Normalized defining polynomial
\( x^{11} - x^{10} + 5x^{9} - 4x^{8} + 10x^{7} - 6x^{6} + 11x^{5} - 7x^{4} + 9x^{3} - 4x^{2} + 2x + 1 \)
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)
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Discriminant: | \(-129891985607\) \(\medspace = -\,167^{5}\) | 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: | \(10.24\) | 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: | $167^{1/2}\approx 12.922847983320086$ | ||
Ramified primes: | \(167\) | 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: | \(\Q(\sqrt{-167}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17}a^{10}+\frac{1}{17}a^{9}+\frac{7}{17}a^{8}-\frac{7}{17}a^{7}-\frac{4}{17}a^{6}+\frac{3}{17}a^{5}-\frac{7}{17}a^{3}-\frac{5}{17}a^{2}+\frac{3}{17}a+\frac{8}{17}$
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: | $\frac{16}{17}a^{10}-\frac{1}{17}a^{9}+\frac{61}{17}a^{8}-\frac{10}{17}a^{7}+\frac{89}{17}a^{6}-\frac{20}{17}a^{5}+5a^{4}-\frac{44}{17}a^{3}+\frac{39}{17}a^{2}-\frac{3}{17}a+\frac{9}{17}$, $\frac{3}{17}a^{10}+\frac{3}{17}a^{9}+\frac{4}{17}a^{8}+\frac{13}{17}a^{7}-\frac{12}{17}a^{6}+\frac{26}{17}a^{5}-a^{4}+\frac{30}{17}a^{3}-\frac{15}{17}a^{2}+\frac{26}{17}a-\frac{10}{17}$, $\frac{15}{17}a^{10}-\frac{2}{17}a^{9}+\frac{54}{17}a^{8}-\frac{3}{17}a^{7}+\frac{76}{17}a^{6}+\frac{11}{17}a^{5}+4a^{4}-\frac{3}{17}a^{3}+\frac{27}{17}a^{2}+\frac{28}{17}a-\frac{16}{17}$, $\frac{6}{17}a^{10}-\frac{11}{17}a^{9}+\frac{25}{17}a^{8}-\frac{42}{17}a^{7}+\frac{44}{17}a^{6}-\frac{67}{17}a^{5}+3a^{4}-\frac{76}{17}a^{3}+\frac{55}{17}a^{2}-\frac{33}{17}a+\frac{14}{17}$, $\frac{7}{17}a^{10}-\frac{10}{17}a^{9}+\frac{32}{17}a^{8}-\frac{32}{17}a^{7}+\frac{57}{17}a^{6}-\frac{30}{17}a^{5}+3a^{4}-\frac{32}{17}a^{3}+\frac{33}{17}a^{2}-\frac{13}{17}a-\frac{12}{17}$ | 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: | \( 9.67372434998 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 9.67372434998 \cdot 1}{2\cdot\sqrt{129891985607}}\cr\approx \mathstrut & 0.262846302466 \end{aligned}\]
Galois group
A solvable group of order 22 |
The 7 conjugacy class representatives for $D_{11}$ |
Character table for $D_{11}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Galois closure: | 22.0.2817611963463158891460983.1 |
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 | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.11.0.1}{11} }$ | ${\href{/padicField/5.2.0.1}{2} }^{5}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.11.0.1}{11} }$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.2.0.1}{2} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.2.0.1}{2} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.2.0.1}{2} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(167\) | $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |