Normalized defining polynomial
\( x^{10} + 3x^{8} + 18x^{6} - 12x^{5} - 18x^{4} - 27x^{2} + 12x - 9 \)
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: | \(15542770074624\) \(\medspace = 2^{10}\cdot 3^{12}\cdot 13^{4}\) | 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: | \(20.85\) | 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: | $2^{7/6}3^{25/18}13^{1/2}\approx 37.22560348698383$ | ||
Ramified primes: | \(2\), \(3\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{3}{8}$, $\frac{1}{3504}a^{9}-\frac{35}{1168}a^{8}-\frac{15}{146}a^{7}+\frac{11}{292}a^{6}-\frac{117}{584}a^{5}-\frac{127}{584}a^{4}+\frac{23}{292}a^{3}+\frac{35}{73}a^{2}-\frac{117}{1168}a-\frac{267}{1168}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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)
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Fundamental units: | $\frac{8}{219}a^{9}+\frac{23}{584}a^{8}+\frac{29}{292}a^{7}+\frac{21}{292}a^{6}+\frac{177}{292}a^{5}+\frac{12}{73}a^{4}-\frac{487}{292}a^{3}-\frac{257}{292}a^{2}-\frac{459}{292}a-\frac{79}{584}$, $\frac{167}{3504}a^{9}+\frac{141}{1168}a^{8}+\frac{27}{292}a^{7}+\frac{85}{292}a^{6}+\frac{463}{584}a^{5}+\frac{837}{584}a^{4}-\frac{226}{73}a^{3}-\frac{355}{146}a^{2}+\frac{2361}{1168}a-\frac{1811}{1168}$, $\frac{41}{1168}a^{9}+\frac{75}{1168}a^{8}+\frac{33}{292}a^{7}+\frac{39}{292}a^{6}+\frac{355}{584}a^{5}+\frac{439}{584}a^{4}-\frac{114}{73}a^{3}-\frac{148}{73}a^{2}-\frac{83}{1168}a+\frac{155}{1168}$, $\frac{1}{16}a^{9}+\frac{1}{16}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{11}{8}a^{5}+\frac{5}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{35}{16}a-\frac{7}{16}$, $\frac{41}{146}a^{9}-\frac{65}{584}a^{8}+\frac{191}{292}a^{7}-\frac{53}{292}a^{6}+\frac{1347}{292}a^{5}-\frac{364}{73}a^{4}-\frac{1969}{292}a^{3}+\frac{2053}{292}a^{2}-\frac{1115}{292}a+\frac{985}{584}$ | 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: | \( 1164.19186525 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 1164.19186525 \cdot 2}{2\cdot\sqrt{15542770074624}}\cr\approx \mathstrut & 1.84094076157 \end{aligned}\]
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: | 6.2.7884864.1, 6.2.1971216.1 |
Degree 15 siblings: | deg 15, 15.3.30638157055420022784.1 |
Degree 20 sibling: | 20.4.241577701592627342528741376.1 |
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: | 6.2.1971216.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.5 | $x^{4} + 2 x + 2$ | $4$ | $1$ | $4$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ |
2.6.6.8 | $x^{6} + 2 x + 2$ | $6$ | $1$ | $6$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.9.12.21 | $x^{9} + 6 x^{4} + 3$ | $9$ | $1$ | $12$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |