Normalized defining polynomial
\( x^{9} - 4x^{8} - 3x^{7} + 29x^{6} - 26x^{5} - 24x^{4} + 34x^{3} - 2x^{2} - 5x + 1 \)
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: | \(17515230173\) \(\medspace = 7^{6}\cdot 53^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.75\) | 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: | $7^{2/3}53^{1/2}\approx 26.640147687438905$ | ||
Ramified primes: | \(7\), \(53\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{53}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $6a^{8}-23a^{7}-23a^{6}+173a^{5}-119a^{4}-185a^{3}+168a^{2}+38a-23$, $6a^{8}-23a^{7}-23a^{6}+173a^{5}-119a^{4}-185a^{3}+168a^{2}+38a-22$, $a^{8}-3a^{7}-6a^{6}+23a^{5}-3a^{4}-27a^{3}+7a^{2}+5a$, $27a^{8}-100a^{7}-111a^{6}+751a^{5}-477a^{4}-796a^{3}+681a^{2}+155a-89$, $4a^{8}-16a^{7}-14a^{6}+120a^{5}-89a^{4}-126a^{3}+120a^{2}+24a-15$, $10a^{8}-36a^{7}-43a^{6}+270a^{5}-162a^{4}-284a^{3}+234a^{2}+53a-30$, $3a^{8}-10a^{7}-15a^{6}+76a^{5}-33a^{4}-86a^{3}+55a^{2}+20a-8$, $17a^{8}-61a^{7}-74a^{6}+458a^{5}-269a^{4}-485a^{3}+393a^{2}+93a-51$ | 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: | \( 92.2723177847 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 92.2723177847 \cdot 1}{2\cdot\sqrt{17515230173}}\cr\approx \mathstrut & 0.178485718004 \end{aligned}\]
Galois group
$C_3\times S_3$ (as 9T4):
A solvable group of order 18 |
The 9 conjugacy class representatives for $S_3\times C_3$ |
Character table for $S_3\times C_3$ |
Intermediate fields
\(\Q(\zeta_{7})^+\), 3.3.2597.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 6 sibling: | 6.6.7294973.1 |
Minimal sibling: | 6.6.7294973.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
\(53\) | 53.3.0.1 | $x^{3} + 3 x + 51$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
53.6.3.1 | $x^{6} + 7314 x^{5} + 17831697 x^{4} + 14491896314 x^{3} + 998947242 x^{2} + 44403234204 x + 739971484841$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.53.2t1.a.a | $1$ | $ 53 $ | \(\Q(\sqrt{53}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
1.371.6t1.b.a | $1$ | $ 7 \cdot 53 $ | 6.6.357453677.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
* | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
1.371.6t1.b.b | $1$ | $ 7 \cdot 53 $ | 6.6.357453677.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
* | 2.2597.3t2.a.a | $2$ | $ 7^{2} \cdot 53 $ | 3.3.2597.1 | $S_3$ (as 3T2) | $1$ | $2$ |
* | 2.371.6t5.b.a | $2$ | $ 7 \cdot 53 $ | 9.9.17515230173.1 | $S_3\times C_3$ (as 9T4) | $0$ | $2$ |
* | 2.371.6t5.b.b | $2$ | $ 7 \cdot 53 $ | 9.9.17515230173.1 | $S_3\times C_3$ (as 9T4) | $0$ | $2$ |