!\cdots\!48}{12\!\cdots\!57}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
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: | $\frac{2335439503002}{12\!\cdots\!57}a^{8}-\frac{3947967663468}{12\!\cdots\!57}a^{7}-\frac{460304001254696}{12\!\cdots\!57}a^{6}+\frac{83481472660399}{12\!\cdots\!57}a^{5}+\frac{24\!\cdots\!00}{12\!\cdots\!57}a^{4}+\frac{38\!\cdots\!58}{12\!\cdots\!57}a^{3}-\frac{29\!\cdots\!21}{12\!\cdots\!57}a^{2}-\frac{79\!\cdots\!56}{12\!\cdots\!57}a-\frac{46\!\cdots\!64}{12\!\cdots\!57}$, $\frac{3580472426003}{12\!\cdots\!57}a^{8}-\frac{9896722714953}{12\!\cdots\!57}a^{7}-\frac{698362349306116}{12\!\cdots\!57}a^{6}+\frac{912063065824864}{12\!\cdots\!57}a^{5}+\frac{36\!\cdots\!94}{12\!\cdots\!57}a^{4}+\frac{14\!\cdots\!96}{12\!\cdots\!57}a^{3}-\frac{50\!\cdots\!65}{12\!\cdots\!57}a^{2}-\frac{60\!\cdots\!84}{12\!\cdots\!57}a+\frac{25\!\cdots\!26}{12\!\cdots\!57}$, $\frac{14\!\cdots\!98}{25\!\cdots\!29}a^{8}-\frac{59\!\cdots\!35}{25\!\cdots\!29}a^{7}-\frac{28\!\cdots\!09}{25\!\cdots\!29}a^{6}+\frac{68\!\cdots\!08}{25\!\cdots\!29}a^{5}+\frac{14\!\cdots\!81}{25\!\cdots\!29}a^{4}-\frac{42\!\cdots\!62}{25\!\cdots\!29}a^{3}-\frac{20\!\cdots\!07}{25\!\cdots\!29}a^{2}-\frac{25\!\cdots\!72}{25\!\cdots\!29}a+\frac{18\!\cdots\!51}{12\!\cdots\!57}$, $\frac{51311707260832}{25\!\cdots\!29}a^{8}-\frac{644701892170253}{25\!\cdots\!29}a^{7}-\frac{24\!\cdots\!02}{25\!\cdots\!29}a^{6}+\frac{38\!\cdots\!60}{25\!\cdots\!29}a^{5}+\frac{20\!\cdots\!53}{25\!\cdots\!29}a^{4}+\frac{71\!\cdots\!33}{25\!\cdots\!29}a^{3}-\frac{32\!\cdots\!93}{25\!\cdots\!29}a^{2}-\frac{44\!\cdots\!35}{25\!\cdots\!29}a+\frac{37\!\cdots\!63}{12\!\cdots\!57}$, $\frac{11\!\cdots\!81}{25\!\cdots\!29}a^{8}-\frac{91\!\cdots\!62}{25\!\cdots\!29}a^{7}-\frac{15\!\cdots\!27}{25\!\cdots\!29}a^{6}+\frac{11\!\cdots\!50}{25\!\cdots\!29}a^{5}+\frac{39\!\cdots\!17}{25\!\cdots\!29}a^{4}-\frac{20\!\cdots\!66}{25\!\cdots\!29}a^{3}-\frac{47\!\cdots\!38}{25\!\cdots\!29}a^{2}+\frac{68\!\cdots\!71}{25\!\cdots\!29}a+\frac{72\!\cdots\!38}{12\!\cdots\!57}$, $\frac{49\!\cdots\!21}{25\!\cdots\!29}a^{8}-\frac{19\!\cdots\!02}{25\!\cdots\!29}a^{7}-\frac{91\!\cdots\!29}{25\!\cdots\!29}a^{6}+\frac{21\!\cdots\!91}{25\!\cdots\!29}a^{5}+\frac{43\!\cdots\!51}{25\!\cdots\!29}a^{4}-\frac{62\!\cdots\!54}{25\!\cdots\!29}a^{3}-\frac{54\!\cdots\!59}{25\!\cdots\!29}a^{2}-\frac{70\!\cdots\!58}{25\!\cdots\!29}a+\frac{52\!\cdots\!13}{12\!\cdots\!57}$, $\frac{23\!\cdots\!95}{25\!\cdots\!29}a^{8}-\frac{17\!\cdots\!54}{25\!\cdots\!29}a^{7}-\frac{39\!\cdots\!53}{25\!\cdots\!29}a^{6}+\frac{24\!\cdots\!91}{25\!\cdots\!29}a^{5}+\frac{15\!\cdots\!71}{25\!\cdots\!29}a^{4}-\frac{64\!\cdots\!41}{25\!\cdots\!29}a^{3}-\frac{15\!\cdots\!92}{25\!\cdots\!29}a^{2}+\frac{40\!\cdots\!10}{25\!\cdots\!29}a-\frac{13\!\cdots\!65}{12\!\cdots\!57}$, $\frac{56\!\cdots\!44}{25\!\cdots\!29}a^{8}-\frac{24\!\cdots\!78}{25\!\cdots\!29}a^{7}-\frac{10\!\cdots\!60}{25\!\cdots\!29}a^{6}+\frac{28\!\cdots\!22}{25\!\cdots\!29}a^{5}+\frac{48\!\cdots\!61}{25\!\cdots\!29}a^{4}-\frac{17\!\cdots\!55}{25\!\cdots\!29}a^{3}-\frac{58\!\cdots\!36}{25\!\cdots\!29}a^{2}-\frac{75\!\cdots\!84}{25\!\cdots\!29}a+\frac{51\!\cdots\!14}{12\!\cdots\!57}$ (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: | \( 785284.164362 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 785284.164362 \cdot 3}{2\cdot\sqrt{15072974715383053921}}\cr\approx \mathstrut & 0.155341886320 \end{aligned}\] (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
3.3.361.1 |
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 | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.9.0.1}{9} }$ | R | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | R | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(19\) | 19.9.8.6 | $x^{9} + 133$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
\(31\) | 31.3.2.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
31.3.2.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
31.3.2.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |