Normalized defining polynomial
\( x^{16} - 6 x^{14} - 20 x^{13} - 18 x^{12} + 36 x^{11} + 166 x^{10} + 352 x^{9} + 518 x^{8} + 576 x^{7} + 502 x^{6} + 348 x^{5} + 198 x^{4} + 92 x^{3} + 34 x^{2} + 8 x + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1658721111359094784\) \(\medspace = 2^{36}\cdot 17^{6}\) | 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: | \(13.76\) | 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: | $2^{9/4}17^{1/2}\approx 19.61290618356907$ | ||
Ramified primes: | \(2\), \(17\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{5}$, $\frac{1}{34}a^{14}-\frac{3}{34}a^{13}+\frac{3}{17}a^{12}+\frac{2}{17}a^{11}+\frac{5}{34}a^{10}-\frac{1}{34}a^{9}-\frac{3}{34}a^{8}-\frac{8}{17}a^{7}+\frac{13}{34}a^{6}+\frac{13}{34}a^{5}-\frac{4}{17}a^{4}-\frac{7}{17}a^{3}-\frac{5}{34}a^{2}-\frac{3}{34}a+\frac{11}{34}$, $\frac{1}{487934}a^{15}-\frac{736}{243967}a^{14}+\frac{57181}{487934}a^{13}-\frac{8094}{243967}a^{12}-\frac{25673}{243967}a^{11}+\frac{52035}{487934}a^{10}-\frac{23560}{243967}a^{9}-\frac{5871}{243967}a^{8}+\frac{192701}{487934}a^{7}-\frac{39768}{243967}a^{6}-\frac{127123}{487934}a^{5}-\frac{48871}{243967}a^{4}+\frac{111456}{243967}a^{3}-\frac{220373}{487934}a^{2}-\frac{14808}{243967}a-\frac{44842}{243967}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{259881}{487934} a^{15} + \frac{38477}{487934} a^{14} - \frac{1611437}{487934} a^{13} - \frac{5339603}{487934} a^{12} - \frac{5269823}{487934} a^{11} + \frac{4651518}{243967} a^{10} + \frac{44445819}{487934} a^{9} + \frac{2831010}{14351} a^{8} + \frac{142986239}{487934} a^{7} + \frac{160078999}{487934} a^{6} + \frac{140535689}{487934} a^{5} + \frac{97065813}{487934} a^{4} + \frac{54431221}{487934} a^{3} + \frac{12339536}{243967} a^{2} + \frac{8200939}{487934} a + \frac{834596}{243967} \) (order $8$) | 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: | $\frac{145077}{243967}a^{15}-\frac{96570}{243967}a^{14}-\frac{894779}{243967}a^{13}-\frac{4479919}{487934}a^{12}-\frac{624569}{243967}a^{11}+\frac{7014102}{243967}a^{10}+\frac{19946745}{243967}a^{9}+\frac{34081893}{243967}a^{8}+\frac{40544651}{243967}a^{7}+\frac{34519678}{243967}a^{6}+\frac{21344427}{243967}a^{5}+\frac{17506887}{487934}a^{4}+\frac{147717}{14351}a^{3}+\frac{23850}{14351}a^{2}-\frac{298509}{243967}a-\frac{239452}{243967}$, $\frac{9821}{243967}a^{15}+\frac{124104}{243967}a^{14}-\frac{109188}{243967}a^{13}-\frac{992189}{243967}a^{12}-\frac{4585225}{487934}a^{11}-\frac{606117}{243967}a^{10}+\frac{7660260}{243967}a^{9}+\frac{44170935}{487934}a^{8}+\frac{38065950}{243967}a^{7}+\frac{46398657}{243967}a^{6}+\frac{41608162}{243967}a^{5}+\frac{27625532}{243967}a^{4}+\frac{27429569}{487934}a^{3}+\frac{5602318}{243967}a^{2}+\frac{2073876}{243967}a+\frac{801427}{487934}$, $\frac{669}{2159}a^{15}+\frac{625}{2159}a^{14}-\frac{9957}{4318}a^{13}-\frac{16484}{2159}a^{12}-\frac{18818}{2159}a^{11}+\frac{27893}{2159}a^{10}+\frac{277291}{4318}a^{9}+\frac{296478}{2159}a^{8}+\frac{435768}{2159}a^{7}+\frac{474713}{2159}a^{6}+\frac{789323}{4318}a^{5}+\frac{258680}{2159}a^{4}+\frac{133887}{2159}a^{3}+\frac{58103}{2159}a^{2}+\frac{35765}{4318}a+\frac{3316}{2159}$, $\frac{105849}{243967}a^{15}-\frac{43974}{243967}a^{14}-\frac{1472017}{487934}a^{13}-\frac{3464419}{487934}a^{12}-\frac{929095}{487934}a^{11}+\frac{11746107}{487934}a^{10}+\frac{31355063}{487934}a^{9}+\frac{25161051}{243967}a^{8}+\frac{28030193}{243967}a^{7}+\frac{21017036}{243967}a^{6}+\frac{19157779}{487934}a^{5}+\frac{137749}{28702}a^{4}-\frac{2994957}{487934}a^{3}-\frac{3033595}{487934}a^{2}-\frac{3068553}{487934}a-\frac{650627}{243967}$, $\frac{190745}{487934}a^{15}-\frac{193401}{243967}a^{14}-\frac{746023}{487934}a^{13}-\frac{827457}{243967}a^{12}+\frac{1813267}{487934}a^{11}+\frac{3526249}{243967}a^{10}+\frac{7081089}{243967}a^{9}+\frac{10449567}{243967}a^{8}+\frac{21492975}{487934}a^{7}+\frac{9489974}{243967}a^{6}+\frac{16106231}{487934}a^{5}+\frac{6136297}{243967}a^{4}+\frac{517817}{28702}a^{3}+\frac{103617}{14351}a^{2}+\frac{600251}{243967}a+\frac{124432}{243967}$, $\frac{238620}{243967}a^{15}-\frac{352339}{243967}a^{14}-\frac{2250611}{487934}a^{13}-\frac{2792090}{243967}a^{12}+\frac{859141}{243967}a^{11}+\frac{9764723}{243967}a^{10}+\frac{1426922}{14351}a^{9}+\frac{39148377}{243967}a^{8}+\frac{44374143}{243967}a^{7}+\frac{38304277}{243967}a^{6}+\frac{52923971}{487934}a^{5}+\frac{13869173}{243967}a^{4}+\frac{7402606}{243967}a^{3}+\frac{2054716}{243967}a^{2}+\frac{782772}{243967}a+\frac{202597}{243967}$, $\frac{682267}{487934}a^{15}-\frac{57097}{487934}a^{14}-\frac{4416347}{487934}a^{13}-\frac{6447897}{243967}a^{12}-\frac{4721789}{243967}a^{11}+\frac{29564287}{487934}a^{10}+\frac{55265857}{243967}a^{9}+\frac{216636663}{487934}a^{8}+\frac{17538839}{28702}a^{7}+\frac{306195329}{487934}a^{6}+\frac{243584441}{487934}a^{5}+\frac{76461918}{243967}a^{4}+\frac{38721023}{243967}a^{3}+\frac{32309807}{487934}a^{2}+\frac{4557425}{243967}a+\frac{1231013}{487934}$ | 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: | \( 1382.13723796 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1382.13723796 \cdot 1}{8\cdot\sqrt{1658721111359094784}}\cr\approx \mathstrut & 0.325846786101 \end{aligned}\]
Galois group
A solvable group of order 32 |
The 11 conjugacy class representatives for $D_8:C_2$ |
Character table for $D_8:C_2$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), 4.0.272.1, 4.0.4352.2, \(\Q(\zeta_{8})\), 8.0.18939904.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.36.1 | $x^{16} + 8 x^{12} + 8 x^{11} + 24 x^{10} + 44 x^{8} - 16 x^{7} + 16 x^{6} - 56 x^{4} + 112 x^{3} + 196$ | $8$ | $2$ | $36$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ |
\(17\) | $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |