Normalized defining polynomial
\( x^{10} - 3x^{7} - 45x^{6} - 108x^{5} - 78x^{4} + 36x^{3} + 108x^{2} + 53x + 9 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(230446741890423969\) \(\medspace = 3^{18}\cdot 29^{6}\) | 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: | \(54.48\) | 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: | $3^{37/18}29^{2/3}\approx 90.29864728073873$ | ||
Ramified primes: | \(3\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{6}+\frac{1}{9}a^{5}+\frac{1}{9}a^{4}-\frac{4}{9}a^{3}-\frac{4}{9}a^{2}-\frac{4}{9}a$, $\frac{1}{9}a^{7}+\frac{1}{9}a^{4}-\frac{2}{9}a$, $\frac{1}{27}a^{8}-\frac{1}{27}a^{7}+\frac{1}{27}a^{5}-\frac{1}{27}a^{4}-\frac{1}{3}a^{3}+\frac{7}{27}a^{2}-\frac{7}{27}a+\frac{1}{3}$, $\frac{1}{27}a^{9}-\frac{1}{27}a^{7}+\frac{1}{27}a^{6}-\frac{1}{27}a^{4}-\frac{2}{27}a^{3}-\frac{7}{27}a+\frac{1}{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}$, which has order $4$
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: | $\frac{4}{27}a^{9}-\frac{1}{27}a^{8}-\frac{11}{27}a^{6}-\frac{178}{27}a^{5}-\frac{128}{9}a^{4}-\frac{209}{27}a^{3}+\frac{188}{27}a^{2}+\frac{116}{9}a+3$, $\frac{4}{27}a^{9}-\frac{1}{9}a^{8}+\frac{8}{27}a^{7}-\frac{23}{27}a^{6}-\frac{55}{9}a^{5}-\frac{316}{27}a^{4}-\frac{332}{27}a^{3}+\frac{2}{9}a^{2}+\frac{596}{27}a+\frac{19}{3}$, $\frac{1}{27}a^{9}-\frac{1}{27}a^{7}-\frac{2}{27}a^{6}-\frac{16}{9}a^{5}-\frac{103}{27}a^{4}-\frac{17}{27}a^{3}+\frac{28}{9}a^{2}+\frac{50}{27}a+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{2}{27}a^{8}-\frac{1}{27}a^{7}-\frac{8}{9}a^{6}-\frac{404}{27}a^{5}-\frac{871}{27}a^{4}-\frac{154}{9}a^{3}+\frac{487}{27}a^{2}+\frac{854}{27}a+\frac{25}{3}$, $\frac{5}{27}a^{8}-\frac{11}{27}a^{7}+\frac{2}{3}a^{6}-\frac{40}{27}a^{5}-\frac{155}{27}a^{4}-\frac{16}{3}a^{3}+\frac{134}{27}a^{2}+\frac{94}{27}a+\frac{2}{3}$ | 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: | \( 23018.4871573 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 23018.4871573 \cdot 4}{2\cdot\sqrt{230446741890423969}}\cr\approx \mathstrut & 0.597862001035 \end{aligned}\]
Galois group
A non-solvable group of order 360 |
The 7 conjugacy class representatives for $\PSL(2,9)$ |
Character table for $\PSL(2,9)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 6 siblings: | 6.2.49660209.2, 6.2.4640470641.4 |
Degree 15 siblings: | data not computed |
Degree 20 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 sibling: | data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.2.49660209.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.9.18.2 | $x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$ | $9$ | $1$ | $18$ | $C_3^2:C_2$ | $[3/2, 5/2]_{2}$ | |
\(29\) | $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
29.3.2.1 | $x^{3} + 29$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
29.6.4.1 | $x^{6} + 72 x^{5} + 1734 x^{4} + 14170 x^{3} + 5556 x^{2} + 50052 x + 397569$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |