Normalized defining polynomial
\( x^{11} - 6x^{9} + 27x^{7} - 36x^{6} - 18x^{5} + 36x^{4} - 72x^{2} + 108 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(12754584000000000\)
\(\medspace = 2^{12}\cdot 3^{13}\cdot 5^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(29.12\) | 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))
| |
Galois root discriminant: | $2^{31/14}3^{25/18}5^{71/60}\approx 143.33403716385226$ | ||
Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{15}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{6}a^{7}-\frac{1}{6}a^{5}$, $\frac{1}{6}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{18}a^{9}-\frac{1}{6}a^{5}$, $\frac{1}{273168}a^{10}-\frac{1135}{91056}a^{9}-\frac{5207}{91056}a^{8}+\frac{4267}{91056}a^{7}-\frac{949}{15176}a^{6}+\frac{4153}{45528}a^{5}+\frac{760}{1897}a^{4}+\frac{2627}{7588}a^{3}+\frac{1317}{7588}a^{2}+\frac{121}{7588}a-\frac{2253}{7588}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: |
$\frac{3385}{91056}a^{10}+\frac{8329}{273168}a^{9}-\frac{6359}{30352}a^{8}-\frac{26447}{91056}a^{7}+\frac{29347}{45528}a^{6}-\frac{15565}{45528}a^{5}-\frac{289}{3794}a^{4}+\frac{5365}{7588}a^{3}+\frac{11667}{7588}a^{2}-\frac{8089}{7588}a-\frac{8983}{7588}$, $\frac{865}{68292}a^{10}+\frac{871}{22764}a^{9}-\frac{577}{22764}a^{8}-\frac{1467}{7588}a^{7}-\frac{349}{11382}a^{6}+\frac{662}{5691}a^{5}-\frac{1181}{3794}a^{4}-\frac{251}{1897}a^{3}+\frac{1005}{1897}a^{2}+\frac{330}{1897}a-\frac{626}{1897}$, $\frac{79}{3794}a^{10}+\frac{757}{17073}a^{9}-\frac{565}{5691}a^{8}-\frac{1363}{11382}a^{7}+\frac{3676}{5691}a^{6}+\frac{205}{11382}a^{5}-\frac{6457}{3794}a^{4}+\frac{4195}{1897}a^{3}+\frac{435}{1897}a^{2}-\frac{6256}{1897}a+\frac{4061}{1897}$, $\frac{8587}{136584}a^{10}+\frac{5417}{136584}a^{9}-\frac{19189}{45528}a^{8}-\frac{5633}{15176}a^{7}+\frac{43081}{22764}a^{6}-\frac{1861}{7588}a^{5}-\frac{11519}{3794}a^{4}-\frac{4869}{3794}a^{3}+\frac{10547}{3794}a^{2}+\frac{7059}{3794}a-\frac{905}{3794}$, $\frac{1181}{39024}a^{10}+\frac{335}{39024}a^{9}-\frac{1019}{13008}a^{8}-\frac{1273}{13008}a^{7}+\frac{2429}{6504}a^{6}-\frac{3659}{6504}a^{5}+\frac{279}{271}a^{4}-\frac{4257}{1084}a^{3}+\frac{4173}{1084}a^{2}-\frac{4523}{1084}a-\frac{2825}{1084}$, $\frac{589}{30352}a^{10}+\frac{9535}{273168}a^{9}-\frac{27515}{91056}a^{8}+\frac{22303}{91056}a^{7}+\frac{7783}{15176}a^{6}-\frac{28087}{45528}a^{5}-\frac{2833}{3794}a^{4}+\frac{1747}{7588}a^{3}+\frac{8045}{7588}a^{2}-\frac{3559}{7588}a+\frac{359}{7588}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 82398.6846529 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 82398.6846529 \cdot 1}{2\cdot\sqrt{12754584000000000}}\cr\approx \mathstrut & 4.54848319624 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 39916800 |
The 56 conjugacy class representatives for $S_{11}$ |
Character table for $S_{11}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 22 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
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 | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.9.12.21 | $x^{9} + 6 x^{4} + 3$ | $9$ | $1$ | $12$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
\(5\)
| 5.5.5.1 | $x^{5} + 20 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ |
5.6.4.2 | $x^{6} + 10 x^{3} - 25$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |