Normalized defining polynomial
\( x^{9} - 3x^{8} + 5x^{6} + 2x^{5} - 9x^{4} - 8x^{3} + 35x^{2} - 35x + 11 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-35904339899\) \(\medspace = -\,3299^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.89\) | 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: | $3299^{1/2}\approx 57.436921923097515$ | ||
Ramified primes: | \(3299\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3299}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{487}a^{8}+\frac{224}{487}a^{7}+\frac{200}{487}a^{6}+\frac{114}{487}a^{5}+\frac{69}{487}a^{4}+\frac{70}{487}a^{3}-\frac{189}{487}a^{2}-\frac{12}{487}a+\frac{163}{487}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{82}{487}a^{8}-\frac{138}{487}a^{7}-\frac{158}{487}a^{6}+\frac{95}{487}a^{5}+\frac{301}{487}a^{4}-\frac{104}{487}a^{3}-\frac{401}{487}a^{2}+\frac{1451}{487}a-\frac{1244}{487}$, $a^{8}-2a^{7}-2a^{6}+3a^{5}+5a^{4}-4a^{3}-12a^{2}+23a-12$, $\frac{371}{487}a^{8}-\frac{660}{487}a^{7}-\frac{798}{487}a^{6}+\frac{899}{487}a^{5}+\frac{1736}{487}a^{4}-\frac{1302}{487}a^{3}-\frac{4374}{487}a^{2}+\frac{7723}{487}a-\frac{3324}{487}$, $\frac{8}{487}a^{8}-\frac{156}{487}a^{7}+\frac{139}{487}a^{6}+\frac{425}{487}a^{5}+\frac{65}{487}a^{4}-\frac{901}{487}a^{3}-\frac{51}{487}a^{2}+\frac{1852}{487}a-\frac{2105}{487}$, $\frac{469}{487}a^{8}-\frac{1110}{487}a^{7}-\frac{678}{487}a^{6}+\frac{1844}{487}a^{5}+\frac{2167}{487}a^{4}-\frac{2721}{487}a^{3}-\frac{5851}{487}a^{2}+\frac{12878}{487}a-\frac{8291}{487}$ | 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: | \( 56.4248753472 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 56.4248753472 \cdot 1}{2\cdot\sqrt{35904339899}}\cr\approx \mathstrut & 0.295458665150 \end{aligned}\]
Galois group
$C_3^2:S_3$ (as 9T12):
A solvable group of order 54 |
The 10 conjugacy class representatives for $(C_3^2:C_3):C_2$ |
Character table for $(C_3^2:C_3):C_2$ |
Intermediate fields
3.1.3299.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 sibling: | data not computed |
Minimal sibling: | 9.3.35904339899.4 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.3.0.1}{3} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3299\) | $\Q_{3299}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{3299}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{3299}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3299.2t1.a.a | $1$ | $ 3299 $ | \(\Q(\sqrt{-3299}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.3299.3t2.d.a | $2$ | $ 3299 $ | 3.1.3299.4 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.3299.3t2.a.a | $2$ | $ 3299 $ | 3.1.3299.3 | $S_3$ (as 3T2) | $1$ | $0$ | |
* | 2.3299.3t2.b.a | $2$ | $ 3299 $ | 3.1.3299.2 | $S_3$ (as 3T2) | $1$ | $0$ |
2.3299.3t2.c.a | $2$ | $ 3299 $ | 3.1.3299.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
* | 3.3299.9t12.d.a | $3$ | $ 3299 $ | 9.3.35904339899.2 | $(C_3^2:C_3):C_2$ (as 9T12) | $0$ | $1$ |
* | 3.3299.9t12.d.b | $3$ | $ 3299 $ | 9.3.35904339899.2 | $(C_3^2:C_3):C_2$ (as 9T12) | $0$ | $1$ |
3.10883401.18t24.d.a | $3$ | $ 3299^{2}$ | 9.3.35904339899.2 | $(C_3^2:C_3):C_2$ (as 9T12) | $0$ | $-1$ | |
3.10883401.18t24.d.b | $3$ | $ 3299^{2}$ | 9.3.35904339899.2 | $(C_3^2:C_3):C_2$ (as 9T12) | $0$ | $-1$ |