Normalized defining polynomial
\( x^{16} + 288 x^{14} + 32916 x^{12} + 1621984 x^{10} + 17935164 x^{8} - 768301248 x^{6} + 6889679824 x^{4} - 24932542464 x^{2} + 35571468816 \)
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: | \(96909452923685986042145406976000000000000=2^{44}\cdot 5^{12}\cdot 41^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $364.46$ | 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: | $2, 5, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3280=2^{4}\cdot 5\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3280}(1,·)$, $\chi_{3280}(2059,·)$, $\chi_{3280}(2049,·)$, $\chi_{3280}(2633,·)$, $\chi_{3280}(1803,·)$, $\chi_{3280}(337,·)$, $\chi_{3280}(83,·)$, $\chi_{3280}(1731,·)$, $\chi_{3280}(329,·)$, $\chi_{3280}(2787,·)$, $\chi_{3280}(1713,·)$, $\chi_{3280}(811,·)$, $\chi_{3280}(1067,·)$, $\chi_{3280}(1139,·)$, $\chi_{3280}(2697,·)$, $\chi_{3280}(1721,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{5} + \frac{1}{3} a$, $\frac{1}{54} a^{6} + \frac{1}{54} a^{4} - \frac{10}{27} a^{2} + \frac{1}{3}$, $\frac{1}{54} a^{7} + \frac{1}{54} a^{5} - \frac{1}{27} a^{3}$, $\frac{1}{432} a^{8} - \frac{1}{108} a^{6} - \frac{17}{216} a^{4} - \frac{1}{54} a^{2} + \frac{5}{12}$, $\frac{1}{40176} a^{9} - \frac{11}{3348} a^{7} - \frac{73}{2232} a^{5} + \frac{679}{5022} a^{3} + \frac{157}{1116} a$, $\frac{1}{40176} a^{10} - \frac{13}{13392} a^{8} - \frac{11}{2232} a^{6} - \frac{1469}{20088} a^{4} + \frac{161}{3348} a^{2} + \frac{1}{12}$, $\frac{1}{40176} a^{11} - \frac{23}{6696} a^{7} - \frac{263}{5022} a^{5} + \frac{23}{372} a^{3} - \frac{3}{31} a$, $\frac{1}{927824544} a^{12} - \frac{125}{463912272} a^{10} + \frac{169963}{154637424} a^{8} - \frac{783487}{231956136} a^{6} - \frac{1503053}{231956136} a^{4} + \frac{805169}{12886452} a^{2} + \frac{9799}{23094}$, $\frac{1}{12061719072} a^{13} - \frac{5339}{463912272} a^{11} + \frac{1231}{154637424} a^{9} + \frac{1661123}{231956136} a^{7} - \frac{16764617}{231956136} a^{5} - \frac{1261133}{12886452} a^{3} - \frac{1421237}{3102294} a$, $\frac{1}{318750237289834272} a^{14} + \frac{5542099}{24519249022294944} a^{12} + \frac{57057443503}{6129812255573736} a^{10} + \frac{6472779160685}{6129812255573736} a^{8} - \frac{1667610210929}{766226531946717} a^{6} - \frac{68263892950859}{3064906127786868} a^{4} - \frac{1752337705849625}{4427086629025476} a^{2} + \frac{16229116177}{39373930548}$, $\frac{1}{4143753084767845536} a^{15} - \frac{33659117}{4143753084767845536} a^{13} + \frac{949536612475}{79687559322458568} a^{11} - \frac{1176473707217}{159375118644917136} a^{9} + \frac{146295631508821}{39843779661229284} a^{7} - \frac{5270203930690807}{79687559322458568} a^{5} + \frac{5621655362149933}{57552126177331188} a^{3} - \frac{1108606218685}{103140011070486} a$
Class group and class number
$C_{4}\times C_{20}\times C_{20}\times C_{40}\times C_{7400}$, which has order $473600000$ (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: | \( 52989569.66560047 \) (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 $C_4^2$ |
| Character table for $C_4^2$ |
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 | ${\href{/LocalNumberField/3.1.0.1}{1} }^{16}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $41$ | 41.8.6.1 | $x^{8} - 9881 x^{4} + 34857216$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 41.8.6.1 | $x^{8} - 9881 x^{4} + 34857216$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |