Normalized defining polynomial
\( x^{16} + 101 x^{14} + 3986 x^{12} + 78850 x^{10} + 834858 x^{8} + 4716149 x^{6} + 13451454 x^{4} + \cdots + 8260705 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(577233192643205020288000000000\) \(\medspace = 2^{16}\cdot 5^{9}\cdot 1652141^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(72.46\) | 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: | \(2\), \(5\), \(1652141\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{8260705}$) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{48\!\cdots\!07}a^{14}-\frac{23\!\cdots\!53}{48\!\cdots\!07}a^{12}-\frac{22\!\cdots\!45}{48\!\cdots\!07}a^{10}-\frac{62\!\cdots\!32}{48\!\cdots\!07}a^{8}+\frac{10\!\cdots\!97}{48\!\cdots\!07}a^{6}-\frac{74\!\cdots\!66}{48\!\cdots\!07}a^{4}-\frac{23\!\cdots\!44}{48\!\cdots\!07}a^{2}-\frac{61\!\cdots\!69}{48\!\cdots\!07}$, $\frac{1}{48\!\cdots\!07}a^{15}-\frac{23\!\cdots\!53}{48\!\cdots\!07}a^{13}-\frac{22\!\cdots\!45}{48\!\cdots\!07}a^{11}-\frac{62\!\cdots\!32}{48\!\cdots\!07}a^{9}+\frac{10\!\cdots\!97}{48\!\cdots\!07}a^{7}-\frac{74\!\cdots\!66}{48\!\cdots\!07}a^{5}-\frac{23\!\cdots\!44}{48\!\cdots\!07}a^{3}-\frac{61\!\cdots\!69}{48\!\cdots\!07}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{9024}$, which has order $18048$ (assuming GRH)
Unit group
Rank: | $7$ | 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: | $\frac{14361980}{2063246614417}a^{14}+\frac{1339676501}{2063246614417}a^{12}+\frac{46962180543}{2063246614417}a^{10}+\frac{776659958429}{2063246614417}a^{8}+\frac{6275079338216}{2063246614417}a^{6}+\frac{24334897890563}{2063246614417}a^{4}+\frac{43994995246751}{2063246614417}a^{2}+\frac{26555226486474}{2063246614417}$, $\frac{17\!\cdots\!91}{48\!\cdots\!07}a^{14}+\frac{18\!\cdots\!64}{48\!\cdots\!07}a^{12}+\frac{72\!\cdots\!08}{48\!\cdots\!07}a^{10}+\frac{14\!\cdots\!16}{48\!\cdots\!07}a^{8}+\frac{14\!\cdots\!94}{48\!\cdots\!07}a^{6}+\frac{69\!\cdots\!89}{48\!\cdots\!07}a^{4}+\frac{14\!\cdots\!28}{48\!\cdots\!07}a^{2}+\frac{89\!\cdots\!78}{48\!\cdots\!07}$, $\frac{17\!\cdots\!91}{48\!\cdots\!07}a^{14}+\frac{18\!\cdots\!64}{48\!\cdots\!07}a^{12}+\frac{72\!\cdots\!08}{48\!\cdots\!07}a^{10}+\frac{14\!\cdots\!16}{48\!\cdots\!07}a^{8}+\frac{14\!\cdots\!94}{48\!\cdots\!07}a^{6}+\frac{69\!\cdots\!89}{48\!\cdots\!07}a^{4}+\frac{14\!\cdots\!28}{48\!\cdots\!07}a^{2}+\frac{85\!\cdots\!71}{48\!\cdots\!07}$, $\frac{16\!\cdots\!89}{48\!\cdots\!07}a^{14}+\frac{13\!\cdots\!07}{48\!\cdots\!07}a^{12}+\frac{38\!\cdots\!45}{48\!\cdots\!07}a^{10}+\frac{41\!\cdots\!43}{48\!\cdots\!07}a^{8}+\frac{58\!\cdots\!42}{48\!\cdots\!07}a^{6}-\frac{12\!\cdots\!16}{48\!\cdots\!07}a^{4}-\frac{36\!\cdots\!07}{48\!\cdots\!07}a^{2}-\frac{27\!\cdots\!24}{48\!\cdots\!07}$, $\frac{16\!\cdots\!89}{48\!\cdots\!07}a^{14}+\frac{13\!\cdots\!07}{48\!\cdots\!07}a^{12}+\frac{38\!\cdots\!45}{48\!\cdots\!07}a^{10}+\frac{41\!\cdots\!43}{48\!\cdots\!07}a^{8}+\frac{58\!\cdots\!42}{48\!\cdots\!07}a^{6}-\frac{12\!\cdots\!16}{48\!\cdots\!07}a^{4}-\frac{36\!\cdots\!07}{48\!\cdots\!07}a^{2}-\frac{22\!\cdots\!17}{48\!\cdots\!07}$, $\frac{22\!\cdots\!39}{48\!\cdots\!07}a^{14}+\frac{20\!\cdots\!19}{48\!\cdots\!07}a^{12}+\frac{65\!\cdots\!48}{48\!\cdots\!07}a^{10}+\frac{95\!\cdots\!83}{48\!\cdots\!07}a^{8}+\frac{62\!\cdots\!81}{48\!\cdots\!07}a^{6}+\frac{16\!\cdots\!25}{48\!\cdots\!07}a^{4}+\frac{17\!\cdots\!44}{48\!\cdots\!07}a^{2}+\frac{93\!\cdots\!87}{48\!\cdots\!07}$, $\frac{23\!\cdots\!13}{48\!\cdots\!07}a^{14}+\frac{21\!\cdots\!50}{48\!\cdots\!07}a^{12}+\frac{70\!\cdots\!08}{48\!\cdots\!07}a^{10}+\frac{10\!\cdots\!63}{48\!\cdots\!07}a^{8}+\frac{66\!\cdots\!13}{48\!\cdots\!07}a^{6}+\frac{14\!\cdots\!84}{48\!\cdots\!07}a^{4}+\frac{16\!\cdots\!00}{48\!\cdots\!07}a^{2}-\frac{11\!\cdots\!99}{48\!\cdots\!07}$ (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: | \( 6960.86418224 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 6960.86418224 \cdot 18048}{2\cdot\sqrt{577233192643205020288000000000}}\cr\approx \mathstrut & 0.200828592886 \end{aligned}\] (assuming GRH)
Galois group
$C_2^6.S_4^2:C_4$ (as 16T1888):
A solvable group of order 147456 |
The 130 conjugacy class representatives for $C_2^6.S_4^2:C_4$ |
Character table for $C_2^6.S_4^2:C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 8.8.1032588125.1 |
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 | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $16$ | $2$ | $8$ | $16$ | |||
\(5\) | 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(1652141\) | $\Q_{1652141}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1652141}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1652141}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1652141}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |