Normalized defining polynomial
\( x^{10} - 4x^{9} + 7x^{8} - 15x^{6} + 13x^{4} + 8x^{3} + 32x^{2} - 64x + 20 \)
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: | \(38806720086016\) \(\medspace = 2^{18}\cdot 23^{6}\) | 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: | \(22.85\) | 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: | $2^{13/6}23^{3/4}\approx 47.15497666602499$ | ||
Ramified primes: | \(2\), \(23\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{59308}a^{9}+\frac{1581}{59308}a^{8}+\frac{129}{59308}a^{7}+\frac{26541}{59308}a^{6}+\frac{3271}{59308}a^{5}+\frac{24739}{59308}a^{4}+\frac{23567}{59308}a^{3}-\frac{10337}{59308}a^{2}-\frac{139}{29654}a-\frac{12769}{29654}$
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)
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Fundamental units: | $\frac{10609}{29654}a^{9}-\frac{13081}{14827}a^{8}+\frac{34131}{29654}a^{7}+\frac{26610}{14827}a^{6}-\frac{82103}{29654}a^{5}-\frac{57646}{14827}a^{4}-\frac{49879}{29654}a^{3}+\frac{5043}{14827}a^{2}+\frac{171146}{14827}a-\frac{66157}{14827}$, $\frac{6741}{29654}a^{9}-\frac{21011}{59308}a^{8}+\frac{9623}{29654}a^{7}+\frac{94733}{59308}a^{6}-\frac{12765}{29654}a^{5}-\frac{209745}{59308}a^{4}-\frac{109947}{29654}a^{3}-\frac{123077}{59308}a^{2}+\frac{100891}{14827}a+\frac{93601}{29654}$, $\frac{7}{14827}a^{9}-\frac{213}{59308}a^{8}+\frac{903}{14827}a^{7}-\frac{13029}{59308}a^{6}+\frac{8070}{14827}a^{5}-\frac{33831}{59308}a^{4}+\frac{1872}{14827}a^{3}-\frac{7723}{59308}a^{2}-\frac{1946}{14827}a+\frac{13143}{29654}$, $\frac{19}{59308}a^{9}+\frac{385}{59308}a^{8}+\frac{2451}{59308}a^{7}+\frac{161}{59308}a^{6}+\frac{2841}{59308}a^{5}+\frac{25231}{59308}a^{4}+\frac{32617}{59308}a^{3}-\frac{48133}{59308}a^{2}-\frac{61949}{29654}a+\frac{53929}{29654}$, $\frac{9883}{14827}a^{9}-\frac{49751}{29654}a^{8}+\frac{29439}{14827}a^{7}+\frac{104281}{29654}a^{6}-\frac{84529}{14827}a^{5}-\frac{255445}{29654}a^{4}-\frac{49163}{14827}a^{3}+\frac{98707}{29654}a^{2}+\frac{395850}{14827}a-\frac{66168}{14827}$ | 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: | \( 2439.80405674 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 2439.80405674 \cdot 1}{2\cdot\sqrt{38806720086016}}\cr\approx \mathstrut & 1.22081807075 \end{aligned}\]
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: | 6.2.286557184.1, 6.2.71639296.1 |
Degree 15 siblings: | deg 15, deg 15 |
Degree 20 sibling: | deg 20 |
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: | 6.2.71639296.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.8.7 | $x^{4} + 4 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ |
2.6.10.4 | $x^{6} + 2 x^{5} + 4 x^{3} + 4 x + 2$ | $6$ | $1$ | $10$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
\(23\) | 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
23.4.3.2 | $x^{4} + 115$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
23.4.3.1 | $x^{4} + 23$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |