Normalized defining polynomial
\( x^{14} - 3 x^{13} + 5 x^{12} - 5 x^{11} + 3 x^{10} - 6 x^{9} + 12 x^{8} - 21 x^{7} + 25 x^{6} - 19 x^{5} + \cdots + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-1667731600838171\) \(\medspace = -\,11^{9}\cdot 29^{4}\) | 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: | \(12.23\) | 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: | $11^{5/6}29^{2/3}\approx 69.62404259846498$ | ||
Ramified primes: | \(11\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{85}a^{13}-\frac{9}{85}a^{12}-\frac{26}{85}a^{11}-\frac{19}{85}a^{10}+\frac{32}{85}a^{9}-\frac{28}{85}a^{8}+\frac{2}{17}a^{7}+\frac{4}{85}a^{6}+\frac{1}{85}a^{5}-\frac{5}{17}a^{4}-\frac{6}{85}a^{3}+\frac{31}{85}a^{2}-\frac{12}{85}a-\frac{14}{85}$
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)
| |
Fundamental units: | $\frac{84}{85}a^{13}-\frac{161}{85}a^{12}+\frac{111}{85}a^{11}+\frac{19}{85}a^{10}-\frac{32}{85}a^{9}-\frac{482}{85}a^{8}+\frac{117}{17}a^{7}-\frac{429}{85}a^{6}+\frac{339}{85}a^{5}+\frac{5}{17}a^{4}+\frac{261}{85}a^{3}-\frac{31}{85}a^{2}+\frac{12}{85}a+\frac{99}{85}$, $a$, $\frac{98}{85}a^{13}-\frac{287}{85}a^{12}+\frac{427}{85}a^{11}-\frac{332}{85}a^{10}+\frac{76}{85}a^{9}-\frac{449}{85}a^{8}+\frac{230}{17}a^{7}-\frac{1818}{85}a^{6}+\frac{1798}{85}a^{5}-\frac{184}{17}a^{4}+\frac{602}{85}a^{3}-\frac{192}{85}a^{2}+\frac{269}{85}a-\frac{97}{85}$, $\frac{171}{85}a^{13}-\frac{264}{85}a^{12}+\frac{59}{85}a^{11}+\frac{321}{85}a^{10}-\frac{393}{85}a^{9}-\frac{793}{85}a^{8}+\frac{172}{17}a^{7}-\frac{336}{85}a^{6}-\frac{509}{85}a^{5}+\frac{335}{17}a^{4}-\frac{431}{85}a^{3}+\frac{711}{85}a^{2}-\frac{12}{85}a+\frac{241}{85}$, $\frac{243}{85}a^{13}-\frac{487}{85}a^{12}+\frac{397}{85}a^{11}-\frac{27}{85}a^{10}-\frac{129}{85}a^{9}-\frac{1279}{85}a^{8}+\frac{350}{17}a^{7}-\frac{1663}{85}a^{6}+\frac{1348}{85}a^{5}+\frac{26}{17}a^{4}+\frac{582}{85}a^{3}+\frac{138}{85}a^{2}+\frac{144}{85}a+\frac{83}{85}$, $\frac{72}{85}a^{13}-\frac{138}{85}a^{12}+\frac{168}{85}a^{11}-\frac{178}{85}a^{10}+\frac{179}{85}a^{9}-\frac{486}{85}a^{8}+\frac{93}{17}a^{7}-\frac{732}{85}a^{6}+\frac{1092}{85}a^{5}-\frac{156}{17}a^{4}+\frac{928}{85}a^{3}-\frac{233}{85}a^{2}+\frac{156}{85}a+\frac{12}{85}$ | 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: | \( 84.1958965155 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 84.1958965155 \cdot 1}{2\cdot\sqrt{1667731600838171}}\cr\approx \mathstrut & 0.398526247103 \end{aligned}\]
Galois group
$C_2\times A_7$ (as 14T47):
A non-solvable group of order 5040 |
The 18 conjugacy class representatives for $A_7\times C_2$ |
Character table for $A_7\times C_2$ |
Intermediate fields
\(\Q(\sqrt{-11}) \), 7.3.12313081.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 30 siblings: | data not computed |
Degree 42 siblings: | 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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.14.0.1}{14} }$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.6.5.2 | $x^{6} + 11$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |
11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(29\) | 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
29.6.4.1 | $x^{6} + 72 x^{5} + 1734 x^{4} + 14170 x^{3} + 5556 x^{2} + 50052 x + 397569$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |