Normalized defining polynomial
\( x^{14} - 2x^{13} + 3x^{12} - 6x^{11} + 6x^{10} - 4x^{9} + 3x^{8} + x^{7} - x^{5} + 3x^{4} + 3x^{2} + x + 1 \)
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: | \(-51902598731283\) \(\medspace = -\,3^{11}\cdot 17117^{2}\) | 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: | \(9.54\) | 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: | $3^{7/6}17117^{1/2}\approx 471.36278284837806$ | ||
Ramified primes: | \(3\), \(17117\) | 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: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{9}a^{11}-\frac{2}{9}a^{10}+\frac{1}{3}a^{9}+\frac{1}{9}a^{8}+\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{1}{9}a^{5}-\frac{1}{9}a^{4}+\frac{2}{9}a^{3}-\frac{1}{3}a^{2}-\frac{4}{9}a-\frac{4}{9}$, $\frac{1}{27}a^{12}-\frac{1}{27}a^{10}+\frac{7}{27}a^{9}-\frac{2}{9}a^{8}+\frac{1}{27}a^{7}-\frac{10}{27}a^{6}-\frac{8}{27}a^{5}+\frac{1}{3}a^{4}+\frac{10}{27}a^{3}-\frac{1}{27}a^{2}-\frac{4}{9}a-\frac{8}{27}$, $\frac{1}{27}a^{13}-\frac{1}{27}a^{11}+\frac{7}{27}a^{10}-\frac{2}{9}a^{9}+\frac{1}{27}a^{8}-\frac{10}{27}a^{7}-\frac{8}{27}a^{6}+\frac{1}{3}a^{5}+\frac{10}{27}a^{4}-\frac{1}{27}a^{3}-\frac{4}{9}a^{2}-\frac{8}{27}a$
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: | \( \frac{10}{27} a^{13} - \frac{13}{27} a^{12} + \frac{17}{27} a^{11} - \frac{52}{27} a^{10} + \frac{38}{27} a^{9} - \frac{20}{27} a^{8} + \frac{49}{27} a^{7} - \frac{4}{27} a^{6} + \frac{5}{27} a^{5} - \frac{17}{27} a^{4} + \frac{22}{27} a^{3} + \frac{1}{27} a^{2} + \frac{49}{27} a + \frac{23}{27} \) (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)
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Fundamental units: | $\frac{7}{27}a^{13}-\frac{13}{27}a^{12}+\frac{8}{27}a^{11}-\frac{22}{27}a^{10}+\frac{20}{27}a^{9}+\frac{19}{27}a^{8}-\frac{14}{27}a^{7}+\frac{5}{27}a^{6}-\frac{7}{27}a^{5}-\frac{8}{27}a^{4}+\frac{1}{27}a^{3}+\frac{19}{27}a^{2}-\frac{14}{27}a-\frac{10}{27}$, $\frac{10}{27}a^{13}-\frac{1}{27}a^{12}-\frac{10}{27}a^{11}-\frac{10}{27}a^{10}-\frac{40}{27}a^{9}+\frac{70}{27}a^{8}-\frac{20}{27}a^{7}+\frac{38}{27}a^{6}-\frac{10}{27}a^{5}+\frac{10}{27}a^{4}+\frac{7}{27}a^{3}+\frac{70}{27}a^{2}+\frac{40}{27}a+\frac{35}{27}$, $\frac{2}{9}a^{13}-\frac{8}{9}a^{12}+\frac{16}{9}a^{11}-\frac{23}{9}a^{10}+\frac{31}{9}a^{9}-\frac{31}{9}a^{8}+\frac{8}{9}a^{7}+\frac{10}{9}a^{6}-\frac{8}{9}a^{5}+\frac{2}{9}a^{4}+\frac{17}{9}a^{3}-\frac{16}{9}a^{2}+\frac{8}{9}a+\frac{1}{9}$, $\frac{1}{9}a^{13}+\frac{5}{27}a^{12}-\frac{4}{9}a^{11}+\frac{7}{27}a^{10}-\frac{37}{27}a^{9}+\frac{5}{3}a^{8}-\frac{7}{27}a^{7}+\frac{16}{27}a^{6}+\frac{5}{27}a^{5}-\frac{8}{9}a^{4}+\frac{29}{27}a^{3}+\frac{13}{27}a^{2}+\frac{11}{9}a+\frac{23}{27}$, $\frac{2}{9}a^{13}+\frac{1}{27}a^{12}-\frac{2}{3}a^{11}+\frac{11}{27}a^{10}-\frac{38}{27}a^{9}+\frac{23}{9}a^{8}-\frac{17}{27}a^{7}-\frac{19}{27}a^{6}+\frac{34}{27}a^{5}-a^{4}+\frac{7}{27}a^{3}+\frac{71}{27}a^{2}+\frac{5}{9}a+\frac{13}{27}$, $\frac{1}{9}a^{13}-\frac{7}{27}a^{12}+\frac{4}{9}a^{11}-\frac{29}{27}a^{10}+\frac{32}{27}a^{9}-\frac{7}{9}a^{8}+\frac{32}{27}a^{7}+\frac{4}{27}a^{6}-\frac{37}{27}a^{5}+\frac{2}{9}a^{4}-\frac{16}{27}a^{3}+\frac{7}{27}a^{2}+a+\frac{23}{27}$ | 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: | \( 27.6082700412 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 27.6082700412 \cdot 1}{6\cdot\sqrt{51902598731283}}\cr\approx \mathstrut & 0.246917840876 \end{aligned}\]
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
\(\Q(\sqrt{-3}) \), 7.1.462159.1 |
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 | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.7.0.1}{7} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{7}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.7.4 | $x^{6} + 3 x^{2} + 3$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(17117\) | Deg $4$ | $2$ | $2$ | $2$ | |||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |