Normalized defining polynomial
\( x^{16} - 32 x^{14} + 524 x^{12} - 4686 x^{10} + 26732 x^{8} - 98244 x^{6} + 292049 x^{4} - 672308 x^{2} + 1449616 \)
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: | \(42098158229810084367630336=2^{16}\cdot 3^{8}\cdot 7^{8}\cdot 19^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $39.95$ | 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: | $2, 3, 7, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1596=2^{2}\cdot 3\cdot 7\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1596}(1,·)$, $\chi_{1596}(1217,·)$, $\chi_{1596}(265,·)$, $\chi_{1596}(1483,·)$, $\chi_{1596}(911,·)$, $\chi_{1596}(533,·)$, $\chi_{1596}(1177,·)$, $\chi_{1596}(797,·)$, $\chi_{1596}(799,·)$, $\chi_{1596}(1595,·)$, $\chi_{1596}(419,·)$, $\chi_{1596}(1063,·)$, $\chi_{1596}(685,·)$, $\chi_{1596}(113,·)$, $\chi_{1596}(1331,·)$, $\chi_{1596}(379,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{140} a^{10} + \frac{13}{140} a^{8} - \frac{9}{35} a^{6} - \frac{23}{140} a^{4} - \frac{59}{140} a^{2} + \frac{1}{5}$, $\frac{1}{140} a^{11} + \frac{13}{140} a^{9} + \frac{17}{70} a^{7} - \frac{23}{140} a^{5} - \frac{59}{140} a^{3} - \frac{3}{10} a$, $\frac{1}{420} a^{12} - \frac{1}{420} a^{10} + \frac{9}{140} a^{8} + \frac{61}{420} a^{6} + \frac{41}{140} a^{4} + \frac{17}{60} a^{2} + \frac{1}{15}$, $\frac{1}{36120} a^{13} + \frac{1}{7224} a^{11} - \frac{17}{344} a^{9} - \frac{283}{7224} a^{7} + \frac{1147}{2408} a^{5} + \frac{667}{7224} a^{3} + \frac{296}{645} a$, $\frac{1}{378540506106840} a^{14} - \frac{6362068927}{25236033740456} a^{12} - \frac{634482942073}{378540506106840} a^{10} - \frac{41217253534019}{378540506106840} a^{8} + \frac{4677910614053}{34412773282440} a^{6} - \frac{133334981841361}{378540506106840} a^{4} + \frac{14286611380719}{31545042175570} a^{2} - \frac{38339669398}{157201206855}$, $\frac{1}{2649783542747880} a^{15} + \frac{10109977193}{883261180915960} a^{13} - \frac{1082679926093}{529956708549576} a^{11} + \frac{8507909895665}{529956708549576} a^{9} + \frac{1239695002981}{48177882595416} a^{7} - \frac{250420906157405}{529956708549576} a^{5} + \frac{75825643824223}{220815295228990} a^{3} + \frac{19353475451714}{47317563263355} a$
Class group and class number
$C_{4}\times C_{8}$, which has order $32$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{4516837}{3659922020370} a^{15} - \frac{223870411}{4879896027160} a^{13} + \frac{11237008757}{14639688081480} a^{11} - \frac{98632083719}{14639688081480} a^{9} + \frac{38748725873}{1330880734680} a^{7} - \frac{857468972971}{14639688081480} a^{5} + \frac{77719733507}{975979205432} a^{3} - \frac{213697252649}{522846002910} a \) (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: | \( 925240.331074 \) (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 | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $19$ | 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |