Normalized defining polynomial
\( x^{16} + 20 x^{14} + 92 x^{12} - 222 x^{10} - 868 x^{8} + 5004 x^{6} - 799 x^{4} - 41752 x^{2} + 71824 \)
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: | \(531343986448422341246976=2^{16}\cdot 3^{8}\cdot 7^{8}\cdot 11^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $30.40$ | 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: | $2, 3, 7, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(924=2^{2}\cdot 3\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{924}(1,·)$, $\chi_{924}(197,·)$, $\chi_{924}(769,·)$, $\chi_{924}(265,·)$, $\chi_{924}(923,·)$, $\chi_{924}(461,·)$, $\chi_{924}(463,·)$, $\chi_{924}(659,·)$, $\chi_{924}(727,·)$, $\chi_{924}(155,·)$, $\chi_{924}(419,·)$, $\chi_{924}(617,·)$, $\chi_{924}(43,·)$, $\chi_{924}(881,·)$, $\chi_{924}(307,·)$, $\chi_{924}(505,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{20} a^{10} - \frac{1}{10} a^{8} - \frac{2}{5} a^{6} + \frac{9}{20} a^{4} + \frac{1}{10} a^{2} + \frac{2}{5}$, $\frac{1}{20} a^{11} - \frac{1}{10} a^{9} + \frac{1}{10} a^{7} + \frac{9}{20} a^{5} + \frac{1}{10} a^{3} - \frac{1}{10} a$, $\frac{1}{2640} a^{12} - \frac{17}{2640} a^{10} + \frac{19}{880} a^{8} + \frac{509}{2640} a^{6} - \frac{131}{880} a^{4} - \frac{47}{2640} a^{2} + \frac{53}{132}$, $\frac{1}{2640} a^{13} - \frac{17}{2640} a^{11} + \frac{19}{880} a^{9} + \frac{509}{2640} a^{7} - \frac{131}{880} a^{5} - \frac{47}{2640} a^{3} + \frac{53}{132} a$, $\frac{1}{35273615520} a^{14} - \frac{1205381}{17636807760} a^{12} + \frac{14535601}{1603346160} a^{10} + \frac{15326003}{400836540} a^{8} + \frac{2855969041}{17636807760} a^{6} + \frac{7808571539}{17636807760} a^{4} - \frac{2086444949}{7054723104} a^{2} - \frac{304843261}{1763680776}$, $\frac{1}{4726664479680} a^{15} + \frac{1006177}{73854132495} a^{13} - \frac{7416663259}{393888706640} a^{11} + \frac{117217568137}{2363332239840} a^{9} - \frac{84874352309}{393888706640} a^{7} + \frac{283938713527}{1181666119920} a^{5} - \frac{196719307555}{945332895936} a^{3} + \frac{26945944551}{78777741328} a$
Class group and class number
$C_{2}\times C_{12}$, which has order $24$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{60487}{1275752896} a^{15} - \frac{927271}{956814672} a^{13} - \frac{1512131}{299004585} a^{11} + \frac{12623003}{3189382240} a^{9} + \frac{22216981}{598009170} a^{7} - \frac{11144223}{99668195} a^{5} - \frac{2403695573}{19136293440} a^{3} + \frac{4699153867}{4784073360} a \) (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: | \( 183925.11064 \) (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 | R | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $11$ | 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |