Normalized defining polynomial
\( x^{14} - 18x^{12} + 114x^{10} - 320x^{8} + 425x^{6} - 266x^{4} + 76x^{2} - 8 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[14, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(152387536786230345728\) \(\medspace = 2^{19}\cdot 4129^{4}\) | 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: | \(27.65\) | 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: | $2^{127/48}4129^{1/2}\approx 402.15885738070904$ | ||
Ramified primes: | \(2\), \(4129\) | 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: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{4}$, $\frac{1}{8}a^{13}+\frac{3}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $13$ | 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: | $\frac{15}{8}a^{13}-\frac{1}{2}a^{12}-\frac{133}{4}a^{11}+\frac{35}{4}a^{10}+\frac{819}{4}a^{9}-\frac{105}{2}a^{8}-\frac{2173}{4}a^{7}+\frac{527}{4}a^{6}+\frac{5127}{8}a^{5}-\frac{273}{2}a^{4}-\frac{613}{2}a^{3}+\frac{93}{2}a^{2}+\frac{93}{2}a-4$, $\frac{15}{8}a^{13}+\frac{1}{2}a^{12}-\frac{133}{4}a^{11}-\frac{35}{4}a^{10}+\frac{819}{4}a^{9}+\frac{105}{2}a^{8}-\frac{2173}{4}a^{7}-\frac{527}{4}a^{6}+\frac{5127}{8}a^{5}+\frac{273}{2}a^{4}-\frac{613}{2}a^{3}-\frac{93}{2}a^{2}+\frac{93}{2}a+4$, $\frac{1}{2}a^{13}-\frac{17}{2}a^{11}+\frac{1}{4}a^{10}+\frac{193}{4}a^{9}-\frac{17}{4}a^{8}-\frac{215}{2}a^{7}+\frac{97}{4}a^{6}+\frac{323}{4}a^{5}-\frac{223}{4}a^{4}+\frac{7}{2}a^{3}+50a^{2}-9a-12$, $\frac{15}{8}a^{13}-33a^{11}+\frac{401}{2}a^{9}-519a^{7}+\frac{4681}{8}a^{5}-\frac{513}{2}a^{3}-\frac{1}{2}a^{2}+\frac{67}{2}a+2$, $\frac{1}{4}a^{13}-\frac{11}{4}a^{12}-\frac{17}{4}a^{11}+\frac{97}{2}a^{10}+\frac{97}{4}a^{9}-296a^{8}-\frac{223}{4}a^{7}+\frac{1549}{2}a^{6}+\frac{101}{2}a^{5}-\frac{3591}{4}a^{4}-16a^{3}+\frac{849}{2}a^{2}+3a-67$, $\frac{1}{4}a^{13}+\frac{11}{4}a^{12}-\frac{17}{4}a^{11}-\frac{97}{2}a^{10}+\frac{97}{4}a^{9}+296a^{8}-\frac{223}{4}a^{7}-\frac{1549}{2}a^{6}+\frac{101}{2}a^{5}+\frac{3591}{4}a^{4}-16a^{3}-\frac{849}{2}a^{2}+3a+67$, $\frac{17}{8}a^{13}-\frac{75}{2}a^{11}+229a^{9}-599a^{7}+\frac{5531}{8}a^{5}-323a^{3}-\frac{1}{2}a^{2}+\frac{103}{2}a$, $\frac{17}{8}a^{13}+\frac{1}{4}a^{12}-\frac{149}{4}a^{11}-\frac{9}{2}a^{10}+\frac{899}{4}a^{9}+\frac{57}{2}a^{8}-\frac{2299}{4}a^{7}-80a^{6}+\frac{5085}{8}a^{5}+\frac{423}{4}a^{4}-\frac{545}{2}a^{3}-\frac{123}{2}a^{2}+\frac{75}{2}a+11$, $\frac{7}{8}a^{13}-\frac{1}{2}a^{12}-\frac{61}{4}a^{11}+\frac{35}{4}a^{10}+\frac{365}{4}a^{9}-\frac{211}{4}a^{8}-\frac{925}{4}a^{7}+\frac{543}{4}a^{6}+\frac{2055}{8}a^{5}-\frac{625}{4}a^{4}-\frac{235}{2}a^{3}+\frac{155}{2}a^{2}+\frac{35}{2}a-13$, $\frac{7}{8}a^{13}+\frac{1}{2}a^{12}-\frac{61}{4}a^{11}-\frac{35}{4}a^{10}+\frac{365}{4}a^{9}+\frac{211}{4}a^{8}-\frac{925}{4}a^{7}-\frac{543}{4}a^{6}+\frac{2055}{8}a^{5}+\frac{625}{4}a^{4}-\frac{235}{2}a^{3}-\frac{155}{2}a^{2}+\frac{35}{2}a+13$, $\frac{17}{8}a^{13}-\frac{75}{2}a^{11}+229a^{9}-599a^{7}+\frac{5531}{8}a^{5}-323a^{3}+\frac{1}{2}a^{2}+\frac{103}{2}a$, $\frac{25}{4}a^{13}+\frac{5}{2}a^{12}-\frac{441}{4}a^{11}-44a^{10}+673a^{9}+\frac{535}{2}a^{8}-\frac{7043}{4}a^{7}-\frac{1389}{2}a^{6}+\frac{8151}{4}a^{5}+\frac{1583}{2}a^{4}-957a^{3}-\frac{717}{2}a^{2}+147a+52$, $\frac{19}{8}a^{13}-\frac{1}{4}a^{12}-\frac{83}{2}a^{11}+4a^{10}+249a^{9}-20a^{8}-\frac{1261}{2}a^{7}+\frac{63}{2}a^{6}+\frac{5489}{8}a^{5}+\frac{21}{4}a^{4}-\frac{577}{2}a^{3}-34a^{2}+\frac{81}{2}a+10$ | 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: | \( 581975.493503 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{0}\cdot 581975.493503 \cdot 1}{2\cdot\sqrt{152387536786230345728}}\cr\approx \mathstrut & 0.386206800122 \end{aligned}\]
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 $C_2^7.\GL(3,2)$ is not computed |
Intermediate fields
7.7.1091113024.1 |
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 | R | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.14.0.1}{14} }$ | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.7.0.1}{7} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.11.3 | $x^{6} + 4 x^{2} + 4 x + 10$ | $6$ | $1$ | $11$ | $S_4\times C_2$ | $[8/3, 8/3, 3]_{3}^{2}$ |
2.8.8.11 | $x^{8} + 4 x^{7} + 14 x^{6} + 32 x^{5} + 55 x^{4} + 60 x^{3} + 36 x^{2} + 18 x + 9$ | $4$ | $2$ | $8$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
\(4129\) | $\Q_{4129}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{4129}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |