Normalized defining polynomial
\( x^{9} - x^{8} + 3x^{7} + 4x^{6} + 14x^{5} - 3x^{4} - 9x^{3} + 7x^{2} - 2x - 1 \)
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: | \(-10021812416\) \(\medspace = -\,2^{6}\cdot 7^{6}\cdot 11^{3}\) | 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: | \(12.92\) | 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: | $2^{2/3}7^{2/3}11^{1/2}\approx 19.26556276580016$ | ||
Ramified primes: | \(2\), \(7\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{1257}a^{8}+\frac{167}{1257}a^{7}-\frac{14}{1257}a^{6}-\frac{253}{1257}a^{5}-\frac{57}{419}a^{4}-\frac{239}{1257}a^{3}+\frac{21}{419}a^{2}+\frac{116}{1257}a-\frac{69}{419}$
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: | $\frac{52}{419}a^{8}+\frac{74}{1257}a^{7}+\frac{110}{419}a^{6}+\frac{1175}{1257}a^{5}+\frac{1164}{419}a^{4}+\frac{980}{419}a^{3}-\frac{1066}{1257}a^{2}-\frac{1597}{1257}a+\frac{130}{419}$, $\frac{51}{419}a^{8}-\frac{8}{1257}a^{7}+\frac{124}{419}a^{6}+\frac{1096}{1257}a^{5}+\frac{916}{419}a^{4}+\frac{800}{419}a^{3}-\frac{836}{1257}a^{2}+\frac{988}{1257}a-\frac{82}{419}$, $\frac{79}{419}a^{8}-\frac{226}{1257}a^{7}+\frac{872}{1257}a^{6}+\frac{794}{1257}a^{5}+\frac{3887}{1257}a^{4}-\frac{26}{419}a^{3}+\frac{266}{1257}a^{2}+\frac{676}{1257}a-\frac{2131}{1257}$, $\frac{446}{419}a^{8}-\frac{519}{419}a^{7}+\frac{4313}{1257}a^{6}+\frac{1549}{419}a^{5}+\frac{17993}{1257}a^{4}-\frac{2263}{419}a^{3}-\frac{3327}{419}a^{2}+\frac{11072}{1257}a-\frac{5035}{1257}$, $\frac{631}{1257}a^{8}-\frac{210}{419}a^{7}+\frac{2060}{1257}a^{6}+\frac{697}{419}a^{5}+\frac{9838}{1257}a^{4}-\frac{2483}{1257}a^{3}-\frac{3404}{1257}a^{2}+\frac{709}{1257}a-\frac{308}{1257}$ | 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: | \( 30.3003873223 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 30.3003873223 \cdot 1}{2\cdot\sqrt{10021812416}}\cr\approx \mathstrut & 0.300313352129 \end{aligned}\]
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
\(\Q(\zeta_{7})^+\), 3.1.44.1 |
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: | 6.0.51131696.1 |
Minimal sibling: | 6.0.51131696.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | R | R | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(7\) | 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(11\) | 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
1.77.6t1.a.a | $1$ | $ 7 \cdot 11 $ | 6.0.3195731.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.77.6t1.a.b | $1$ | $ 7 \cdot 11 $ | 6.0.3195731.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
* | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 2.44.3t2.b.a | $2$ | $ 2^{2} \cdot 11 $ | 3.1.44.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.2156.6t5.a.a | $2$ | $ 2^{2} \cdot 7^{2} \cdot 11 $ | 9.3.10021812416.1 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |
* | 2.2156.6t5.a.b | $2$ | $ 2^{2} \cdot 7^{2} \cdot 11 $ | 9.3.10021812416.1 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |