Normalized defining polynomial
\( x^{14} - 6 x^{13} + 9 x^{12} + 10 x^{11} - 35 x^{10} + 34 x^{9} - 39 x^{8} + 12 x^{7} + 15 x^{6} + \cdots - 1 \)
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: | \(9524221049139396608\) \(\medspace = 2^{15}\cdot 4129^{4}\) | 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: | \(22.68\) | 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: | $2^{25/12}4129^{1/2}\approx 272.312931678814$ | ||
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{8}a^{2}-\frac{1}{2}a+\frac{3}{8}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}+\frac{1}{16}a^{9}+\frac{1}{16}a^{8}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{16}a^{3}-\frac{5}{16}a^{2}+\frac{1}{16}a-\frac{1}{16}$, $\frac{1}{32}a^{12}+\frac{1}{16}a^{9}+\frac{1}{32}a^{8}-\frac{1}{16}a^{7}-\frac{1}{8}a^{6}+\frac{3}{16}a^{5}+\frac{7}{32}a^{4}-\frac{3}{16}a^{3}+\frac{3}{8}a^{2}+\frac{15}{32}$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{12}+\frac{1}{32}a^{10}-\frac{1}{64}a^{9}-\frac{3}{64}a^{8}-\frac{1}{32}a^{7}+\frac{5}{32}a^{6}+\frac{1}{64}a^{5}+\frac{19}{64}a^{4}-\frac{7}{32}a^{3}+\frac{5}{16}a^{2}+\frac{15}{64}a+\frac{17}{64}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$
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: | $a$, $\frac{1}{64}a^{13}-\frac{5}{64}a^{12}+\frac{1}{16}a^{11}+\frac{7}{32}a^{10}-\frac{21}{64}a^{9}+\frac{13}{64}a^{8}-\frac{13}{32}a^{7}-\frac{7}{32}a^{6}+\frac{1}{64}a^{5}+\frac{55}{64}a^{4}+\frac{51}{32}a^{3}+2a^{2}+\frac{131}{64}a+\frac{65}{64}$, $\frac{11}{64}a^{13}-\frac{45}{64}a^{12}-\frac{7}{8}a^{11}+\frac{243}{32}a^{10}-\frac{503}{64}a^{9}-\frac{611}{64}a^{8}+\frac{799}{32}a^{7}-\frac{1061}{32}a^{6}+\frac{1455}{64}a^{5}+\frac{915}{64}a^{4}+\frac{301}{32}a^{3}+\frac{53}{16}a^{2}+\frac{221}{64}a+\frac{77}{64}$, $\frac{7}{8}a^{13}-\frac{85}{16}a^{12}+\frac{131}{16}a^{11}+\frac{137}{16}a^{10}-\frac{509}{16}a^{9}+\frac{251}{8}a^{8}-\frac{275}{8}a^{7}+\frac{89}{8}a^{6}+\frac{123}{8}a^{5}+\frac{713}{16}a^{4}+\frac{611}{16}a^{3}+\frac{205}{16}a^{2}-\frac{7}{16}a-\frac{17}{8}$, $\frac{181}{64}a^{13}-\frac{1173}{64}a^{12}+\frac{269}{8}a^{11}+\frac{521}{32}a^{10}-\frac{7341}{64}a^{9}+\frac{9513}{64}a^{8}-\frac{5013}{32}a^{7}+\frac{2353}{32}a^{6}+\frac{3045}{64}a^{5}+\frac{6863}{64}a^{4}+\frac{2441}{32}a^{3}+\frac{187}{16}a^{2}-\frac{829}{64}a-\frac{259}{64}$, $\frac{3}{8}a^{13}-\frac{69}{32}a^{12}+\frac{43}{16}a^{11}+\frac{85}{16}a^{10}-13a^{9}+\frac{233}{32}a^{8}-\frac{107}{16}a^{7}-\frac{1}{2}a^{6}+\frac{101}{16}a^{5}+\frac{757}{32}a^{4}+\frac{83}{4}a^{3}+\frac{79}{16}a^{2}-\frac{23}{16}a-\frac{17}{32}$, $\frac{1}{8}a^{13}-\frac{21}{32}a^{12}+\frac{5}{8}a^{11}+\frac{7}{4}a^{10}-\frac{49}{16}a^{9}+\frac{59}{32}a^{8}-\frac{55}{16}a^{7}-\frac{9}{8}a^{6}+\frac{19}{16}a^{5}+\frac{253}{32}a^{4}+\frac{193}{16}a^{3}+\frac{105}{8}a^{2}+\frac{13}{2}a+\frac{69}{32}$ | 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: | \( 24626.8840382 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 24626.8840382 \cdot 2}{2\cdot\sqrt{9524221049139396608}}\cr\approx \mathstrut & 1.96396585802 \end{aligned}\]
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
7.7.1091113024.2 |
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 | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.3.0.1}{3} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }$ | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.7.0.1}{7} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.6.6.7 | $x^{6} + 2 x^{2} + 2 x + 2$ | $6$ | $1$ | $6$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
2.6.6.7 | $x^{6} + 2 x^{2} + 2 x + 2$ | $6$ | $1$ | $6$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
\(4129\) | $\Q_{4129}$ | $x$ | $1$ | $1$ | $0$ | 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 $4$ | $2$ | $2$ | $2$ |