Normalized defining polynomial
\( x^{16} - 2 x^{15} + 3 x^{14} + 6 x^{13} - 9 x^{12} + 14 x^{11} + 6 x^{10} - 6 x^{9} + 21 x^{8} + \cdots + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-36973431874322432\) \(\medspace = -\,2^{16}\cdot 47\cdot 331^{4}\) | 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: | \(10.85\) | 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: | not computed | ||
Ramified primes: | \(2\), \(47\), \(331\) | 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: | \(\Q(\sqrt{-47}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{691}a^{14}-\frac{333}{691}a^{13}-\frac{335}{691}a^{12}-\frac{27}{691}a^{11}+\frac{280}{691}a^{10}-\frac{45}{691}a^{9}+\frac{110}{691}a^{8}+\frac{252}{691}a^{7}+\frac{110}{691}a^{6}-\frac{45}{691}a^{5}+\frac{280}{691}a^{4}-\frac{27}{691}a^{3}-\frac{335}{691}a^{2}-\frac{333}{691}a+\frac{1}{691}$, $\frac{1}{691}a^{15}+\frac{27}{691}a^{13}-\frac{331}{691}a^{12}+\frac{272}{691}a^{11}-\frac{90}{691}a^{10}+\frac{327}{691}a^{9}+\frac{259}{691}a^{8}-\frac{276}{691}a^{7}-\frac{38}{691}a^{6}-\frac{194}{691}a^{5}-\frac{72}{691}a^{4}-\frac{343}{691}a^{3}+\frac{54}{691}a^{2}-\frac{328}{691}a+\frac{333}{691}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{42}{691}a^{15}-\frac{217}{691}a^{14}+\frac{149}{691}a^{13}+\frac{58}{691}a^{12}-\frac{1374}{691}a^{11}-\frac{277}{691}a^{10}-\frac{1377}{691}a^{9}-\frac{1936}{691}a^{8}-\frac{2013}{691}a^{7}-\frac{4045}{691}a^{6}-\frac{1838}{691}a^{5}-\frac{2285}{691}a^{4}-\frac{2328}{691}a^{3}-\frac{1047}{691}a^{2}-\frac{1632}{691}a-\frac{51}{691}$, $\frac{1136}{691}a^{15}-\frac{1493}{691}a^{14}+\frac{1990}{691}a^{13}+\frac{8742}{691}a^{12}-\frac{5180}{691}a^{11}+\frac{9717}{691}a^{10}+\frac{15074}{691}a^{9}-\frac{1296}{691}a^{8}+\frac{19194}{691}a^{7}+\frac{4739}{691}a^{6}+\frac{2276}{691}a^{5}+\frac{16345}{691}a^{4}-\frac{3146}{691}a^{3}+\frac{407}{691}a^{2}+\frac{3636}{691}a-\frac{1182}{691}$, $a^{15}-2a^{14}+3a^{13}+6a^{12}-9a^{11}+14a^{10}+6a^{9}-6a^{8}+21a^{7}-6a^{6}+6a^{5}+14a^{4}-9a^{3}+6a^{2}+3a-2$, $\frac{192}{691}a^{15}-\frac{248}{691}a^{14}+\frac{702}{691}a^{13}+\frac{871}{691}a^{12}+\frac{185}{691}a^{11}+\frac{3801}{691}a^{10}+\frac{698}{691}a^{9}+\frac{4482}{691}a^{8}+\frac{5437}{691}a^{7}+\frac{2047}{691}a^{6}+\frac{6389}{691}a^{5}+\frac{2420}{691}a^{4}+\frac{3030}{691}a^{3}+\frac{2927}{691}a^{2}-\frac{431}{691}a+\frac{807}{691}$, $\frac{113}{691}a^{15}-\frac{920}{691}a^{14}+\frac{1225}{691}a^{13}-\frac{766}{691}a^{12}-\frac{5923}{691}a^{11}+\frac{3793}{691}a^{10}-\frac{6642}{691}a^{9}-\frac{8361}{691}a^{8}-\frac{448}{691}a^{7}-\frac{13591}{691}a^{6}-\frac{2634}{691}a^{5}-\frac{3847}{691}a^{4}-\frac{9773}{691}a^{3}+\frac{588}{691}a^{2}-\frac{885}{691}a-\frac{1296}{691}$, $\frac{750}{691}a^{15}-\frac{1279}{691}a^{14}+\frac{1844}{691}a^{13}+\frac{4701}{691}a^{12}-\frac{4699}{691}a^{11}+\frac{7637}{691}a^{10}+\frac{4984}{691}a^{9}-\frac{1029}{691}a^{8}+\frac{9672}{691}a^{7}-\frac{2659}{691}a^{6}+\frac{1885}{691}a^{5}+\frac{5244}{691}a^{4}-\frac{3670}{691}a^{3}-\frac{224}{691}a^{2}+\frac{247}{691}a-\frac{289}{691}$, $\frac{178}{691}a^{15}-\frac{438}{691}a^{14}+\frac{713}{691}a^{13}+\frac{746}{691}a^{12}-\frac{1948}{691}a^{11}+\frac{2995}{691}a^{10}-\frac{167}{691}a^{9}-\frac{1387}{691}a^{8}+\frac{2881}{691}a^{7}-\frac{2428}{691}a^{6}+\frac{380}{691}a^{5}+\frac{1362}{691}a^{4}-\frac{3622}{691}a^{3}+\frac{176}{691}a^{2}+\frac{404}{691}a-\frac{1281}{691}$, $\frac{1060}{691}a^{15}-\frac{2231}{691}a^{14}+\frac{3151}{691}a^{13}+\frac{6110}{691}a^{12}-\frac{10072}{691}a^{11}+\frac{13071}{691}a^{10}+\frac{4775}{691}a^{9}-\frac{6802}{691}a^{8}+\frac{16579}{691}a^{7}-\frac{8599}{691}a^{6}+\frac{3242}{691}a^{5}+\frac{10730}{691}a^{4}-\frac{9668}{691}a^{3}+\frac{1683}{691}a^{2}+\frac{2064}{691}a-\frac{1661}{691}$ | 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: | \( 42.6664411408 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{7}\cdot 42.6664411408 \cdot 1}{2\cdot\sqrt{36973431874322432}}\cr\approx \mathstrut & 0.171565757332 \end{aligned}\]
Galois group
$C_2\wr C_2^3.S_4$ (as 16T1848):
A solvable group of order 49152 |
The 116 conjugacy class representatives for $C_2\wr C_2^3.S_4$ |
Character table for $C_2\wr C_2^3.S_4$ |
Intermediate fields
4.2.331.1, 8.4.28047616.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 siblings: | data not computed |
Degree 32 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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $2$ | $8$ | $16$ | |||
\(47\) | 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.4.0.1 | $x^{4} + 8 x^{2} + 40 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
47.4.0.1 | $x^{4} + 8 x^{2} + 40 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
47.4.0.1 | $x^{4} + 8 x^{2} + 40 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(331\) | $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | 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 $4$ | $2$ | $2$ | $2$ |