Normalized defining polynomial
\( x^{16} + 784 x^{14} + 194516 x^{12} + 21761432 x^{10} + 1214654686 x^{8} + 33084444456 x^{6} + \cdots + 135973824098 \)
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: | \(138870981735380017471819830544433646927872\) \(\medspace = 2^{67}\cdot 7^{14}\cdot 193^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(372.75\) | 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: | $2^{305/64}7^{7/8}193^{1/2}\approx 2074.127800586848$ | ||
Ramified primes: | \(2\), \(7\), \(193\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7}a^{8}$, $\frac{1}{7}a^{9}$, $\frac{1}{1351}a^{10}+\frac{12}{1351}a^{8}-\frac{4}{193}a^{6}-\frac{68}{193}a^{4}+\frac{44}{193}a^{2}$, $\frac{1}{1351}a^{11}+\frac{12}{1351}a^{9}-\frac{4}{193}a^{7}-\frac{68}{193}a^{5}+\frac{44}{193}a^{3}$, $\frac{1}{260743}a^{12}+\frac{12}{260743}a^{10}-\frac{993}{260743}a^{8}-\frac{17052}{37249}a^{6}-\frac{5746}{37249}a^{4}+\frac{68}{193}a^{2}$, $\frac{1}{260743}a^{13}+\frac{12}{260743}a^{11}-\frac{993}{260743}a^{9}-\frac{17052}{37249}a^{7}-\frac{5746}{37249}a^{5}+\frac{68}{193}a^{3}$, $\frac{1}{72\!\cdots\!87}a^{14}+\frac{75\!\cdots\!38}{72\!\cdots\!87}a^{12}+\frac{25\!\cdots\!56}{72\!\cdots\!87}a^{10}-\frac{28\!\cdots\!35}{92\!\cdots\!83}a^{8}-\frac{30\!\cdots\!97}{10\!\cdots\!41}a^{6}-\frac{33\!\cdots\!45}{53\!\cdots\!37}a^{4}+\frac{82\!\cdots\!16}{27\!\cdots\!09}a^{2}-\frac{40\!\cdots\!94}{14\!\cdots\!13}$, $\frac{1}{72\!\cdots\!87}a^{15}+\frac{75\!\cdots\!38}{72\!\cdots\!87}a^{13}+\frac{25\!\cdots\!56}{72\!\cdots\!87}a^{11}-\frac{28\!\cdots\!35}{92\!\cdots\!83}a^{9}-\frac{30\!\cdots\!97}{10\!\cdots\!41}a^{7}-\frac{33\!\cdots\!45}{53\!\cdots\!37}a^{5}+\frac{82\!\cdots\!16}{27\!\cdots\!09}a^{3}-\frac{40\!\cdots\!94}{14\!\cdots\!13}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2583840}$, which has order $165365760$ (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: | $\frac{924026151830}{57\!\cdots\!47}a^{14}+\frac{690238807283695}{57\!\cdots\!47}a^{12}+\frac{15\!\cdots\!36}{57\!\cdots\!47}a^{10}+\frac{49\!\cdots\!77}{19\!\cdots\!49}a^{8}+\frac{97\!\cdots\!44}{81\!\cdots\!21}a^{6}+\frac{96\!\cdots\!50}{42\!\cdots\!97}a^{4}+\frac{67\!\cdots\!12}{21\!\cdots\!29}a^{2}-\frac{46\!\cdots\!87}{11\!\cdots\!53}$, $\frac{48\!\cdots\!38}{22\!\cdots\!71}a^{14}+\frac{37\!\cdots\!74}{22\!\cdots\!71}a^{12}+\frac{94\!\cdots\!16}{22\!\cdots\!71}a^{10}+\frac{13\!\cdots\!66}{28\!\cdots\!39}a^{8}+\frac{85\!\cdots\!54}{31\!\cdots\!53}a^{6}+\frac{11\!\cdots\!73}{16\!\cdots\!21}a^{4}+\frac{62\!\cdots\!04}{85\!\cdots\!97}a^{2}+\frac{12\!\cdots\!79}{44\!\cdots\!29}$, $\frac{17\!\cdots\!92}{24\!\cdots\!29}a^{14}+\frac{14\!\cdots\!20}{24\!\cdots\!29}a^{12}+\frac{38\!\cdots\!67}{24\!\cdots\!29}a^{10}+\frac{61\!\cdots\!68}{30\!\cdots\!61}a^{8}+\frac{43\!\cdots\!69}{34\!\cdots\!47}a^{6}+\frac{66\!\cdots\!57}{17\!\cdots\!79}a^{4}+\frac{39\!\cdots\!69}{93\!\cdots\!03}a^{2}+\frac{10\!\cdots\!61}{48\!\cdots\!71}$, $\frac{38\!\cdots\!35}{72\!\cdots\!87}a^{14}+\frac{30\!\cdots\!64}{72\!\cdots\!87}a^{12}+\frac{76\!\cdots\!78}{72\!\cdots\!87}a^{10}+\frac{10\!\cdots\!81}{92\!\cdots\!83}a^{8}+\frac{69\!\cdots\!14}{10\!\cdots\!41}a^{6}+\frac{98\!\cdots\!52}{53\!\cdots\!37}a^{4}+\frac{55\!\cdots\!55}{27\!\cdots\!09}a^{2}+\frac{26\!\cdots\!63}{14\!\cdots\!13}$, $\frac{47\!\cdots\!28}{72\!\cdots\!87}a^{14}+\frac{35\!\cdots\!25}{72\!\cdots\!87}a^{12}+\frac{80\!\cdots\!17}{72\!\cdots\!87}a^{10}+\frac{96\!\cdots\!96}{92\!\cdots\!83}a^{8}+\frac{46\!\cdots\!03}{10\!\cdots\!41}a^{6}+\frac{38\!\cdots\!40}{53\!\cdots\!37}a^{4}+\frac{34\!\cdots\!10}{27\!\cdots\!09}a^{2}+\frac{35\!\cdots\!03}{14\!\cdots\!13}$, $\frac{15\!\cdots\!19}{24\!\cdots\!29}a^{14}+\frac{11\!\cdots\!34}{24\!\cdots\!29}a^{12}+\frac{26\!\cdots\!99}{24\!\cdots\!29}a^{10}+\frac{30\!\cdots\!18}{30\!\cdots\!61}a^{8}+\frac{11\!\cdots\!44}{34\!\cdots\!47}a^{6}+\frac{20\!\cdots\!96}{17\!\cdots\!79}a^{4}-\frac{74\!\cdots\!53}{93\!\cdots\!03}a^{2}-\frac{47\!\cdots\!45}{48\!\cdots\!71}$, $\frac{12\!\cdots\!94}{72\!\cdots\!87}a^{14}+\frac{94\!\cdots\!52}{72\!\cdots\!87}a^{12}+\frac{23\!\cdots\!14}{72\!\cdots\!87}a^{10}+\frac{47\!\cdots\!66}{13\!\cdots\!69}a^{8}+\frac{20\!\cdots\!26}{10\!\cdots\!41}a^{6}+\frac{28\!\cdots\!16}{53\!\cdots\!37}a^{4}+\frac{15\!\cdots\!08}{27\!\cdots\!09}a^{2}+\frac{32\!\cdots\!49}{14\!\cdots\!13}$ (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: | \( 1200202.11347 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1200202.11347 \cdot 165365760}{2\cdot\sqrt{138870981735380017471819830544433646927872}}\cr\approx \mathstrut & 0.646849395628 \end{aligned}\] (assuming GRH)
Galois group
$D_4^2:C_2^2:C_4$ (as 16T1188):
A solvable group of order 1024 |
The 52 conjugacy class representatives for $D_4^2:C_2^2:C_4$ |
Character table for $D_4^2:C_2^2:C_4$ |
Intermediate fields
\(\Q(\sqrt{7}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{7})\), 8.8.3947645370368.1 |
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 | ${\href{/padicField/3.4.0.1}{4} }^{3}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 $16$ | $16$ | $1$ | $67$ | |||
\(7\) | 7.8.7.2 | $x^{8} + 21$ | $8$ | $1$ | $7$ | $D_{8}$ | $[\ ]_{8}^{2}$ |
7.8.7.2 | $x^{8} + 21$ | $8$ | $1$ | $7$ | $D_{8}$ | $[\ ]_{8}^{2}$ | |
\(193\) | $\Q_{193}$ | $x + 188$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{193}$ | $x + 188$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{193}$ | $x + 188$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{193}$ | $x + 188$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{193}$ | $x + 188$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{193}$ | $x + 188$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
193.2.1.1 | $x^{2} + 193$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
193.2.1.2 | $x^{2} + 965$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
193.2.1.1 | $x^{2} + 193$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
193.2.1.1 | $x^{2} + 193$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
193.2.0.1 | $x^{2} + 192 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |