Normalized defining polynomial
\( x^{16} + 816 x^{14} + 246024 x^{12} + 33547392 x^{10} + 2007605412 x^{8} + 43864874784 x^{6} + 378786592080 x^{4} + 1110011465088 x^{2} + 359229669714 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(11352029626874238391016464929669692733284941824=2^{79}\cdot 3^{8}\cdot 17^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $755.86$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3264=2^{6}\cdot 3\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3264}(1,·)$, $\chi_{3264}(2051,·)$, $\chi_{3264}(2569,·)$, $\chi_{3264}(2315,·)$, $\chi_{3264}(3217,·)$, $\chi_{3264}(2387,·)$, $\chi_{3264}(25,·)$, $\chi_{3264}(923,·)$, $\chi_{3264}(2209,·)$, $\chi_{3264}(227,·)$, $\chi_{3264}(2089,·)$, $\chi_{3264}(2411,·)$, $\chi_{3264}(625,·)$, $\chi_{3264}(1523,·)$, $\chi_{3264}(3001,·)$, $\chi_{3264}(2171,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{4} - \frac{1}{9} a^{2}$, $\frac{1}{81} a^{5} + \frac{1}{27} a^{3} - \frac{2}{9} a$, $\frac{1}{243} a^{6} + \frac{1}{81} a^{4} - \frac{2}{27} a^{2}$, $\frac{1}{729} a^{7} - \frac{1}{27} a^{3} + \frac{2}{27} a$, $\frac{1}{72171} a^{8} + \frac{14}{8019} a^{6} + \frac{49}{2673} a^{4} + \frac{377}{2673} a^{2} + \frac{4}{11}$, $\frac{1}{72171} a^{9} + \frac{1}{2673} a^{7} + \frac{16}{2673} a^{5} + \frac{80}{2673} a^{3} - \frac{46}{297} a$, $\frac{1}{1948617} a^{10} + \frac{2}{649539} a^{8} + \frac{83}{72171} a^{6} + \frac{734}{72171} a^{4} + \frac{2173}{24057} a^{2} + \frac{3}{11}$, $\frac{1}{91584999} a^{11} - \frac{79}{30528333} a^{9} + \frac{434}{3392037} a^{7} - \frac{10282}{3392037} a^{5} + \frac{42025}{1130679} a^{3} - \frac{139}{423} a$, $\frac{1}{7418384919} a^{12} - \frac{173}{2472794973} a^{10} - \frac{2693}{824264991} a^{8} + \frac{225188}{274754997} a^{6} + \frac{1088480}{91584999} a^{4} + \frac{2549981}{30528333} a^{2} - \frac{64}{297}$, $\frac{1}{7418384919} a^{13} - \frac{1}{224799543} a^{11} - \frac{370}{74933181} a^{9} + \frac{14728}{24977727} a^{7} - \frac{2636}{8325909} a^{5} + \frac{1226279}{30528333} a^{3} - \frac{176}{423} a$, $\frac{1}{3205873759331000637} a^{14} + \frac{1222780}{32382563225565663} a^{12} - \frac{2169299915}{13192896128934159} a^{10} - \frac{313613708678}{118736065160407431} a^{8} + \frac{17469983895907}{13192896128934159} a^{6} - \frac{25726487717594}{1465877347659351} a^{4} + \frac{333585168253198}{4397632042978053} a^{2} - \frac{21292631108}{42783099777}$, $\frac{1}{3205873759331000637} a^{15} + \frac{1222780}{32382563225565663} a^{13} - \frac{8537300}{13192896128934159} a^{11} + \frac{13093598710}{118736065160407431} a^{9} + \frac{3304888497013}{13192896128934159} a^{7} + \frac{6334907963776}{1465877347659351} a^{5} - \frac{77307611377817}{4397632042978053} a^{3} - \frac{5074249084}{2010805689519} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{8186128}$, which has order $2095648768$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1731702170.9408467 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 16 |
| The 16 conjugacy class representatives for $C_{16}$ |
| Character table for $C_{16}$ |
Intermediate fields
| \(\Q(\sqrt{34}) \), 4.4.10061824.2, 8.8.881195590409519104.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | $16$ | ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ | $16$ | R | $16$ | $16$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | $16$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | $16$ | $16$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ | $16$ | $16$ |
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 | Data not computed | ||||||
| $3$ | 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17 | Data not computed | ||||||