Normalized defining polynomial
\( x^{16} - 8 x^{15} + 17 x^{14} + 21 x^{13} - 111 x^{12} + 29 x^{11} + 272 x^{10} - 172 x^{9} - 373 x^{8} + \cdots + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[12, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(813503926129374655621\)
\(\medspace = 53^{2}\cdot 1229^{2}\cdot 1483^{2}\cdot 87181\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(20.27\) | 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: | $53^{1/2}1229^{1/2}1483^{1/2}87181^{1/2}\approx 2901986.41381227$ | ||
Ramified primes: |
\(53\), \(1229\), \(1483\), \(87181\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{87181}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 $1$
Unit group
Rank: | $13$ | 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: |
$a^{12}-6a^{11}+6a^{10}+25a^{9}-43a^{8}-44a^{7}+91a^{6}+52a^{5}-89a^{4}-43a^{3}+36a^{2}+14a-3$, $a^{14}-7a^{13}+11a^{12}+25a^{11}-74a^{10}-26a^{9}+178a^{8}+5a^{7}-232a^{6}-6a^{5}+169a^{4}+19a^{3}-54a^{2}-9a+3$, $a^{15}-8a^{14}+16a^{13}+28a^{12}-122a^{11}+4a^{10}+347a^{9}-151a^{8}-548a^{7}+282a^{6}+529a^{5}-231a^{4}-290a^{3}+90a^{2}+66a-13$, $a^{12}-6a^{11}+6a^{10}+25a^{9}-42a^{8}-48a^{7}+91a^{6}+66a^{5}-94a^{4}-61a^{3}+40a^{2}+22a-4$, $a^{2}-a-2$, $a^{2}-a-1$, $a^{14}-7a^{13}+11a^{12}+25a^{11}-74a^{10}-26a^{9}+178a^{8}+5a^{7}-232a^{6}-6a^{5}+168a^{4}+21a^{3}-52a^{2}-12a+2$, $a^{3}-2a^{2}-a+1$, $2a^{15}-15a^{14}+27a^{13}+52a^{12}-191a^{11}-22a^{10}+492a^{9}-118a^{8}-700a^{7}+197a^{6}+590a^{5}-126a^{4}-257a^{3}+39a^{2}+42a-6$, $a^{15}-8a^{14}+18a^{13}+14a^{12}-99a^{11}+48a^{10}+204a^{9}-173a^{8}-237a^{7}+226a^{6}+174a^{5}-147a^{4}-73a^{3}+41a^{2}+12a-3$, $a^{14}-9a^{13}+25a^{12}+2a^{11}-118a^{10}+117a^{9}+200a^{8}-305a^{7}-180a^{6}+350a^{5}+95a^{4}-203a^{3}-11a^{2}+45a-6$, $a^{13}-7a^{12}+11a^{11}+25a^{10}-75a^{9}-21a^{8}+175a^{7}-13a^{6}-214a^{5}+25a^{4}+141a^{3}-13a^{2}-36a+5$, $a^{15}-7a^{14}+10a^{13}+32a^{12}-86a^{11}-45a^{10}+245a^{9}+11a^{8}-372a^{7}+23a^{6}+326a^{5}-19a^{4}-148a^{3}+9a^{2}+26a-4$
| 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: | \( 54465.5173785 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 54465.5173785 \cdot 1}{2\cdot\sqrt{813503926129374655621}}\cr\approx \mathstrut & 0.154394382836 \end{aligned}\]
Galois group
$C_2^8.S_8$ (as 16T1948):
A non-solvable group of order 10321920 |
The 185 conjugacy class representatives for $C_2^8.S_8$ |
Character table for $C_2^8.S_8$ |
Intermediate fields
8.6.96598171.1 |
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 | $16$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(53\)
| 53.3.0.1 | $x^{3} + 3 x + 51$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
53.3.0.1 | $x^{3} + 3 x + 51$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
53.3.0.1 | $x^{3} + 3 x + 51$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
53.3.0.1 | $x^{3} + 3 x + 51$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(1229\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(1483\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(87181\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |