Normalized defining polynomial
\( x^{16} + 40 x^{14} + 668 x^{12} + 5394 x^{10} + 19172 x^{8} + 20988 x^{6} + 32129 x^{4} + 79084 x^{2} + 71824 \)
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: | \(17291220918428778836852736=2^{16}\cdot 3^{8}\cdot 7^{8}\cdot 17^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $37.79$ | 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: | $2, 3, 7, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1428=2^{2}\cdot 3\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1428}(1,·)$, $\chi_{1428}(545,·)$, $\chi_{1428}(713,·)$, $\chi_{1428}(715,·)$, $\chi_{1428}(1427,·)$, $\chi_{1428}(407,·)$, $\chi_{1428}(475,·)$, $\chi_{1428}(1121,·)$, $\chi_{1428}(1189,·)$, $\chi_{1428}(169,·)$, $\chi_{1428}(1259,·)$, $\chi_{1428}(307,·)$, $\chi_{1428}(239,·)$, $\chi_{1428}(883,·)$, $\chi_{1428}(953,·)$, $\chi_{1428}(1021,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{20} a^{10} + \frac{1}{20} a^{8} + \frac{2}{5} a^{6} - \frac{3}{20} a^{4} - \frac{3}{20} a^{2} - \frac{1}{5}$, $\frac{1}{20} a^{11} + \frac{1}{20} a^{9} - \frac{1}{10} a^{7} - \frac{3}{20} a^{5} - \frac{3}{20} a^{3} + \frac{3}{10} a$, $\frac{1}{2580} a^{12} + \frac{53}{2580} a^{10} + \frac{21}{172} a^{8} - \frac{587}{2580} a^{6} - \frac{213}{860} a^{4} - \frac{5}{516} a^{2} - \frac{62}{645}$, $\frac{1}{5160} a^{13} + \frac{53}{5160} a^{11} + \frac{21}{344} a^{9} - \frac{587}{5160} a^{7} - \frac{213}{1720} a^{5} - \frac{5}{1032} a^{3} - \frac{31}{645} a$, $\frac{1}{15787854759000} a^{14} + \frac{589329227}{5262618253000} a^{12} - \frac{158009026411}{15787854759000} a^{10} - \frac{861007664957}{15787854759000} a^{8} + \frac{1528295251577}{3157570951800} a^{6} + \frac{7631913241673}{15787854759000} a^{4} + \frac{170457514037}{2631309126500} a^{2} + \frac{468098687492}{1973481844875}$, $\frac{1}{1057786268853000} a^{15} + \frac{25066623427}{352595422951000} a^{13} - \frac{1002475676311}{1057786268853000} a^{11} + \frac{25427606305843}{1057786268853000} a^{9} - \frac{25722276381283}{211557253770600} a^{7} + \frac{385402233277373}{1057786268853000} a^{5} + \frac{56174506643637}{176297711475500} a^{3} + \frac{125449133308609}{264446567213250} a$
Class group and class number
$C_{2}\times C_{20}$, which has order $40$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{223790141}{14490222861000} a^{15} - \frac{2750163457}{4830074287000} a^{13} - \frac{123293004049}{14490222861000} a^{11} - \frac{803778441263}{14490222861000} a^{9} - \frac{291302339887}{2898044572200} a^{7} + \frac{2466205113107}{14490222861000} a^{5} - \frac{1437116340617}{2415037143500} a^{3} - \frac{2031478205219}{3622555715250} a \) (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: | \( 454890.966834 \) (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 | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $17$ | 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |