Normalized defining polynomial
\( x^{14} - 2 x^{13} + 2 x^{12} - 6 x^{11} + 8 x^{10} - 4 x^{9} + 9 x^{8} - 11 x^{7} + 6 x^{6} - 6 x^{5} + \cdots + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-16973485041483\) \(\medspace = -\,3^{7}\cdot 37^{2}\cdot 2381^{2}\) | 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: | \(8.81\) | 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: | $3^{1/2}37^{1/2}2381^{1/2}\approx 514.0924041454026$ | ||
Ramified primes: | \(3\), \(37\), \(2381\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{51}a^{13}+\frac{4}{51}a^{12}-\frac{8}{51}a^{11}-\frac{1}{17}a^{10}-\frac{10}{51}a^{9}+\frac{7}{17}a^{8}-\frac{6}{17}a^{7}+\frac{1}{3}a^{6}+\frac{2}{17}a^{5}-\frac{7}{17}a^{4}+\frac{5}{17}a^{3}+\frac{6}{17}a^{2}+\frac{25}{51}a-\frac{20}{51}$
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)
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Torsion generator: | \( -\frac{2}{17} a^{13} + \frac{10}{51} a^{12} - \frac{20}{51} a^{11} + \frac{52}{51} a^{10} - \frac{76}{51} a^{9} + \frac{112}{51} a^{8} - \frac{49}{17} a^{7} + \frac{10}{3} a^{6} - \frac{206}{51} a^{5} + \frac{109}{51} a^{4} - \frac{124}{51} a^{3} + \frac{130}{51} a^{2} - \frac{16}{17} a + \frac{52}{51} \) (order $6$) | 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: | $\frac{2}{51}a^{13}+\frac{8}{51}a^{12}-\frac{16}{51}a^{11}+\frac{28}{51}a^{10}-\frac{88}{51}a^{9}+\frac{31}{17}a^{8}-\frac{104}{51}a^{7}+\frac{10}{3}a^{6}-\frac{73}{51}a^{5}+\frac{94}{51}a^{4}-\frac{106}{51}a^{3}+\frac{12}{17}a^{2}-\frac{1}{51}a-\frac{2}{17}$, $\frac{53}{51}a^{13}-\frac{94}{51}a^{12}+\frac{86}{51}a^{11}-\frac{278}{51}a^{10}+\frac{320}{51}a^{9}-\frac{37}{17}a^{8}+\frac{355}{51}a^{7}-\frac{23}{3}a^{6}+\frac{233}{51}a^{5}-\frac{212}{51}a^{4}+\frac{149}{51}a^{3}-\frac{39}{17}a^{2}+\frac{50}{51}a-\frac{2}{17}$, $\frac{37}{51}a^{13}-\frac{56}{51}a^{12}+\frac{10}{51}a^{11}-\frac{128}{51}a^{10}+\frac{41}{17}a^{9}+\frac{55}{17}a^{8}+\frac{31}{51}a^{7}-2a^{6}-\frac{220}{51}a^{5}+\frac{175}{51}a^{4}-\frac{142}{51}a^{3}+\frac{35}{17}a^{2}-\frac{95}{51}a+\frac{110}{51}$, $\frac{40}{51}a^{13}-\frac{95}{51}a^{12}+\frac{35}{17}a^{11}-\frac{239}{51}a^{10}+\frac{331}{51}a^{9}-\frac{163}{51}a^{8}+\frac{232}{51}a^{7}-\frac{17}{3}a^{6}+\frac{63}{17}a^{5}-\frac{59}{17}a^{4}+\frac{22}{51}a^{3}-\frac{15}{17}a^{2}+\frac{16}{17}a+\frac{11}{17}$, $\frac{4}{51}a^{13}-\frac{35}{51}a^{12}+\frac{53}{51}a^{11}-\frac{21}{17}a^{10}+\frac{181}{51}a^{9}-\frac{188}{51}a^{8}+\frac{27}{17}a^{7}-5a^{6}+\frac{245}{51}a^{5}-\frac{50}{51}a^{4}+\frac{94}{51}a^{3}-\frac{44}{17}a^{2}+\frac{32}{51}a-\frac{4}{17}$, $\frac{8}{17}a^{13}-\frac{40}{51}a^{12}+\frac{46}{51}a^{11}-\frac{157}{51}a^{10}+\frac{185}{51}a^{9}-\frac{125}{51}a^{8}+\frac{94}{17}a^{7}-5a^{6}+\frac{48}{17}a^{5}-\frac{113}{51}a^{4}+\frac{35}{17}a^{3}-\frac{112}{51}a^{2}+\frac{5}{51}a+\frac{10}{17}$ | 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: | \( 11.1445691239 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 11.1445691239 \cdot 1}{6\cdot\sqrt{16973485041483}}\cr\approx \mathstrut & 0.174295269175 \end{aligned}\]
Galois group
$C_{7236}$ (as 14T49):
A non-solvable group of order 10080 |
The 30 conjugacy class representatives for $S_7\times C_2$ |
Character table for $S_7\times C_2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 7.3.792873.1 |
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 sibling: | 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 | ${\href{/padicField/2.14.0.1}{14} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(37\) | $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(2381\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |