Normalized defining polynomial
\( x^{11} - 3x^{10} + 6x^{9} - 4x^{8} + 2x^{7} + 4x^{6} - 3x^{5} + x^{4} + 2x^{3} + 5x^{2} - 2x - 1 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(337570274092\) \(\medspace = 2^{2}\cdot 89\cdot 4801\cdot 197507\) | 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: | \(11.17\) | 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: | $2\cdot 89^{1/2}4801^{1/2}197507^{1/2}\approx 581007.9810914821$ | ||
Ramified primes: | \(2\), \(89\), \(4801\), \(197507\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{84392568523}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5978}a^{10}+\frac{194}{427}a^{9}+\frac{990}{2989}a^{8}-\frac{3}{7}a^{7}-\frac{853}{2989}a^{6}+\frac{159}{2989}a^{5}-\frac{2171}{5978}a^{4}-\frac{1331}{2989}a^{3}+\frac{691}{2989}a^{2}-\frac{51}{122}a+\frac{2203}{5978}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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)
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Fundamental units: | $\frac{3285}{5978}a^{10}-\frac{648}{427}a^{9}+\frac{9085}{2989}a^{8}-\frac{13}{7}a^{7}+\frac{4566}{2989}a^{6}+\frac{5218}{2989}a^{5}-\frac{5959}{5978}a^{4}+\frac{3561}{2989}a^{3}+\frac{1284}{2989}a^{2}+\frac{337}{122}a-\frac{2503}{5978}$, $\frac{1605}{5978}a^{10}-\frac{340}{427}a^{9}+\frac{4780}{2989}a^{8}-\frac{6}{7}a^{7}-\frac{103}{2989}a^{6}+\frac{7108}{2989}a^{5}-\frac{11237}{5978}a^{4}+\frac{3869}{2989}a^{3}+\frac{3125}{2989}a^{2}+\frac{7}{122}a+\frac{2817}{5978}$, $\frac{1292}{2989}a^{10}-\frac{429}{427}a^{9}+\frac{5554}{2989}a^{8}-\frac{3}{7}a^{7}+\frac{1730}{2989}a^{6}+\frac{4352}{2989}a^{5}+\frac{1739}{2989}a^{4}-\frac{1954}{2989}a^{3}+\frac{1111}{2989}a^{2}+\frac{171}{61}a+\frac{748}{2989}$, $\frac{1565}{5978}a^{10}-\frac{414}{427}a^{9}+\frac{7026}{2989}a^{8}-\frac{19}{7}a^{7}+\frac{7116}{2989}a^{6}+\frac{748}{2989}a^{5}-\frac{8089}{5978}a^{4}+\frac{6296}{2989}a^{3}-\frac{603}{2989}a^{2}+\frac{95}{122}a-\frac{7589}{5978}$, $a$, $\frac{947}{5978}a^{10}-\frac{319}{427}a^{9}+\frac{4962}{2989}a^{8}-\frac{13}{7}a^{7}+\frac{2228}{2989}a^{6}+\frac{1123}{2989}a^{5}-\frac{5483}{5978}a^{4}+\frac{901}{2989}a^{3}+\frac{2775}{2989}a^{2}+\frac{15}{122}a-\frac{6059}{5978}$ | 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: | \( 34.4297607586 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 34.4297607586 \cdot 1}{2\cdot\sqrt{337570274092}}\cr\approx \mathstrut & 0.369429329349 \end{aligned}\]
Galois group
A non-solvable group of order 39916800 |
The 56 conjugacy class representatives for $S_{11}$ are not computed |
Character table for $S_{11}$ is not computed |
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 | ${\href{/padicField/3.11.0.1}{11} }$ | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(89\) | 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
89.3.0.1 | $x^{3} + 3 x + 86$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
89.6.0.1 | $x^{6} + x^{4} + 82 x^{3} + 80 x^{2} + 15 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(4801\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
\(197507\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |