Normalized defining polynomial
\( x^{14} - x^{13} - 3x^{12} - x^{11} + x^{10} + 2x^{8} + 7x^{7} + 3x^{6} + 4x^{5} + 9x^{4} + 7x^{3} + x^{2} + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-270118421833507\) \(\medspace = -\,13^{4}\cdot 67\cdot 109^{4}\) | 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: | \(10.74\) | 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: | $13^{1/2}67^{1/2}109^{1/2}\approx 308.1217291915648$ | ||
Ramified primes: | \(13\), \(67\), \(109\) | 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: | \(\Q(\sqrt{-67}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13}a^{12}-\frac{6}{13}a^{11}+\frac{5}{13}a^{9}+\frac{2}{13}a^{8}-\frac{2}{13}a^{7}-\frac{3}{13}a^{6}-\frac{2}{13}a^{5}+\frac{3}{13}a^{4}+\frac{4}{13}a^{3}-\frac{1}{13}a^{2}-\frac{5}{13}a+\frac{1}{13}$, $\frac{1}{17771}a^{13}-\frac{6}{17771}a^{12}+\frac{633}{1367}a^{11}-\frac{1503}{17771}a^{10}+\frac{7516}{17771}a^{9}+\frac{3430}{17771}a^{8}-\frac{744}{17771}a^{7}+\frac{5094}{17771}a^{6}+\frac{3240}{17771}a^{5}+\frac{2942}{17771}a^{4}-\frac{7866}{17771}a^{3}+\frac{1061}{17771}a^{2}+\frac{4265}{17771}a+\frac{673}{1367}$
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: | $\frac{7294}{17771}a^{13}-\frac{13690}{17771}a^{12}-\frac{10846}{17771}a^{11}+\frac{1825}{17771}a^{10}+\frac{6371}{17771}a^{9}+\frac{3687}{17771}a^{8}+\frac{1702}{1367}a^{7}+\frac{30650}{17771}a^{6}-\frac{9705}{17771}a^{5}+\frac{28489}{17771}a^{4}+\frac{21625}{17771}a^{3}-\frac{3754}{17771}a^{2}-\frac{16313}{17771}a-\frac{5923}{17771}$, $\frac{3447}{17771}a^{13}-\frac{7012}{17771}a^{12}-\frac{8089}{17771}a^{11}+\frac{8291}{17771}a^{10}+\frac{967}{1367}a^{9}-\frac{2707}{17771}a^{8}+\frac{2658}{17771}a^{7}+\frac{13573}{17771}a^{6}-\frac{19248}{17771}a^{5}-\frac{699}{17771}a^{4}+\frac{447}{1367}a^{3}+\frac{542}{17771}a^{2}-\frac{27970}{17771}a-\frac{3685}{17771}$, $a$, $\frac{3609}{17771}a^{13}-\frac{13452}{17771}a^{12}+\frac{7221}{17771}a^{11}+\frac{13599}{17771}a^{10}-\frac{5605}{17771}a^{9}-\frac{8884}{17771}a^{8}+\frac{17463}{17771}a^{7}+\frac{169}{1367}a^{6}-\frac{2641}{1367}a^{5}+\frac{32997}{17771}a^{4}-\frac{10841}{17771}a^{3}-\frac{2720}{1367}a^{2}-\frac{213}{1367}a+\frac{4276}{17771}$, $\frac{777}{17771}a^{13}-\frac{1928}{17771}a^{12}-\frac{2260}{17771}a^{11}+\frac{5055}{17771}a^{10}+\frac{6943}{17771}a^{9}-\frac{12843}{17771}a^{8}+\frac{2887}{17771}a^{7}+\frac{4674}{17771}a^{6}-\frac{11470}{17771}a^{5}+\frac{129}{1367}a^{4}+\frac{12278}{17771}a^{3}+\frac{4197}{17771}a^{2}-\frac{5171}{17771}a+\frac{12185}{17771}$, $\frac{3581}{17771}a^{13}-\frac{3715}{17771}a^{12}-\frac{1080}{1367}a^{11}+\frac{2370}{17771}a^{10}+\frac{9502}{17771}a^{9}+\frac{3069}{17771}a^{8}+\frac{1386}{17771}a^{7}+\frac{26339}{17771}a^{6}-\frac{2023}{17771}a^{5}-\frac{2901}{17771}a^{4}+\frac{34431}{17771}a^{3}+\frac{14218}{17771}a^{2}-\frac{10095}{17771}a-\frac{1375}{1367}$, $\frac{1015}{17771}a^{13}+\frac{4846}{17771}a^{12}-\frac{12238}{17771}a^{11}-\frac{15010}{17771}a^{10}+\frac{6348}{17771}a^{9}+\frac{2435}{17771}a^{8}+\frac{4892}{17771}a^{7}+\frac{19554}{17771}a^{6}+\frac{32406}{17771}a^{5}-\frac{164}{1367}a^{4}+\frac{38933}{17771}a^{3}+\frac{35261}{17771}a^{2}+\frac{9255}{17771}a+\frac{5671}{17771}$, $\frac{662}{17771}a^{13}+\frac{129}{17771}a^{12}+\frac{2837}{17771}a^{11}-\frac{17581}{17771}a^{10}+\frac{2446}{17771}a^{9}+\frac{4174}{17771}a^{8}-\frac{3142}{17771}a^{7}+\frac{1206}{17771}a^{6}+\frac{21929}{17771}a^{5}+\frac{22868}{17771}a^{4}-\frac{1756}{17771}a^{3}+\frac{58525}{17771}a^{2}+\frac{12878}{17771}a+\frac{2593}{17771}$ | 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: | \( 33.079338429 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{5}\cdot 33.079338429 \cdot 1}{2\cdot\sqrt{270118421833507}}\cr\approx \mathstrut & 0.15767725192 \end{aligned}\]
Galois group
$C_2^7.\GL(3,2)$ (as 14T51):
A non-solvable group of order 21504 |
The 48 conjugacy class representatives for $C_2^7.\GL(3,2)$ |
Character table for $C_2^7.\GL(3,2)$ |
Intermediate fields
7.3.2007889.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 14 sibling: | data not computed |
Degree 28 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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(67\) | 67.2.1.1 | $x^{2} + 134$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
67.3.0.1 | $x^{3} + 6 x + 65$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
67.3.0.1 | $x^{3} + 6 x + 65$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
67.6.0.1 | $x^{6} + 63 x^{3} + 49 x^{2} + 55 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(109\) | $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
109.2.1.1 | $x^{2} + 109$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.1 | $x^{2} + 109$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.4.0.1 | $x^{4} + 11 x^{2} + 98 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |