Normalized defining polynomial
\( x^{11} - x^{10} - 4x^{9} + x^{8} + 5x^{7} + 7x^{6} - 7x^{5} - 5x^{4} + 7x^{3} - 3x^{2} - x + 1 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[7, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(610429790897\) | 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: | \(11.79\) | 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: | $610429790897^{1/2}\approx 781300.0645699449$ | ||
Ramified primes: | \(610429790897\) | 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: | $\Q(\sqrt{610429790897}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{97}a^{10}+\frac{38}{97}a^{9}+\frac{23}{97}a^{8}+\frac{25}{97}a^{7}+\frac{10}{97}a^{6}+\frac{9}{97}a^{5}-\frac{44}{97}a^{4}+\frac{25}{97}a^{3}+\frac{12}{97}a^{2}-\frac{20}{97}a-\frac{5}{97}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $a$, $\frac{196}{97}a^{10}-\frac{21}{97}a^{9}-\frac{827}{97}a^{8}-\frac{532}{97}a^{7}+\frac{602}{97}a^{6}+\frac{1958}{97}a^{5}+\frac{300}{97}a^{4}-\frac{920}{97}a^{3}+\frac{509}{97}a^{2}-\frac{137}{97}a-\frac{301}{97}$, $\frac{225}{97}a^{10}-\frac{83}{97}a^{9}-\frac{936}{97}a^{8}-\frac{389}{97}a^{7}+\frac{795}{97}a^{6}+\frac{2122}{97}a^{5}-\frac{103}{97}a^{4}-\frac{1068}{97}a^{3}+\frac{760}{97}a^{2}-\frac{329}{97}a-\frac{252}{97}$, $\frac{57}{97}a^{10}-\frac{65}{97}a^{9}-\frac{241}{97}a^{8}+\frac{67}{97}a^{7}+\frac{376}{97}a^{6}+\frac{513}{97}a^{5}-\frac{471}{97}a^{4}-\frac{515}{97}a^{3}+\frac{199}{97}a^{2}-\frac{73}{97}a+\frac{6}{97}$, $\frac{89}{97}a^{10}-\frac{13}{97}a^{9}-\frac{378}{97}a^{8}-\frac{200}{97}a^{7}+\frac{308}{97}a^{6}+\frac{801}{97}a^{5}-\frac{36}{97}a^{4}-\frac{491}{97}a^{3}+\frac{486}{97}a^{2}+\frac{63}{97}a-\frac{154}{97}$, $\frac{42}{97}a^{10}+\frac{44}{97}a^{9}-\frac{198}{97}a^{8}-\frac{308}{97}a^{7}+\frac{32}{97}a^{6}+\frac{572}{97}a^{5}+\frac{577}{97}a^{4}-\frac{211}{97}a^{3}-\frac{78}{97}a^{2}+\frac{33}{97}a-\frac{113}{97}$, $\frac{183}{97}a^{10}-\frac{127}{97}a^{9}-\frac{738}{97}a^{8}-\frac{81}{97}a^{7}+\frac{763}{97}a^{6}+\frac{1550}{97}a^{5}-\frac{680}{97}a^{4}-\frac{857}{97}a^{3}+\frac{838}{97}a^{2}-\frac{362}{97}a-\frac{42}{97}$, $\frac{196}{97}a^{10}-\frac{21}{97}a^{9}-\frac{827}{97}a^{8}-\frac{532}{97}a^{7}+\frac{602}{97}a^{6}+\frac{1958}{97}a^{5}+\frac{300}{97}a^{4}-\frac{1017}{97}a^{3}+\frac{509}{97}a^{2}+\frac{57}{97}a-\frac{301}{97}$ | 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: | \( 49.9574794147 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{2}\cdot 49.9574794147 \cdot 1}{2\cdot\sqrt{610429790897}}\cr\approx \mathstrut & 0.161555730955 \end{aligned}\]
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 | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.11.0.1}{11} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(610429790897\) | $\Q_{610429790897}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |