Normalized defining polynomial
\( x^{11} - 4x^{10} + 6x^{9} + 2x^{8} - 14x^{7} + 10x^{6} + 7x^{5} - 16x^{4} + 2x^{3} + 8x^{2} - x - 8 \)
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: | \(-77265229938688\) \(\medspace = -\,2^{15}\cdot 11^{9}\) | 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: | \(18.30\) | 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))
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Galois root discriminant: | $2^{3/2}11^{9/10}\approx 24.479267048299516$ | ||
Ramified primes: | \(2\), \(11\) | 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: | \(\Q(\sqrt{-22}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17467}a^{10}+\frac{3984}{17467}a^{9}-\frac{6772}{17467}a^{8}-\frac{2752}{17467}a^{7}-\frac{5714}{17467}a^{6}+\frac{7013}{17467}a^{5}+\frac{3184}{17467}a^{4}-\frac{733}{17467}a^{3}-\frac{6213}{17467}a^{2}+\frac{8237}{17467}a-\frac{6272}{17467}$
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: | $\frac{1643}{17467}a^{10}-\frac{4413}{17467}a^{9}+\frac{83}{17467}a^{8}+\frac{19884}{17467}a^{7}-\frac{25790}{17467}a^{6}-\frac{5861}{17467}a^{5}+\frac{26146}{17467}a^{4}-\frac{16563}{17467}a^{3}-\frac{7231}{17467}a^{2}+\frac{13933}{17467}a+\frac{18101}{17467}$, $\frac{1227}{17467}a^{10}-\frac{2392}{17467}a^{9}+\frac{5048}{17467}a^{8}-\frac{5573}{17467}a^{7}+\frac{10656}{17467}a^{6}+\frac{11187}{17467}a^{5}-\frac{23307}{17467}a^{4}-\frac{8574}{17467}a^{3}+\frac{9728}{17467}a^{2}-\frac{41528}{17467}a-\frac{27731}{17467}$, $\frac{1870}{17467}a^{10}-\frac{8329}{17467}a^{9}+\frac{17402}{17467}a^{8}-\frac{10942}{17467}a^{7}-\frac{12843}{17467}a^{6}+\frac{31527}{17467}a^{5}-\frac{19634}{17467}a^{4}-\frac{8284}{17467}a^{3}+\frac{14712}{17467}a^{2}-\frac{2704}{17467}a-\frac{8283}{17467}$, $\frac{986}{17467}a^{10}-\frac{1851}{17467}a^{9}-\frac{4798}{17467}a^{8}+\frac{28847}{17467}a^{7}-\frac{44564}{17467}a^{6}+\frac{15353}{17467}a^{5}+\frac{47765}{17467}a^{4}-\frac{76459}{17467}a^{3}+\frac{39833}{17467}a^{2}+\frac{16994}{17467}a-\frac{18341}{17467}$, $\frac{2159}{17467}a^{10}-\frac{9775}{17467}a^{9}+\frac{16598}{17467}a^{8}-\frac{2788}{17467}a^{7}-\frac{22291}{17467}a^{6}+\frac{14645}{17467}a^{5}+\frac{9725}{17467}a^{4}-\frac{10517}{17467}a^{3}+\frac{789}{17467}a^{2}+\frac{2277}{17467}a-\frac{4323}{17467}$ | 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: | \( 900.940644336 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 900.940644336 \cdot 1}{2\cdot\sqrt{77265229938688}}\cr\approx \mathstrut & 1.00369895245 \end{aligned}\]
Galois group
A solvable group of order 110 |
The 11 conjugacy class representatives for $F_{11}$ |
Character table for $F_{11}$ |
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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.10.15.5 | $x^{10} + 46 x^{8} + 808 x^{6} + 6768 x^{4} + 27216 x^{2} + 9568$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |