Normalized defining polynomial
\( x^{11} - 44x^{8} - 393216 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-4775687280042915590049890304\) \(\medspace = -\,2^{33}\cdot 3^{11}\cdot 11^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(328.30\) | 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: | $2^{97/24}3^{13/8}11^{64/55}\approx 1598.7749512075377$ | ||
Ramified primes: | \(2\), \(3\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-6}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{4}a^{3}$, $\frac{1}{8}a^{4}-\frac{1}{2}a$, $\frac{1}{32}a^{5}+\frac{1}{8}a^{2}$, $\frac{1}{128}a^{6}-\frac{3}{32}a^{3}$, $\frac{1}{512}a^{7}+\frac{5}{128}a^{4}-\frac{1}{2}a$, $\frac{1}{2048}a^{8}+\frac{5}{512}a^{5}+\frac{1}{8}a^{2}$, $\frac{1}{8192}a^{9}+\frac{5}{2048}a^{6}-\frac{3}{32}a^{3}$, $\frac{1}{32768}a^{10}+\frac{5}{8192}a^{7}+\frac{5}{128}a^{4}-\frac{1}{2}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $\frac{1199020293527}{32768}a^{10}-\frac{142508336737}{512}a^{9}+\frac{27489289555}{256}a^{8}+\frac{9689284894611}{8192}a^{7}+\frac{778172431169}{128}a^{6}-\frac{1510740474897}{64}a^{5}-\frac{2920755739403}{128}a^{4}+\frac{4706972826359}{16}a^{3}-\frac{567134253543}{2}a^{2}-\frac{5292662267709}{2}a+7794372519089$, $\frac{235358059383}{4096}a^{10}-\frac{553037917521}{4096}a^{9}+\frac{62963942435}{256}a^{8}-\frac{2770113389571}{1024}a^{7}+\frac{4841927105571}{1024}a^{6}-\frac{155909934061}{64}a^{5}-\frac{3651155525615}{128}a^{4}+\frac{5824207949081}{32}a^{3}-\frac{3073748125881}{4}a^{2}+2694437652601a-8294223297187$, $\frac{2491397313587}{32768}a^{10}-\frac{7012973855015}{8192}a^{9}+\frac{1478426736387}{2048}a^{8}+\frac{37199376067935}{8192}a^{7}+\frac{32277956198445}{2048}a^{6}-\frac{39431086674465}{512}a^{5}-\frac{3935734288259}{128}a^{4}+\frac{3525267825635}{4}a^{3}-\frac{2505913309923}{2}a^{2}-\frac{14406628121555}{2}a+26376549135425$, $\frac{1660728023837}{2048}a^{10}-\frac{46223394193675}{8192}a^{9}+\frac{50146481918685}{2048}a^{8}-\frac{31483584564017}{256}a^{7}+\frac{10\!\cdots\!49}{2048}a^{6}-\frac{936516082812831}{512}a^{5}+\frac{721621250357745}{128}a^{4}-\frac{491880923253571}{32}a^{3}+\frac{144420704367047}{4}a^{2}-\frac{131610649136809}{2}a+47743102188593$, $\frac{12\!\cdots\!79}{16384}a^{10}-\frac{51\!\cdots\!43}{8192}a^{9}-\frac{14\!\cdots\!55}{2048}a^{8}+\frac{23\!\cdots\!03}{4096}a^{7}-\frac{77\!\cdots\!35}{2048}a^{6}+\frac{42\!\cdots\!69}{512}a^{5}+\frac{10\!\cdots\!57}{64}a^{4}-\frac{13\!\cdots\!79}{8}a^{3}+\frac{41\!\cdots\!55}{8}a^{2}+\frac{25\!\cdots\!79}{2}a-74\!\cdots\!79$ (assuming GRH) | 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: | \( 16881878101.6 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 16881878101.6 \cdot 1}{2\cdot\sqrt{4775687280042915590049890304}}\cr\approx \mathstrut & 2.39222575355 \end{aligned}\] (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 | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.8.31.25 | $x^{8} + 8 x^{6} + 16 x^{3} + 50$ | $8$ | $1$ | $31$ | $Z_8 : Z_8^\times$ | $[2, 3, 4, 5]^{2}$ | |
\(3\) | 3.3.4.1 | $x^{3} + 6 x^{2} + 21$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
3.8.7.1 | $x^{8} + 3$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ | |
\(11\) | 11.11.12.3 | $x^{11} + 77 x^{2} + 11$ | $11$ | $1$ | $12$ | $C_{11}:C_5$ | $[6/5]_{5}$ |