Normalized defining polynomial
\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 159 x^{12} - 226 x^{11} + 239 x^{10} - 172 x^{9} + 48 x^{8} + \cdots + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(284048656000000000000\) \(\medspace = 2^{16}\cdot 5^{12}\cdot 41^{2}\cdot 59\cdot 179\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.98\) | 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))
| |
Galois root discriminant: | not computed | ||
Ramified primes: | \(2\), \(5\), \(41\), \(59\), \(179\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{10561}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $a^{15}-7a^{14}+25a^{13}-59a^{12}+100a^{11}-126a^{10}+113a^{9}-59a^{8}-11a^{7}+59a^{6}-66a^{5}+44a^{4}-14a^{3}+4a$, $a^{15}-8a^{14}+32a^{13}-84a^{12}+159a^{11}-226a^{10}+239a^{9}-172a^{8}+48a^{7}+70a^{6}-125a^{5}+110a^{4}-58a^{3}+14a^{2}+4a-4$, $2a^{14}-14a^{13}+49a^{12}-112a^{11}+181a^{10}-212a^{9}+167a^{8}-56a^{7}-61a^{6}+120a^{5}-103a^{4}+52a^{3}-5a^{2}-8a+4$, $a^{14}-7a^{13}+24a^{12}-53a^{11}+82a^{10}-91a^{9}+66a^{8}-15a^{7}-33a^{6}+52a^{5}-40a^{4}+18a^{3}-4a$, $3a^{15}-23a^{14}+88a^{13}-220a^{12}+393a^{11}-520a^{10}+496a^{9}-289a^{8}-9a^{7}+238a^{6}-292a^{5}+205a^{4}-75a^{3}-4a^{2}+17a-6$, $3a^{14}-21a^{13}+74a^{12}-171a^{11}+281a^{10}-338a^{9}+280a^{8}-115a^{7}-72a^{6}+179a^{5}-168a^{4}+94a^{3}-17a^{2}-9a+8$, $4a^{15}-32a^{14}+126a^{13}-322a^{12}+587a^{11}-792a^{10}+775a^{9}-476a^{8}+25a^{7}+336a^{6}-439a^{5}+320a^{4}-129a^{3}+5a^{2}+20a-7$, $4a^{14}-28a^{13}+98a^{12}-224a^{11}+363a^{10}-429a^{9}+346a^{8}-130a^{7}-105a^{6}+231a^{5}-208a^{4}+112a^{3}-17a^{2}-13a+9$, $2a^{15}-15a^{14}+57a^{13}-143a^{12}+258a^{11}-347a^{10}+339a^{9}-209a^{8}+13a^{7}+145a^{6}-190a^{5}+140a^{4}-54a^{3}+2a^{2}+11a-4$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 6664.38613462 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 6664.38613462 \cdot 1}{2\cdot\sqrt{284048656000000000000}}\cr\approx \mathstrut & 0.194640396194 \end{aligned}\] (assuming GRH)
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
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.4.164000000.1 |
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | $16$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | $16$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ | $16$ | $16$ | R |
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 $16$ | $2$ | $8$ | $16$ | |||
\(5\) | 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(59\) | $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
59.2.1.1 | $x^{2} + 118$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.4.0.1 | $x^{4} + 2 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(179\) | $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
179.2.1.1 | $x^{2} + 358$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |