Normalized defining polynomial
\( x^{9} - 2x^{8} + 4x^{7} - 5x^{6} - 2x^{5} - 20x^{4} + 75x^{3} - 36x^{2} + 8x - 1 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(3709075402816\) \(\medspace = 2^{6}\cdot 7^{4}\cdot 17^{6}\) | 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: | \(24.92\) | 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: | $2^{2/3}7^{1/2}17^{3/4}\approx 35.16190326674663$ | ||
Ramified primes: | \(2\), \(7\), \(17\) | 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)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}+\frac{3}{7}a^{5}-\frac{1}{7}a^{4}+\frac{1}{7}a^{3}-\frac{2}{7}a^{2}-\frac{2}{7}a+\frac{1}{7}$, $\frac{1}{21}a^{7}+\frac{1}{21}a^{6}+\frac{1}{7}a^{4}+\frac{10}{21}a^{3}-\frac{5}{21}a^{2}+\frac{5}{21}a+\frac{5}{21}$, $\frac{1}{147}a^{8}+\frac{2}{147}a^{7}-\frac{2}{147}a^{6}-\frac{16}{49}a^{5}+\frac{37}{147}a^{4}-\frac{40}{147}a^{3}+\frac{16}{49}a^{2}-\frac{26}{147}a+\frac{23}{147}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $4$ | 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)
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Fundamental units: | $a$, $\frac{23}{49}a^{8}-\frac{142}{147}a^{7}+\frac{275}{147}a^{6}-\frac{117}{49}a^{5}-\frac{52}{49}a^{4}-\frac{1339}{147}a^{3}+\frac{5237}{147}a^{2}-\frac{2522}{147}a+\frac{292}{147}$, $\frac{8}{7}a^{8}-\frac{43}{21}a^{7}+\frac{86}{21}a^{6}-\frac{33}{7}a^{5}-\frac{24}{7}a^{4}-\frac{70}{3}a^{3}+\frac{1703}{21}a^{2}-\frac{479}{21}a+\frac{1}{3}$, $\frac{148}{147}a^{8}-\frac{250}{147}a^{7}+\frac{502}{147}a^{6}-\frac{191}{49}a^{5}-\frac{446}{147}a^{4}-\frac{3274}{147}a^{3}+\frac{3509}{49}a^{2}-\frac{2210}{147}a-\frac{187}{147}$ | 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: | \( 576.974862739 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 576.974862739 \cdot 3}{2\cdot\sqrt{3709075402816}}\cr\approx \mathstrut & 1.40076341224 \end{aligned}\]
Galois group
$C_3^2:C_4$ (as 9T9):
A solvable group of order 36 |
The 6 conjugacy class representatives for $C_3^2:C_4$ |
Character table for $C_3^2:C_4$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Galois closure: | data not computed |
Degree 6 siblings: | 6.2.16370116.1, 6.2.65480464.1 |
Degree 12 siblings: | deg 12, deg 12 |
Degree 18 sibling: | deg 18 |
Minimal sibling: | 6.2.16370116.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(17\) | $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
17.4.3.2 | $x^{4} + 34$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
17.4.3.2 | $x^{4} + 34$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |