Normalized defining polynomial
\( x^{16} - 28 x^{14} + 308 x^{12} - 1710 x^{10} + 5156 x^{8} - 7740 x^{6} + 9473 x^{4} + 368 x^{2} + 256 \)
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: | \(36004060626969600000000=2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 11^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $25.69$ | 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: | $2, 3, 5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(660=2^{2}\cdot 3\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(131,·)$, $\chi_{660}(199,·)$, $\chi_{660}(329,·)$, $\chi_{660}(331,·)$, $\chi_{660}(461,·)$, $\chi_{660}(529,·)$, $\chi_{660}(659,·)$, $\chi_{660}(89,·)$, $\chi_{660}(221,·)$, $\chi_{660}(419,·)$, $\chi_{660}(551,·)$, $\chi_{660}(109,·)$, $\chi_{660}(241,·)$, $\chi_{660}(439,·)$, $\chi_{660}(571,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4} a^{7} + \frac{1}{4} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{5}$, $\frac{1}{3720} a^{12} - \frac{317}{3720} a^{10} - \frac{151}{1240} a^{8} + \frac{289}{744} a^{6} + \frac{181}{1240} a^{4} - \frac{377}{3720} a^{2} + \frac{98}{465}$, $\frac{1}{3720} a^{13} - \frac{317}{3720} a^{11} - \frac{151}{1240} a^{9} - \frac{83}{744} a^{7} + \frac{181}{1240} a^{5} - \frac{377}{3720} a^{3} - \frac{269}{930} a$, $\frac{1}{13201602960} a^{14} - \frac{73784}{825100185} a^{12} - \frac{64096881}{550066790} a^{10} - \frac{59049737}{6600801480} a^{8} - \frac{9375377}{39290485} a^{6} - \frac{412719781}{1650200370} a^{4} + \frac{49515223}{145072560} a^{2} + \frac{19414524}{275033395}$, $\frac{1}{52806411840} a^{15} + \frac{493091}{4400534320} a^{13} - \frac{947068939}{13201602960} a^{11} + \frac{1633736449}{26403205920} a^{9} - \frac{217701629}{1885943280} a^{7} - \frac{632467243}{2640320592} a^{5} + \frac{30882337}{193430080} a^{3} - \frac{622086317}{1650200370} a$
Class group and class number
$C_{2}\times C_{4}$, which has order $8$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{170447}{580290240} a^{15} - \frac{1233107}{145072560} a^{13} + \frac{14273813}{145072560} a^{11} - \frac{172001269}{290145120} a^{9} + \frac{296025349}{145072560} a^{7} - \frac{567868999}{145072560} a^{5} + \frac{3038615671}{580290240} a^{3} - \frac{82680871}{36268140} a \) (order $12$) | 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: | \( 101728.538468 \) (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 $C_2^4$ |
| Character table for $C_2^4$ |
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 | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $11$ | 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |