Normalized defining polynomial
\( x^{16} + 816 x^{14} + 236232 x^{12} + 30712608 x^{10} + 2042084115 x^{8} + 71545582152 x^{6} + 1241503524972 x^{4} + 8539740283104 x^{2} + 8022993935121 \)
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: | \(262686006513622818702917369856000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $387.90$ | 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: | $2, 3, 5, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4080=2^{4}\cdot 3\cdot 5\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4080}(1,·)$, $\chi_{4080}(1067,·)$, $\chi_{4080}(1801,·)$, $\chi_{4080}(2189,·)$, $\chi_{4080}(2449,·)$, $\chi_{4080}(4067,·)$, $\chi_{4080}(1109,·)$, $\chi_{4080}(1883,·)$, $\chi_{4080}(803,·)$, $\chi_{4080}(3367,·)$, $\chi_{4080}(169,·)$, $\chi_{4080}(103,·)$, $\chi_{4080}(3821,·)$, $\chi_{4080}(1903,·)$, $\chi_{4080}(2741,·)$, $\chi_{4080}(1087,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{153} a^{4}$, $\frac{1}{153} a^{5}$, $\frac{1}{459} a^{6}$, $\frac{1}{459} a^{7}$, $\frac{1}{117045} a^{8} - \frac{2}{2295} a^{6} - \frac{2}{765} a^{4} - \frac{1}{15} a^{2} - \frac{1}{5}$, $\frac{1}{1287495} a^{9} - \frac{1}{935} a^{7} + \frac{2}{935} a^{5} + \frac{3}{55} a^{3} - \frac{6}{55} a$, $\frac{1}{3862485} a^{10} + \frac{1}{429165} a^{8} - \frac{26}{25245} a^{6} - \frac{1}{8415} a^{4} + \frac{9}{55} a^{2} - \frac{2}{5}$, $\frac{1}{42487335} a^{11} - \frac{1}{14162445} a^{9} + \frac{1}{10285} a^{7} + \frac{49}{30855} a^{5} - \frac{64}{1815} a^{3} + \frac{167}{605} a$, $\frac{1}{32502811275} a^{12} + \frac{14}{212436675} a^{10} + \frac{89}{42487335} a^{8} - \frac{533}{833085} a^{6} - \frac{686}{277695} a^{4} + \frac{818}{27225} a^{2} - \frac{8}{75}$, $\frac{1}{32502811275} a^{13} - \frac{1}{212436675} a^{11} - \frac{1}{42487335} a^{9} + \frac{82}{833085} a^{7} - \frac{161}{277695} a^{5} - \frac{757}{27225} a^{3} + \frac{3562}{9075} a$, $\frac{1}{91774450373920875} a^{14} + \frac{41161}{10197161152657875} a^{12} - \frac{7968182}{599833008979875} a^{10} + \frac{641021}{470457261945} a^{8} - \frac{29226952}{27037773675} a^{6} - \frac{2738554964}{1306825727625} a^{4} + \frac{195581224}{25624033875} a^{2} - \frac{11206559}{23529875}$, $\frac{1}{91774450373920875} a^{15} + \frac{41161}{10197161152657875} a^{13} + \frac{6149743}{599833008979875} a^{11} - \frac{2091134}{7997773453065} a^{9} - \frac{27669112}{27037773675} a^{7} + \frac{2287426336}{1306825727625} a^{5} - \frac{3503315126}{25624033875} a^{3} + \frac{51092886}{2847114875} a$
Class group and class number
$C_{2}\times C_{4}\times C_{60}\times C_{60}\times C_{120}\times C_{120}$, which has order $414720000$ (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: | \( 1671686.3034391652 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_4^2$ |
| Character table for $C_4^2$ |
Intermediate fields
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 | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 17 | Data not computed | ||||||