Normalized defining polynomial
\( x^{14} - 2 x^{13} - 8 x^{12} + 22 x^{11} - 11 x^{10} - 13 x^{8} + 26 x^{7} - 13 x^{6} - 11 x^{4} + \cdots + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[6, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(3593221476070719488\) \(\medspace = 2^{23}\cdot 809^{4}\) | 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: | \(21.15\) | 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: | $2^{31/12}809^{1/2}\approx 170.46494509614888$ | ||
Ramified primes: | \(2\), \(809\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{149}a^{12}+\frac{47}{149}a^{11}+\frac{59}{149}a^{10}+\frac{35}{149}a^{9}+\frac{6}{149}a^{8}-\frac{39}{149}a^{7}+\frac{7}{149}a^{6}-\frac{39}{149}a^{5}+\frac{6}{149}a^{4}+\frac{35}{149}a^{3}+\frac{59}{149}a^{2}+\frac{47}{149}a+\frac{1}{149}$, $\frac{1}{1192}a^{13}-\frac{3}{1192}a^{12}-\frac{205}{1192}a^{11}+\frac{363}{1192}a^{10}-\frac{127}{596}a^{9}+\frac{203}{596}a^{8}-\frac{427}{1192}a^{7}-\frac{91}{1192}a^{6}-\frac{65}{596}a^{5}-\frac{207}{596}a^{4}+\frac{395}{1192}a^{3}-\frac{221}{1192}a^{2}+\frac{333}{1192}a-\frac{199}{1192}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $\frac{1817}{596}a^{13}-\frac{2635}{596}a^{12}-\frac{16037}{596}a^{11}+\frac{31247}{596}a^{10}-\frac{1189}{298}a^{9}-\frac{1161}{298}a^{8}-\frac{24463}{596}a^{7}+\frac{33761}{596}a^{6}-\frac{2263}{298}a^{5}-\frac{2025}{298}a^{4}-\frac{21105}{596}a^{3}+\frac{28019}{596}a^{2}+\frac{1353}{596}a-\frac{4147}{596}$, $a$, $\frac{12683}{1192}a^{13}-\frac{18905}{1192}a^{12}-\frac{111311}{1192}a^{11}+\frac{222809}{1192}a^{10}-\frac{12201}{596}a^{9}-\frac{8795}{596}a^{8}-\frac{170073}{1192}a^{7}+\frac{242183}{1192}a^{6}-\frac{19411}{596}a^{5}-\frac{12897}{596}a^{4}-\frac{149055}{1192}a^{3}+\frac{201569}{1192}a^{2}+\frac{3567}{1192}a-\frac{28989}{1192}$, $\frac{8757}{1192}a^{13}-\frac{13583}{1192}a^{12}-\frac{75985}{1192}a^{11}+\frac{158279}{1192}a^{10}-\frac{13383}{596}a^{9}-\frac{4449}{596}a^{8}-\frac{118095}{1192}a^{7}+\frac{174009}{1192}a^{6}-\frac{19173}{596}a^{5}-\frac{6899}{596}a^{4}-\frac{102025}{1192}a^{3}+\frac{147143}{1192}a^{2}-\frac{6367}{1192}a-\frac{18243}{1192}$, $\frac{6969}{1192}a^{13}-\frac{10603}{1192}a^{12}-\frac{61085}{1192}a^{11}+\frac{124307}{1192}a^{10}-\frac{7423}{596}a^{9}-\frac{5045}{596}a^{8}-\frac{96043}{1192}a^{7}+\frac{134077}{1192}a^{6}-\frac{12021}{596}a^{5}-\frac{7495}{596}a^{4}-\frac{83549}{1192}a^{3}+\frac{111979}{1192}a^{2}+\frac{189}{1192}a-\frac{16455}{1192}$, $\frac{1173}{1192}a^{13}-\frac{11}{8}a^{12}-\frac{10257}{1192}a^{11}+\frac{19391}{1192}a^{10}-\frac{7}{4}a^{9}+\frac{591}{596}a^{8}-\frac{16335}{1192}a^{7}+\frac{22041}{1192}a^{6}-\frac{1453}{596}a^{5}-\frac{559}{596}a^{4}-\frac{15609}{1192}a^{3}+\frac{18567}{1192}a^{2}-\frac{1407}{1192}a-\frac{1491}{1192}$, $\frac{2877}{596}a^{13}-\frac{4487}{596}a^{12}-\frac{24901}{596}a^{11}+\frac{52147}{596}a^{10}-\frac{4693}{298}a^{9}-\frac{1027}{298}a^{8}-\frac{39559}{596}a^{7}+\frac{56857}{596}a^{6}-\frac{6169}{298}a^{5}-\frac{1709}{298}a^{4}-\frac{35109}{596}a^{3}+\frac{47931}{596}a^{2}-\frac{2239}{596}a-\frac{5755}{596}$, $\frac{10363}{596}a^{13}-\frac{16121}{596}a^{12}-\frac{90051}{596}a^{11}+\frac{187993}{596}a^{10}-\frac{15333}{298}a^{9}-\frac{6841}{298}a^{8}-\frac{140029}{596}a^{7}+\frac{206531}{596}a^{6}-\frac{21407}{298}a^{5}-\frac{9469}{298}a^{4}-\frac{122127}{596}a^{3}+\frac{174077}{596}a^{2}-\frac{6301}{596}a-\frac{22657}{596}$, $\frac{1211}{149}a^{13}-\frac{1741}{149}a^{12}-\frac{10629}{149}a^{11}+\frac{20632}{149}a^{10}-\frac{2080}{149}a^{9}-\frac{751}{149}a^{8}-\frac{15894}{149}a^{7}+\frac{22839}{149}a^{6}-\frac{3546}{149}a^{5}-\frac{1729}{149}a^{4}-\frac{14334}{149}a^{3}+124a^{2}+\frac{40}{149}a-\frac{2485}{149}$ | 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: | \( 17505.5082711 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{4}\cdot 17505.5082711 \cdot 1}{2\cdot\sqrt{3593221476070719488}}\cr\approx \mathstrut & 0.460576932905 \end{aligned}\]
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
7.7.670188544.1 |
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 | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.14.0.1}{14} }$ | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.7.0.1}{7} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.12.20.37 | $x^{12} + 10 x^{11} + 51 x^{10} + 176 x^{9} + 450 x^{8} + 870 x^{7} + 1299 x^{6} + 1516 x^{5} + 1250 x^{4} + 542 x^{3} + 67 x^{2} - 56 x + 7$ | $6$ | $2$ | $20$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
\(809\) | $\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |