Normalized defining polynomial
\( x^{9} - 819x^{7} - 819x^{6} + 105651x^{5} + 14742x^{4} - 2502318x^{3} + 1938573x^{2} + 5214573x - 4850209 \)
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: | \(147570226003740066617015289\) \(\medspace = 3^{22}\cdot 7^{8}\cdot 13^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(808.47\) | 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: | $3^{22/9}7^{8/9}13^{8/9}\approx 808.4748732595972$ | ||
Ramified primes: | \(3\), \(7\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{4}a^{3}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{16}a^{6}-\frac{1}{8}a^{4}-\frac{1}{8}a^{3}+\frac{1}{16}a^{2}+\frac{1}{8}a+\frac{1}{16}$, $\frac{1}{16}a^{7}-\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{1}{16}a^{3}+\frac{1}{8}a^{2}+\frac{1}{16}a$, $\frac{1}{13\!\cdots\!00}a^{8}+\frac{39\!\cdots\!01}{13\!\cdots\!00}a^{7}+\frac{24\!\cdots\!07}{13\!\cdots\!00}a^{6}+\frac{92\!\cdots\!43}{83\!\cdots\!25}a^{5}+\frac{35\!\cdots\!89}{13\!\cdots\!00}a^{4}-\frac{15\!\cdots\!19}{13\!\cdots\!00}a^{3}+\frac{14\!\cdots\!47}{33\!\cdots\!00}a^{2}+\frac{16\!\cdots\!11}{13\!\cdots\!00}a+\frac{33\!\cdots\!09}{13\!\cdots\!00}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{3}\times C_{9}$, which has order $27$ (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: | $\frac{430264597624153}{33\!\cdots\!00}a^{8}-\frac{19\!\cdots\!13}{13\!\cdots\!00}a^{7}-\frac{19\!\cdots\!41}{13\!\cdots\!00}a^{6}+\frac{39\!\cdots\!53}{66\!\cdots\!00}a^{5}+\frac{68\!\cdots\!67}{33\!\cdots\!00}a^{4}+\frac{74\!\cdots\!97}{13\!\cdots\!00}a^{3}-\frac{76\!\cdots\!19}{13\!\cdots\!00}a^{2}-\frac{54\!\cdots\!43}{13\!\cdots\!00}a+\frac{83\!\cdots\!83}{13\!\cdots\!00}$, $\frac{6238707848036}{33\!\cdots\!09}a^{8}-\frac{14\!\cdots\!35}{53\!\cdots\!44}a^{7}-\frac{47\!\cdots\!23}{53\!\cdots\!44}a^{6}+\frac{11\!\cdots\!67}{26\!\cdots\!72}a^{5}+\frac{28\!\cdots\!82}{33\!\cdots\!09}a^{4}-\frac{10\!\cdots\!73}{53\!\cdots\!44}a^{3}-\frac{45\!\cdots\!85}{53\!\cdots\!44}a^{2}+\frac{28\!\cdots\!75}{53\!\cdots\!44}a+\frac{51\!\cdots\!17}{53\!\cdots\!44}$, $\frac{17\!\cdots\!01}{26\!\cdots\!20}a^{8}+\frac{54\!\cdots\!41}{26\!\cdots\!20}a^{7}-\frac{18\!\cdots\!21}{33\!\cdots\!90}a^{6}-\frac{14\!\cdots\!53}{66\!\cdots\!80}a^{5}+\frac{18\!\cdots\!19}{26\!\cdots\!20}a^{4}+\frac{53\!\cdots\!71}{26\!\cdots\!20}a^{3}-\frac{50\!\cdots\!87}{26\!\cdots\!20}a^{2}-\frac{31\!\cdots\!99}{26\!\cdots\!20}a+\frac{39\!\cdots\!27}{13\!\cdots\!60}$, $\frac{14\!\cdots\!71}{33\!\cdots\!00}a^{8}+\frac{66\!\cdots\!21}{33\!\cdots\!00}a^{7}-\frac{22\!\cdots\!31}{66\!\cdots\!00}a^{6}-\frac{64\!\cdots\!27}{33\!\cdots\!00}a^{5}+\frac{12\!\cdots\!69}{33\!\cdots\!00}a^{4}+\frac{14\!\cdots\!44}{83\!\cdots\!25}a^{3}-\frac{17\!\cdots\!29}{66\!\cdots\!00}a^{2}-\frac{63\!\cdots\!47}{16\!\cdots\!50}a+\frac{29\!\cdots\!03}{66\!\cdots\!00}$, $\frac{44\!\cdots\!59}{13\!\cdots\!00}a^{8}+\frac{10\!\cdots\!17}{66\!\cdots\!00}a^{7}-\frac{32\!\cdots\!37}{13\!\cdots\!00}a^{6}-\frac{12\!\cdots\!29}{66\!\cdots\!00}a^{5}+\frac{31\!\cdots\!01}{13\!\cdots\!00}a^{4}+\frac{12\!\cdots\!27}{66\!\cdots\!00}a^{3}+\frac{39\!\cdots\!99}{16\!\cdots\!50}a^{2}-\frac{85\!\cdots\!11}{83\!\cdots\!25}a+\frac{10\!\cdots\!81}{13\!\cdots\!00}$, $\frac{11\!\cdots\!63}{13\!\cdots\!00}a^{8}+\frac{45\!\cdots\!19}{66\!\cdots\!00}a^{7}-\frac{81\!\cdots\!59}{13\!\cdots\!00}a^{6}-\frac{37\!\cdots\!53}{66\!\cdots\!00}a^{5}+\frac{44\!\cdots\!57}{13\!\cdots\!00}a^{4}+\frac{11\!\cdots\!39}{66\!\cdots\!00}a^{3}-\frac{40\!\cdots\!53}{66\!\cdots\!00}a^{2}-\frac{89\!\cdots\!79}{16\!\cdots\!50}a+\frac{13\!\cdots\!67}{13\!\cdots\!00}$, $\frac{70\!\cdots\!73}{26\!\cdots\!20}a^{8}+\frac{70\!\cdots\!13}{26\!\cdots\!20}a^{7}-\frac{49\!\cdots\!69}{26\!\cdots\!20}a^{6}-\frac{69\!\cdots\!97}{33\!\cdots\!90}a^{5}+\frac{14\!\cdots\!77}{26\!\cdots\!20}a^{4}+\frac{13\!\cdots\!13}{26\!\cdots\!20}a^{3}-\frac{38\!\cdots\!39}{66\!\cdots\!80}a^{2}-\frac{28\!\cdots\!77}{26\!\cdots\!20}a+\frac{30\!\cdots\!57}{26\!\cdots\!20}$, $\frac{49\!\cdots\!81}{66\!\cdots\!80}a^{8}+\frac{19\!\cdots\!79}{26\!\cdots\!20}a^{7}-\frac{14\!\cdots\!17}{26\!\cdots\!20}a^{6}-\frac{75\!\cdots\!89}{13\!\cdots\!60}a^{5}+\frac{12\!\cdots\!39}{66\!\cdots\!80}a^{4}+\frac{37\!\cdots\!69}{26\!\cdots\!20}a^{3}-\frac{52\!\cdots\!43}{26\!\cdots\!20}a^{2}-\frac{81\!\cdots\!51}{26\!\cdots\!20}a+\frac{96\!\cdots\!31}{26\!\cdots\!20}$ (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: | \( 1557459742.51 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 1557459742.51 \cdot 27}{2\cdot\sqrt{147570226003740066617015289}}\cr\approx \mathstrut & 0.886178456802 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 27 |
The 11 conjugacy class representatives for $C_9:C_3$ |
Character table for $C_9:C_3$ |
Intermediate fields
3.3.670761.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/11.9.0.1}{9} }$ | R | ${\href{/padicField/17.9.0.1}{9} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.22.11 | $x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$ | $9$ | $1$ | $22$ | $C_9:C_3$ | $[2, 3]^{3}$ |
\(7\) | 7.9.8.2 | $x^{9} + 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
\(13\) | 13.9.8.3 | $x^{9} + 52$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |