Normalized defining polynomial
\( x^{14} - 14x^{12} + 84x^{10} - 322x^{8} + 896x^{6} - 1722x^{4} + 2212x^{2} - 1694 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(338637882455405716008075264\) \(\medspace = 2^{27}\cdot 3^{12}\cdot 7^{15}\) | 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: | \(78.52\) | 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: | $2^{27/14}3^{6/7}7^{47/42}\approx 86.14376989441227$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{14}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{110123}a^{12}-\frac{5065}{110123}a^{10}+\frac{34863}{110123}a^{8}-\frac{6658}{110123}a^{6}+\frac{3303}{8471}a^{4}-\frac{54424}{110123}a^{2}+\frac{30828}{110123}$, $\frac{1}{1211353}a^{13}+\frac{105058}{1211353}a^{11}-\frac{405629}{1211353}a^{9}-\frac{6658}{1211353}a^{7}+\frac{45658}{93181}a^{5}-\frac{605039}{1211353}a^{3}+\frac{361197}{1211353}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $7$ | 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: | $\frac{888}{28171}a^{13}+\frac{10848}{28171}a^{11}-\frac{52196}{28171}a^{9}+\frac{165420}{28171}a^{7}-\frac{30072}{2167}a^{5}+\frac{448056}{28171}a^{3}-\frac{269640}{28171}a+15$, $\frac{207397}{1211353}a^{13}-\frac{32511}{110123}a^{12}+\frac{2315091}{1211353}a^{11}+\frac{364699}{110123}a^{10}-\frac{10707608}{1211353}a^{9}-\frac{1696922}{110123}a^{8}+\frac{35036043}{1211353}a^{7}+\frac{5572693}{110123}a^{6}-\frac{6242590}{93181}a^{5}-\frac{996544}{8471}a^{4}+\frac{111872042}{1211353}a^{3}+\frac{17982472}{110123}a^{2}-\frac{114971871}{1211353}a-\frac{18520349}{110123}$, $\frac{207397}{1211353}a^{13}-\frac{32511}{110123}a^{12}-\frac{2315091}{1211353}a^{11}+\frac{364699}{110123}a^{10}+\frac{10707608}{1211353}a^{9}-\frac{1696922}{110123}a^{8}-\frac{35036043}{1211353}a^{7}+\frac{5572693}{110123}a^{6}+\frac{6242590}{93181}a^{5}-\frac{996544}{8471}a^{4}-\frac{111872042}{1211353}a^{3}+\frac{17982472}{110123}a^{2}+\frac{114971871}{1211353}a-\frac{18520349}{110123}$, $\frac{10048}{110123}a^{12}+\frac{126417}{110123}a^{10}-\frac{662899}{110123}a^{8}+\frac{2147260}{110123}a^{6}-\frac{380361}{8471}a^{4}+\frac{7139529}{110123}a^{2}-\frac{4608911}{110123}$, $\frac{2354}{110123}a^{12}+\frac{29726}{110123}a^{10}-\frac{135990}{110123}a^{8}+\frac{475958}{110123}a^{6}-\frac{83594}{8471}a^{4}+\frac{1582769}{110123}a^{2}-\frac{1539777}{110123}$, $\frac{65114}{1211353}a^{13}-\frac{374}{110123}a^{12}+\frac{975132}{1211353}a^{11}+\frac{22219}{110123}a^{10}-\frac{6270871}{1211353}a^{9}-\frac{264494}{110123}a^{8}+\frac{24091698}{1211353}a^{7}+\frac{1388862}{110123}a^{6}-\frac{4787438}{93181}a^{5}-\frac{320454}{8471}a^{4}+\frac{109908950}{1211353}a^{3}+\frac{8901784}{110123}a^{2}-\frac{107162027}{1211353}a-\frac{11419549}{110123}$, $\frac{17989536}{1211353}a^{13}-\frac{1480173}{110123}a^{12}+\frac{259027583}{1211353}a^{11}+\frac{26442048}{110123}a^{10}-\frac{1476957334}{1211353}a^{9}-\frac{189595674}{110123}a^{8}+\frac{4555278740}{1211353}a^{7}+\frac{718967508}{110123}a^{6}-\frac{667588613}{93181}a^{5}-\frac{124836309}{8471}a^{4}+\frac{9905994485}{1211353}a^{3}+\frac{2261574689}{110123}a^{2}-\frac{4676929404}{1211353}a-\frac{1588290747}{110123}$ (assuming GRH) | 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: | \( 172711264.87665197 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 172711264.87665197 \cdot 1}{2\cdot\sqrt{338637882455405716008075264}}\cr\approx \mathstrut & 1.15494712133179 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times F_7$ (as 14T7):
A solvable group of order 84 |
The 14 conjugacy class representatives for $F_7 \times C_2$ |
Character table for $F_7 \times C_2$ |
Intermediate fields
\(\Q(\sqrt{14}) \), 7.1.38423222208.8 |
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: | 14.0.48376840350772245144010752.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.1.0.1}{1} }^{14}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.14.27.121 | $x^{14} + 6 x^{12} + 4 x^{9} + 4 x^{7} + 4 x^{4} + 6$ | $14$ | $1$ | $27$ | $(C_7:C_3) \times C_2$ | $[3]_{7}^{3}$ |
\(3\) | 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(7\) | 7.14.15.1 | $x^{14} + 14 x^{3} + 7 x^{2} + 21$ | $14$ | $1$ | $15$ | $F_7 \times C_2$ | $[7/6]_{6}^{2}$ |