Normalized defining polynomial
\( x^{16} + 32 x^{14} + 428 x^{12} + 2610 x^{10} + 5324 x^{8} - 4644 x^{6} + 401 x^{4} + 13100 x^{2} + 10000 \)
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: | \(2021999793722499352559616=2^{16}\cdot 3^{8}\cdot 7^{8}\cdot 13^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $33.05$ | 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: | $2, 3, 7, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1092=2^{2}\cdot 3\cdot 7\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1092}(1,·)$, $\chi_{1092}(1091,·)$, $\chi_{1092}(391,·)$, $\chi_{1092}(727,·)$, $\chi_{1092}(911,·)$, $\chi_{1092}(209,·)$, $\chi_{1092}(755,·)$, $\chi_{1092}(155,·)$, $\chi_{1092}(545,·)$, $\chi_{1092}(547,·)$, $\chi_{1092}(337,·)$, $\chi_{1092}(937,·)$, $\chi_{1092}(365,·)$, $\chi_{1092}(883,·)$, $\chi_{1092}(181,·)$, $\chi_{1092}(701,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{4}$, $\frac{1}{20} a^{11} - \frac{1}{20} a^{9} - \frac{1}{5} a^{7} - \frac{3}{20} a^{5} + \frac{3}{20} a^{3} - \frac{2}{5} a$, $\frac{1}{60} a^{12} + \frac{1}{15} a^{10} + \frac{1}{10} a^{8} + \frac{17}{60} a^{6} - \frac{1}{5} a^{4} + \frac{11}{30} a^{2} - \frac{1}{3}$, $\frac{1}{120} a^{13} + \frac{1}{120} a^{11} + \frac{3}{40} a^{9} + \frac{29}{120} a^{7} + \frac{19}{40} a^{5} - \frac{47}{120} a^{3} + \frac{1}{30} a$, $\frac{1}{93104054914200} a^{14} - \frac{255942916931}{31034684971400} a^{12} + \frac{160475251909}{5476709112600} a^{10} - \frac{1075266975593}{18620810982840} a^{8} + \frac{429223473647}{5476709112600} a^{6} - \frac{41630119143419}{93104054914200} a^{4} - \frac{3320711285729}{15517342485700} a^{2} + \frac{222216344302}{465520274571}$, $\frac{1}{465520274571000} a^{15} - \frac{255942916931}{155173424857000} a^{13} + \frac{160475251909}{27383545563000} a^{11} + \frac{3579935770117}{93104054914200} a^{9} + \frac{429223473647}{27383545563000} a^{7} + \frac{51473935770781}{465520274571000} a^{5} - \frac{3739679537501}{19396678107125} a^{3} - \frac{141764840968}{465520274571} a$
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: | \( \frac{47374352}{458189246625} a^{15} + \frac{11668374037}{3665513973000} a^{13} + \frac{2898534173}{71872823000} a^{11} + \frac{160083571687}{733102794600} a^{9} + \frac{17686858659}{71872823000} a^{7} - \frac{3868937870729}{3665513973000} a^{5} + \frac{2608809159691}{3665513973000} a^{3} + \frac{16333342137}{12218379910} a \) (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: | \( 801785.836569 \) (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 | ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $13$ | 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |