Normalized defining polynomial
\( x^{9} - 2x^{8} - x^{7} + 11x^{6} - 25x^{5} - 33x^{4} + 35x^{3} + 18x^{2} + 14x + 29 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(4915625528641\) \(\medspace = 1489^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(25.71\) | 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: | $1489^{1/2}\approx 38.58756276314948$ | ||
Ramified primes: | \(1489\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{222397}a^{8}-\frac{4040}{31771}a^{7}-\frac{37773}{222397}a^{6}-\frac{27886}{222397}a^{5}-\frac{8497}{31771}a^{4}-\frac{41382}{222397}a^{3}-\frac{52783}{222397}a^{2}+\frac{91425}{222397}a+\frac{48989}{222397}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{1391}{31771}a^{8}-\frac{4982}{31771}a^{7}+\frac{6991}{31771}a^{6}+\frac{2965}{31771}a^{5}-\frac{35376}{31771}a^{4}+\frac{6690}{31771}a^{3}+\frac{33399}{31771}a^{2}-\frac{7138}{31771}a-\frac{5096}{31771}$, $\frac{1065}{31771}a^{8}+\frac{708}{31771}a^{7}-\frac{6159}{31771}a^{6}+\frac{7295}{31771}a^{5}+\frac{6239}{31771}a^{4}-\frac{100766}{31771}a^{3}-\frac{74538}{31771}a^{2}+\frac{116594}{31771}a+\frac{100616}{31771}$, $\frac{582}{31771}a^{8}-\frac{1582}{31771}a^{7}+\frac{1646}{31771}a^{6}+\frac{5329}{31771}a^{5}-\frac{18159}{31771}a^{4}-\frac{1906}{31771}a^{3}+\frac{2851}{31771}a^{2}-\frac{7075}{31771}a+\frac{13011}{31771}$, $\frac{22108}{222397}a^{8}-\frac{8039}{31771}a^{7}+\frac{15251}{222397}a^{6}+\frac{203193}{222397}a^{5}-\frac{85066}{31771}a^{4}-\frac{376792}{222397}a^{3}+\frac{657686}{222397}a^{2}+\frac{524758}{222397}a+\frac{197819}{222397}$, $\frac{4136}{31771}a^{8}-\frac{17029}{31771}a^{7}+\frac{20650}{31771}a^{6}+\frac{24005}{31771}a^{5}-\frac{161146}{31771}a^{4}+\frac{121509}{31771}a^{3}+\frac{146908}{31771}a^{2}-\frac{4642}{31771}a+\frac{141921}{31771}$, $\frac{3461}{222397}a^{8}-\frac{3200}{31771}a^{7}+\frac{37083}{222397}a^{6}+\frac{6852}{222397}a^{5}-\frac{19942}{31771}a^{4}+\frac{222963}{222397}a^{3}+\frac{128371}{222397}a^{2}-\frac{49006}{222397}a+\frac{84415}{222397}$ | 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: | \( 1307.61299806 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 1307.61299806 \cdot 1}{2\cdot\sqrt{4915625528641}}\cr\approx \mathstrut & 0.372537120002 \end{aligned}\]
Galois group
$C_3^3:S_4$ (as 9T30):
A solvable group of order 648 |
The 14 conjugacy class representatives for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
Character table for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
Intermediate fields
3.3.1489.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 siblings: | data not computed |
Degree 36 siblings: | 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 | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{5}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(1489\) | $\Q_{1489}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.1489.2t1.a.a | $1$ | $ 1489 $ | \(\Q(\sqrt{1489}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 2.1489.3t2.a.a | $2$ | $ 1489 $ | 3.3.1489.1 | $S_3$ (as 3T2) | $1$ | $2$ |
3.2217121.6t8.a.a | $3$ | $ 1489^{2}$ | 4.0.1489.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.1489.4t5.a.a | $3$ | $ 1489 $ | 4.0.1489.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
6.3301293169.18t217.a.a | $6$ | $ 1489^{3}$ | 9.5.4915625528641.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $2$ | |
* | 6.3301293169.9t30.a.a | $6$ | $ 1489^{3}$ | 9.5.4915625528641.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $2$ |
6.3301293169.36t1121.a.a | $6$ | $ 1489^{3}$ | 9.5.4915625528641.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $-2$ | |
6.3301293169.36t1121.a.b | $6$ | $ 1489^{3}$ | 9.5.4915625528641.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $-2$ | |
8.491...641.12t177.a.a | $8$ | $ 1489^{4}$ | 9.5.4915625528641.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
8.491...641.12t177.b.a | $8$ | $ 1489^{4}$ | 9.5.4915625528641.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
8.491...641.12t178.a.a | $8$ | $ 1489^{4}$ | 9.5.4915625528641.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
12.162...329.36t1123.a.a | $12$ | $ 1489^{7}$ | 9.5.4915625528641.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
12.731...449.18t218.a.a | $12$ | $ 1489^{5}$ | 9.5.4915625528641.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ |