Normalized defining polynomial
\( x^{16} - 288 x^{14} + 39396 x^{12} - 3310304 x^{10} + 185330844 x^{8} - 7050027072 x^{6} + 177390624784 x^{4} - 2681723402496 x^{2} + 18451182794256 \)
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$$\chi_{3280}(1,·)$, $\chi_{3280}(2049,·)$, $\chi_{3280}(2633,·)$, $\chi_{3280}(501,·)$, $\chi_{3280}(2861,·)$, $\chi_{3280}(337,·)$, $\chi_{3280}(1557,·)$, $\chi_{3280}(329,·)$, $\chi_{3280}(3117,·)$, $\chi_{3280}(829,·)$, $\chi_{3280}(1713,·)$, $\chi_{3280}(3189,·)$, $\chi_{3280}(2697,·)$, $\chi_{3280}(1721,·)$, $\chi_{3280}(573,·)$, $\chi_{3280}(2133,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{5} + \frac{1}{3} a$, $\frac{1}{18} a^{6} + \frac{1}{18} a^{4} - \frac{1}{9} a^{2}$, $\frac{1}{18} a^{7} + \frac{1}{18} a^{5} - \frac{1}{9} a^{3}$, $\frac{1}{144} a^{8} - \frac{1}{36} a^{6} - \frac{5}{72} a^{4} + \frac{5}{18} a^{2} + \frac{1}{4}$, $\frac{1}{1296} a^{9} - \frac{1}{108} a^{7} + \frac{17}{216} a^{5} + \frac{1}{162} a^{3} + \frac{13}{36} a$, $\frac{1}{16848} a^{10} + \frac{17}{5616} a^{8} + \frac{11}{2808} a^{6} - \frac{527}{8424} a^{4} - \frac{217}{468} a^{2} - \frac{25}{52}$, $\frac{1}{522288} a^{11} + \frac{5}{29016} a^{9} + \frac{23}{2808} a^{7} + \frac{6113}{130572} a^{5} - \frac{1873}{43524} a^{3} + \frac{853}{2418} a$, $\frac{1}{40738464} a^{12} + \frac{19}{5092308} a^{10} + \frac{59}{109512} a^{8} + \frac{137767}{5092308} a^{6} - \frac{203209}{5092308} a^{4} - \frac{35297}{141453} a^{2} - \frac{329}{676}$, $\frac{1}{40738464} a^{13} - \frac{1}{10184616} a^{11} + \frac{659}{3394872} a^{9} + \frac{27173}{2546154} a^{7} + \frac{168695}{5092308} a^{5} - \frac{15415}{94302} a^{3} + \frac{2957}{20956} a$, $\frac{1}{161264392422348384} a^{14} - \frac{135205025}{12404953263257568} a^{12} - \frac{92915968075}{20158049052793548} a^{10} + \frac{120293539409569}{80632196211174192} a^{8} - \frac{37075931931133}{10079024526396774} a^{6} - \frac{1059589389196309}{40316098105587096} a^{4} - \frac{18802682840197}{172291017545244} a^{2} + \frac{69744074494}{222997135413}$, $\frac{1}{206902215477872976672} a^{15} + \frac{48808165685}{6465694233683530521} a^{13} + \frac{3689365897531}{3978888759189864936} a^{11} - \frac{71180915702237}{7957777518379729872} a^{9} + \frac{73207113127673}{76517091522882018} a^{7} - \frac{192546734625651979}{3978888759189864936} a^{5} + \frac{459164190364065821}{2873641881637124676} a^{3} + \frac{28731015299354569}{106431180801374988} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{40}\times C_{156880}$, which has order $200806400$ (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: | \( 546895017.4404038 \) (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 | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | $x^{8} - 9881 x^{4} + 34857216$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |