Normalized defining polynomial
\( x^{11} + x^{9} - x^{8} + 3x^{7} + 3x^{6} - 6x^{5} + 3x^{4} + 5x^{3} - 6x^{2} + 5x + 1 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-1461660310351\) \(\medspace = -\,271^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.76\) | 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: | $271^{1/2}\approx 16.46207763315433$ | ||
Ramified primes: | \(271\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-271}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{741}a^{10}-\frac{101}{741}a^{9}+\frac{25}{247}a^{8}+\frac{27}{247}a^{7}-\frac{9}{247}a^{6}-\frac{6}{19}a^{5}-\frac{28}{247}a^{4}+\frac{112}{247}a^{3}+\frac{155}{741}a^{2}-\frac{100}{741}a+\frac{75}{247}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
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)
| |
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: | $\frac{34}{741}a^{10}+\frac{8}{247}a^{9}-\frac{167}{741}a^{8}-\frac{70}{247}a^{7}-\frac{59}{247}a^{6}+\frac{5}{19}a^{5}-\frac{211}{247}a^{4}-\frac{391}{247}a^{3}+\frac{824}{741}a^{2}+\frac{184}{247}a-\frac{748}{741}$, $\frac{79}{741}a^{10}-\frac{25}{247}a^{9}-\frac{1}{247}a^{8}-\frac{90}{247}a^{7}+\frac{30}{247}a^{6}+\frac{1}{19}a^{5}-\frac{236}{247}a^{4}+\frac{203}{247}a^{3}+\frac{389}{741}a^{2}-\frac{328}{247}a+\frac{244}{247}$, $\frac{280}{741}a^{10}+\frac{125}{741}a^{9}+\frac{84}{247}a^{8}-\frac{97}{247}a^{7}+\frac{197}{247}a^{6}+\frac{30}{19}a^{5}-\frac{430}{247}a^{4}-\frac{9}{247}a^{3}+\frac{1163}{741}a^{2}-\frac{89}{741}a+\frac{252}{247}$, $\frac{66}{247}a^{10}+\frac{3}{247}a^{9}+\frac{277}{741}a^{8}-\frac{88}{247}a^{7}+\frac{194}{247}a^{6}+\frac{9}{19}a^{5}-\frac{357}{247}a^{4}+\frac{193}{247}a^{3}+\frac{103}{247}a^{2}-\frac{425}{247}a+\frac{1325}{741}$, $a$ | 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: | \( 55.1714638864 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 55.1714638864 \cdot 1}{2\cdot\sqrt{1461660310351}}\cr\approx \mathstrut & 0.446879695079 \end{aligned}\]
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: | 22.0.578978183833808423828407471.1 |
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 | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.2.0.1}{2} }^{5}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.11.0.1}{11} }$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.2.0.1}{2} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.11.0.1}{11} }$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(271\) | $\Q_{271}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |