Normalized defining polynomial
\( x^{16} + 4063 x^{14} + 6007715 x^{12} + 4216341258 x^{10} + 1588534344529 x^{8} + 337055877573450 x^{6} + \cdots + 61\!\cdots\!37 \)
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: | \(683567948564897299391469349182811817443328\) \(\medspace = 2^{16}\cdot 17^{15}\cdot 43691^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(411.79\) | 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: | $2^{7/4}17^{15/16}43691^{1/2}\approx 10012.541950101795$ | ||
Ramified primes: | \(2\), \(17\), \(43691\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{43691}a^{10}+\frac{4063}{43691}a^{8}-\frac{21643}{43691}a^{6}-\frac{15006}{43691}a^{4}+\frac{14421}{43691}a^{2}$, $\frac{1}{43691}a^{11}+\frac{4063}{43691}a^{9}-\frac{21643}{43691}a^{7}-\frac{15006}{43691}a^{5}+\frac{14421}{43691}a^{3}$, $\frac{1}{1908903481}a^{12}+\frac{4063}{1908903481}a^{10}+\frac{6007715}{1908903481}a^{8}+\frac{398534296}{1908903481}a^{6}+\frac{326648337}{1908903481}a^{4}+\frac{18080}{43691}a^{2}$, $\frac{1}{1908903481}a^{13}+\frac{4063}{1908903481}a^{11}+\frac{6007715}{1908903481}a^{9}+\frac{398534296}{1908903481}a^{7}+\frac{326648337}{1908903481}a^{5}+\frac{18080}{43691}a^{3}$, $\frac{1}{16\!\cdots\!27}a^{14}-\frac{21\!\cdots\!64}{16\!\cdots\!27}a^{12}+\frac{58\!\cdots\!66}{16\!\cdots\!27}a^{10}-\frac{42\!\cdots\!33}{16\!\cdots\!27}a^{8}+\frac{33\!\cdots\!96}{16\!\cdots\!27}a^{6}+\frac{56\!\cdots\!98}{36\!\cdots\!97}a^{4}+\frac{71\!\cdots\!13}{84\!\cdots\!67}a^{2}+\frac{48\!\cdots\!76}{19\!\cdots\!37}$, $\frac{1}{16\!\cdots\!27}a^{15}-\frac{21\!\cdots\!64}{16\!\cdots\!27}a^{13}+\frac{58\!\cdots\!66}{16\!\cdots\!27}a^{11}-\frac{42\!\cdots\!33}{16\!\cdots\!27}a^{9}+\frac{33\!\cdots\!96}{16\!\cdots\!27}a^{7}+\frac{56\!\cdots\!98}{36\!\cdots\!97}a^{5}+\frac{71\!\cdots\!13}{84\!\cdots\!67}a^{3}+\frac{48\!\cdots\!76}{19\!\cdots\!37}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1068746342}$, which has order $34199882944$ (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: | $\frac{81\!\cdots\!26}{16\!\cdots\!27}a^{14}+\frac{32\!\cdots\!72}{16\!\cdots\!27}a^{12}+\frac{45\!\cdots\!92}{16\!\cdots\!27}a^{10}+\frac{29\!\cdots\!64}{16\!\cdots\!27}a^{8}+\frac{99\!\cdots\!42}{16\!\cdots\!27}a^{6}+\frac{39\!\cdots\!46}{36\!\cdots\!97}a^{4}+\frac{77\!\cdots\!79}{84\!\cdots\!67}a^{2}+\frac{58\!\cdots\!96}{19\!\cdots\!37}$, $\frac{14\!\cdots\!12}{16\!\cdots\!27}a^{14}+\frac{58\!\cdots\!36}{16\!\cdots\!27}a^{12}+\frac{82\!\cdots\!96}{16\!\cdots\!27}a^{10}+\frac{53\!\cdots\!68}{16\!\cdots\!27}a^{8}+\frac{17\!\cdots\!44}{16\!\cdots\!27}a^{6}+\frac{71\!\cdots\!05}{36\!\cdots\!97}a^{4}+\frac{14\!\cdots\!32}{84\!\cdots\!67}a^{2}+\frac{10\!\cdots\!70}{19\!\cdots\!37}$, $\frac{14\!\cdots\!12}{16\!\cdots\!27}a^{14}+\frac{58\!\cdots\!36}{16\!\cdots\!27}a^{12}+\frac{82\!\cdots\!96}{16\!\cdots\!27}a^{10}+\frac{53\!\cdots\!68}{16\!\cdots\!27}a^{8}+\frac{17\!\cdots\!44}{16\!\cdots\!27}a^{6}+\frac{71\!\cdots\!05}{36\!\cdots\!97}a^{4}+\frac{14\!\cdots\!32}{84\!\cdots\!67}a^{2}+\frac{10\!\cdots\!07}{19\!\cdots\!37}$, $\frac{12\!\cdots\!86}{16\!\cdots\!27}a^{14}+\frac{50\!\cdots\!13}{16\!\cdots\!27}a^{12}+\frac{71\!\cdots\!20}{16\!\cdots\!27}a^{10}+\frac{46\!\cdots\!70}{16\!\cdots\!27}a^{8}+\frac{15\!\cdots\!13}{16\!\cdots\!27}a^{6}+\frac{61\!\cdots\!47}{36\!\cdots\!97}a^{4}+\frac{11\!\cdots\!57}{84\!\cdots\!67}a^{2}+\frac{89\!\cdots\!13}{19\!\cdots\!37}$, $\frac{15\!\cdots\!85}{16\!\cdots\!27}a^{14}+\frac{61\!\cdots\!16}{16\!\cdots\!27}a^{12}+\frac{86\!\cdots\!27}{16\!\cdots\!27}a^{10}+\frac{55\!\cdots\!48}{16\!\cdots\!27}a^{8}+\frac{18\!\cdots\!10}{16\!\cdots\!27}a^{6}+\frac{70\!\cdots\!34}{36\!\cdots\!97}a^{4}+\frac{13\!\cdots\!41}{84\!\cdots\!67}a^{2}+\frac{93\!\cdots\!99}{19\!\cdots\!37}$, $\frac{38\!\cdots\!28}{16\!\cdots\!27}a^{14}+\frac{15\!\cdots\!64}{16\!\cdots\!27}a^{12}+\frac{21\!\cdots\!57}{16\!\cdots\!27}a^{10}+\frac{14\!\cdots\!02}{16\!\cdots\!27}a^{8}+\frac{47\!\cdots\!59}{16\!\cdots\!27}a^{6}+\frac{19\!\cdots\!48}{36\!\cdots\!97}a^{4}+\frac{38\!\cdots\!55}{84\!\cdots\!67}a^{2}+\frac{29\!\cdots\!95}{19\!\cdots\!37}$, $\frac{79\!\cdots\!13}{16\!\cdots\!27}a^{14}+\frac{31\!\cdots\!82}{16\!\cdots\!27}a^{12}+\frac{44\!\cdots\!06}{16\!\cdots\!27}a^{10}+\frac{28\!\cdots\!46}{16\!\cdots\!27}a^{8}+\frac{94\!\cdots\!98}{16\!\cdots\!27}a^{6}+\frac{37\!\cdots\!86}{36\!\cdots\!97}a^{4}+\frac{72\!\cdots\!78}{84\!\cdots\!67}a^{2}+\frac{54\!\cdots\!68}{19\!\cdots\!37}$ (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: | \( 3640.01221338 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.01221338 \cdot 34199882944}{2\cdot\sqrt{683567948564897299391469349182811817443328}}\cr\approx \mathstrut & 0.182871305833 \end{aligned}\] (assuming GRH)
Galois group
$C_2^5.C_8$ (as 16T591):
A solvable group of order 256 |
The 28 conjugacy class representatives for $C_2^5.C_8$ |
Character table for $C_2^5.C_8$ |
Intermediate fields
\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\) |
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 | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.6 | $x^{8} + 2 x^{7} + 24 x^{6} + 84 x^{5} + 264 x^{4} + 408 x^{3} + 384 x^{2} - 208 x + 80$ | $2$ | $4$ | $8$ | $(C_8:C_2):C_2$ | $[2, 2, 2]^{4}$ |
2.8.8.6 | $x^{8} + 2 x^{7} + 24 x^{6} + 84 x^{5} + 264 x^{4} + 408 x^{3} + 384 x^{2} - 208 x + 80$ | $2$ | $4$ | $8$ | $(C_8:C_2):C_2$ | $[2, 2, 2]^{4}$ | |
\(17\) | 17.16.15.5 | $x^{16} + 17$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |
\(43691\) | $\Q_{43691}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{43691}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{43691}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{43691}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |