Normalized defining polynomial
\( x^{9} + 3x^{7} + 9x^{5} - 9x^{4} + 8x^{3} - 18x + 9 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-22785532875\) \(\medspace = -\,3^{12}\cdot 5^{3}\cdot 7^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(14.15\) | 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))
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Galois root discriminant: | $3^{4/3}5^{1/2}7^{1/2}\approx 25.597390575239302$ | ||
Ramified primes: | \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-35}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3831}a^{8}-\frac{70}{1277}a^{7}-\frac{623}{1277}a^{6}+\frac{576}{1277}a^{5}+\frac{358}{1277}a^{4}+\frac{160}{1277}a^{3}-\frac{1186}{3831}a^{2}+\frac{15}{1277}a-\frac{602}{1277}$
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)
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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: | $\frac{364}{3831}a^{8}+\frac{60}{1277}a^{7}+\frac{534}{1277}a^{6}+\frac{236}{1277}a^{5}+\frac{1335}{1277}a^{4}-\frac{502}{1277}a^{3}+\frac{5030}{3831}a^{2}-\frac{925}{1277}a-\frac{761}{1277}$, $\frac{364}{3831}a^{8}+\frac{60}{1277}a^{7}+\frac{534}{1277}a^{6}+\frac{236}{1277}a^{5}+\frac{1335}{1277}a^{4}-\frac{502}{1277}a^{3}+\frac{5030}{3831}a^{2}-\frac{925}{1277}a-\frac{2038}{1277}$, $\frac{170}{1277}a^{8}+\frac{56}{1277}a^{7}+\frac{243}{1277}a^{6}+\frac{50}{1277}a^{5}+\frac{1246}{1277}a^{4}-\frac{128}{1277}a^{3}+\frac{146}{1277}a^{2}+\frac{2542}{1277}a-\frac{3094}{1277}$, $\frac{1021}{3831}a^{8}+\frac{42}{1277}a^{7}+\frac{1140}{1277}a^{6}+\frac{676}{1277}a^{5}+\frac{4127}{1277}a^{4}-\frac{1373}{1277}a^{3}+\frac{11183}{3831}a^{2}+\frac{1268}{1277}a-\frac{6790}{1277}$, $\frac{86}{1277}a^{8}-\frac{182}{1277}a^{7}+\frac{168}{1277}a^{6}-\frac{801}{1277}a^{5}+\frac{420}{1277}a^{4}-\frac{3415}{1277}a^{3}+\frac{1441}{1277}a^{2}-\frac{3792}{1277}a-\frac{799}{1277}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 100.309452929 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 100.309452929 \cdot 1}{2\cdot\sqrt{22785532875}}\cr\approx \mathstrut & 0.659343703838 \end{aligned}\]
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
\(\Q(\zeta_{9})^+\), 3.1.2835.1 |
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: | 6.0.3472875.1 |
Minimal sibling: | 6.0.3472875.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | R | R | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
\(5\) | 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
1.315.6t1.h.a | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.281302875.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.315.6t1.h.b | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.281302875.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
* | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 2.2835.3t2.a.a | $2$ | $ 3^{4} \cdot 5 \cdot 7 $ | 3.1.2835.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.315.6t5.b.a | $2$ | $ 3^{2} \cdot 5 \cdot 7 $ | 9.3.22785532875.2 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |
* | 2.315.6t5.b.b | $2$ | $ 3^{2} \cdot 5 \cdot 7 $ | 9.3.22785532875.2 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |