Normalized defining polynomial
\( x^{16} + 10x^{14} + 40x^{12} + 80x^{10} + 75x^{8} + 10x^{6} - 35x^{4} - 25x^{2} - 5 \)
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: | \(-43107820312500000000\) \(\medspace = -\,2^{8}\cdot 3^{8}\cdot 5^{15}\cdot 29^{2}\) | 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: | \(16.87\) | 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: | $2^{15/8}3^{1/2}5^{15/16}29^{1/2}\approx 154.6946332724261$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(29\) | 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: | \(\Q(\sqrt{-5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{58}a^{14}-\frac{13}{58}a^{12}+\frac{10}{29}a^{10}-\frac{1}{2}a^{9}+\frac{13}{29}a^{8}-\frac{1}{2}a^{7}-\frac{1}{58}a^{6}+\frac{2}{29}a^{4}+\frac{9}{29}a^{2}-\frac{1}{2}a-\frac{2}{29}$, $\frac{1}{58}a^{15}-\frac{13}{58}a^{13}+\frac{10}{29}a^{11}-\frac{1}{2}a^{10}+\frac{13}{29}a^{9}-\frac{1}{2}a^{8}-\frac{1}{58}a^{7}+\frac{2}{29}a^{5}+\frac{9}{29}a^{3}-\frac{1}{2}a^{2}-\frac{2}{29}a$
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: | $\frac{18}{29}a^{14}+\frac{172}{29}a^{12}+\frac{650}{29}a^{10}+\frac{1193}{29}a^{8}+\frac{910}{29}a^{6}-\frac{160}{29}a^{4}-\frac{575}{29}a^{2}-\frac{188}{29}$, $\frac{18}{29}a^{14}+\frac{172}{29}a^{12}+\frac{650}{29}a^{10}+\frac{1193}{29}a^{8}+\frac{910}{29}a^{6}-\frac{160}{29}a^{4}-\frac{604}{29}a^{2}-\frac{246}{29}$, $a^{2}+1$, $\frac{5}{58}a^{15}-\frac{5}{29}a^{14}+\frac{51}{58}a^{13}-\frac{51}{29}a^{12}+\frac{108}{29}a^{11}-\frac{403}{58}a^{10}+\frac{239}{29}a^{9}-\frac{753}{58}a^{8}+\frac{517}{58}a^{7}-\frac{285}{29}a^{6}+\frac{39}{29}a^{5}+\frac{67}{29}a^{4}-\frac{158}{29}a^{3}+\frac{429}{58}a^{2}-\frac{68}{29}a+\frac{78}{29}$, $\frac{1}{58}a^{15}-\frac{6}{29}a^{14}+\frac{8}{29}a^{13}-\frac{105}{58}a^{12}+\frac{39}{29}a^{11}-\frac{327}{58}a^{10}+\frac{71}{29}a^{9}-\frac{185}{29}a^{8}+\frac{14}{29}a^{7}+\frac{64}{29}a^{6}-\frac{199}{58}a^{5}+\frac{503}{58}a^{4}-\frac{127}{58}a^{3}+\frac{37}{29}a^{2}+\frac{83}{58}a-\frac{63}{29}$, $\frac{43}{58}a^{15}-\frac{21}{29}a^{14}+\frac{199}{29}a^{13}-\frac{191}{29}a^{12}+\frac{1411}{58}a^{11}-\frac{1333}{58}a^{10}+\frac{1139}{29}a^{9}-\frac{1068}{29}a^{8}+\frac{1175}{58}a^{7}-\frac{1147}{58}a^{6}-\frac{901}{58}a^{5}+\frac{351}{29}a^{4}-\frac{454}{29}a^{3}+\frac{781}{58}a^{2}-\frac{57}{29}a+\frac{197}{58}$, $\frac{60}{29}a^{15}-\frac{1}{29}a^{14}+\frac{1137}{58}a^{13}-\frac{3}{58}a^{12}+\frac{2099}{29}a^{11}+\frac{38}{29}a^{10}+\frac{3677}{29}a^{9}+\frac{383}{58}a^{8}+\frac{5013}{58}a^{7}+\frac{349}{29}a^{6}-\frac{1579}{58}a^{5}+\frac{369}{58}a^{4}-\frac{3379}{58}a^{3}-\frac{297}{58}a^{2}-\frac{1147}{58}a-\frac{112}{29}$, $\frac{7}{58}a^{15}+\frac{17}{58}a^{14}+\frac{27}{29}a^{13}+\frac{78}{29}a^{12}+\frac{70}{29}a^{11}+\frac{543}{58}a^{10}+\frac{95}{58}a^{9}+\frac{424}{29}a^{8}-\frac{181}{58}a^{7}+\frac{389}{58}a^{6}-\frac{291}{58}a^{5}-\frac{425}{58}a^{4}-\frac{77}{58}a^{3}-\frac{224}{29}a^{2}-\frac{14}{29}a-\frac{63}{29}$ | 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: | \( 2260.21175134 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{7}\cdot 2260.21175134 \cdot 1}{2\cdot\sqrt{43107820312500000000}}\cr\approx \mathstrut & 0.266170653454 \end{aligned}\]
Galois group
$C_4^4.C_2\wr C_4$ (as 16T1771):
A solvable group of order 16384 |
The 190 conjugacy class representatives for $C_4^4.C_2\wr C_4$ |
Character table for $C_4^4.C_2\wr C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.4.183515625.1 |
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 | R | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | $16$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | $16$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.7 | $x^{8} + 8 x^{7} + 40 x^{6} + 120 x^{5} + 232 x^{4} + 240 x^{3} + 160 x^{2} + 64 x + 16$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ |
2.8.0.1 | $x^{8} + x^{4} + x^{3} + x^{2} + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | 5.16.15.1 | $x^{16} + 20$ | $16$ | $1$ | $15$ | 16T125 | $[\ ]_{16}^{4}$ |
\(29\) | $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |