Normalized defining polynomial
\( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 872 x^{12} - 2684 x^{11} + 7338 x^{10} - 16472 x^{9} + 33053 x^{8} - 55468 x^{7} + 76638 x^{6} - 83224 x^{5} + 70130 x^{4} - 44152 x^{3} - 60356 x^{2} + 74504 x + 80644 \)
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: | \(63456228123711897600000000=2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $40.99$ | 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: | \(840=2^{3}\cdot 3\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(391,·)$, $\chi_{840}(209,·)$, $\chi_{840}(211,·)$, $\chi_{840}(559,·)$, $\chi_{840}(349,·)$, $\chi_{840}(419,·)$, $\chi_{840}(379,·)$, $\chi_{840}(41,·)$, $\chi_{840}(71,·)$, $\chi_{840}(29,·)$, $\chi_{840}(239,·)$, $\chi_{840}(181,·)$, $\chi_{840}(169,·)$, $\chi_{840}(251,·)$, $\chi_{840}(701,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{30} a^{12} - \frac{1}{30} a^{11} - \frac{1}{15} a^{10} + \frac{1}{30} a^{8} - \frac{1}{6} a^{7} + \frac{2}{15} a^{6} - \frac{4}{15} a^{5} + \frac{1}{15} a^{3} - \frac{2}{5} a^{2} + \frac{1}{3} a + \frac{1}{5}$, $\frac{1}{30} a^{13} + \frac{1}{15} a^{11} - \frac{1}{15} a^{10} + \frac{1}{30} a^{9} + \frac{1}{30} a^{8} - \frac{1}{5} a^{7} - \frac{7}{15} a^{6} + \frac{1}{15} a^{5} + \frac{7}{30} a^{4} + \frac{1}{3} a^{3} - \frac{2}{5} a^{2} - \frac{2}{15} a + \frac{1}{5}$, $\frac{1}{17192112494850} a^{14} - \frac{7}{17192112494850} a^{13} + \frac{2972516293}{217621677150} a^{12} - \frac{1408972722791}{17192112494850} a^{11} - \frac{109084385629}{3438422498970} a^{10} + \frac{219322086739}{2865352082475} a^{9} - \frac{202941579814}{8596056247425} a^{8} + \frac{1152054225293}{17192112494850} a^{7} - \frac{470195529028}{1719211249485} a^{6} + \frac{1770374625421}{5730704164950} a^{5} - \frac{4712376513739}{17192112494850} a^{4} - \frac{4232135926426}{8596056247425} a^{3} + \frac{9571203907}{343842249897} a^{2} + \frac{142486766527}{8596056247425} a + \frac{2041049939944}{8596056247425}$, $\frac{1}{1391340472095715650} a^{15} + \frac{1759}{60493064004161550} a^{14} - \frac{3995042994342943}{695670236047857825} a^{13} - \frac{8129235222658763}{1391340472095715650} a^{12} + \frac{48901737275215283}{695670236047857825} a^{11} - \frac{5945953754603999}{154593385788412850} a^{10} - \frac{53110370184874787}{1391340472095715650} a^{9} - \frac{10949669043702407}{695670236047857825} a^{8} + \frac{198081981212008607}{1391340472095715650} a^{7} + \frac{115706673043213448}{231890078682619275} a^{6} - \frac{276528493090195987}{1391340472095715650} a^{5} - \frac{202879848353052893}{1391340472095715650} a^{4} + \frac{347340276222432856}{695670236047857825} a^{3} + \frac{103578240890543992}{695670236047857825} a^{2} - \frac{132740055673654058}{695670236047857825} a + \frac{29020428766740332}{231890078682619275}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{16}$, which has order $128$ (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: | \( 31712.813565 \) (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 | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
| 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| $3$ | 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 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |