Normalized defining polynomial
\( x^{16} + 9 x^{14} + 44 x^{12} + 261 x^{10} + 1029 x^{8} + 1044 x^{6} + 704 x^{4} + 576 x^{2} + 256 \)
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: | \(968265199641600000000=2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $20.49$ | 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: | $2, 3, 5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(420=2^{2}\cdot 3\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(391,·)$, $\chi_{420}(139,·)$, $\chi_{420}(209,·)$, $\chi_{420}(211,·)$, $\chi_{420}(251,·)$, $\chi_{420}(281,·)$, $\chi_{420}(349,·)$, $\chi_{420}(419,·)$, $\chi_{420}(41,·)$, $\chi_{420}(71,·)$, $\chi_{420}(29,·)$, $\chi_{420}(239,·)$, $\chi_{420}(181,·)$, $\chi_{420}(169,·)$, $\chi_{420}(379,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{12} a^{8} + \frac{1}{3} a^{4} - \frac{1}{4} a^{2} + \frac{1}{3}$, $\frac{1}{24} a^{9} - \frac{1}{3} a^{5} - \frac{1}{8} a^{3} - \frac{1}{3} a$, $\frac{1}{72} a^{10} - \frac{1}{36} a^{8} - \frac{4}{9} a^{6} - \frac{35}{72} a^{4} - \frac{1}{36} a^{2} - \frac{4}{9}$, $\frac{1}{72} a^{11} + \frac{1}{72} a^{9} + \frac{1}{18} a^{7} + \frac{13}{72} a^{5} - \frac{11}{72} a^{3} - \frac{5}{18} a$, $\frac{1}{720} a^{12} - \frac{1}{144} a^{10} - \frac{1}{36} a^{8} - \frac{299}{720} a^{6} - \frac{61}{144} a^{4} + \frac{5}{36} a^{2} - \frac{2}{15}$, $\frac{1}{1440} a^{13} + \frac{1}{288} a^{11} + \frac{1}{72} a^{9} + \frac{101}{1440} a^{7} + \frac{61}{288} a^{5} - \frac{5}{72} a^{3} + \frac{17}{45} a$, $\frac{1}{5875200} a^{14} - \frac{3083}{5875200} a^{12} - \frac{257}{48960} a^{10} + \frac{16247}{1958400} a^{8} - \frac{404981}{1958400} a^{6} - \frac{259}{765} a^{4} - \frac{35629}{91800} a^{2} + \frac{45677}{91800}$, $\frac{1}{11750400} a^{15} - \frac{3083}{11750400} a^{13} - \frac{257}{97920} a^{11} + \frac{16247}{3916800} a^{9} - \frac{404981}{3916800} a^{7} - \frac{259}{1530} a^{5} + \frac{56171}{183600} a^{3} - \frac{46123}{183600} a$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{22991}{11750400} a^{15} + \frac{159707}{11750400} a^{13} + \frac{5273}{97920} a^{11} + \frac{1438777}{3916800} a^{9} + \frac{4390949}{3916800} a^{7} - \frac{13807}{12240} a^{5} + \frac{138061}{183600} a^{3} + \frac{108667}{183600} a \) (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: | \( 31712.813565 \) (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 | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $7$ | 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 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |