Normalized defining polynomial
\( x^{16} + 656 x^{14} + 155144 x^{12} + 16296352 x^{10} + 793516132 x^{8} + 16786170144 x^{6} + 140306830128 x^{4} + 284382596736 x^{2} + 161476189281 \)
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: | \(4590539770923916035789355095649376383046516736=2^{64}\cdot 3^{8}\cdot 41^{14}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $714.28$ | 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: | $2, 3, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3936=2^{5}\cdot 3\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3936}(1,·)$, $\chi_{3936}(2627,·)$, $\chi_{3936}(1093,·)$, $\chi_{3936}(647,·)$, $\chi_{3936}(3277,·)$, $\chi_{3936}(3863,·)$, $\chi_{3936}(2651,·)$, $\chi_{3936}(3037,·)$, $\chi_{3936}(1321,·)$, $\chi_{3936}(875,·)$, $\chi_{3936}(1967,·)$, $\chi_{3936}(1393,·)$, $\chi_{3936}(2867,·)$, $\chi_{3936}(3253,·)$, $\chi_{3936}(2041,·)$, $\chi_{3936}(575,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{41} a^{8}$, $\frac{1}{1353} a^{9} - \frac{5}{33} a^{7} + \frac{13}{33} a^{5} + \frac{8}{33} a^{3} - \frac{1}{33} a$, $\frac{1}{4059} a^{10} + \frac{26}{4059} a^{8} + \frac{13}{99} a^{6} + \frac{41}{99} a^{4} - \frac{34}{99} a^{2}$, $\frac{1}{12177} a^{11} - \frac{1}{12177} a^{9} + \frac{49}{297} a^{7} - \frac{112}{297} a^{5} - \frac{52}{297} a^{3} + \frac{1}{11} a$, $\frac{1}{2666763} a^{12} - \frac{190}{2666763} a^{10} + \frac{26795}{2666763} a^{8} - \frac{15934}{65043} a^{6} - \frac{18493}{65043} a^{4} + \frac{10}{803} a^{2} - \frac{5}{73}$, $\frac{1}{88003179} a^{13} - \frac{3475}{88003179} a^{11} + \frac{18254}{88003179} a^{9} - \frac{703156}{2146419} a^{7} - \frac{714913}{2146419} a^{5} - \frac{3821}{21681} a^{3} - \frac{1807}{26499} a$, $\frac{1}{5276199310282569120024871881} a^{14} + \frac{447482981213267449517}{5276199310282569120024871881} a^{12} + \frac{218795583695803308819938}{5276199310282569120024871881} a^{10} + \frac{43340268601987243897115653}{5276199310282569120024871881} a^{8} + \frac{51049249381244600871757679}{128687788055672417561582241} a^{6} + \frac{275804394993047018100859}{1299876647026994116783659} a^{4} - \frac{174943094413203599495662}{529579374714701306837787} a^{2} + \frac{45709020136502300343}{132626940825119285459}$, $\frac{1}{15828597930847707360074615643} a^{15} + \frac{87755114317738428683}{15828597930847707360074615643} a^{13} + \frac{168973274130772539434429}{15828597930847707360074615643} a^{11} + \frac{2977003296974410114436683}{15828597930847707360074615643} a^{9} + \frac{158407872889013863965539975}{386063364167017252684746723} a^{7} - \frac{264563182113703032806182}{3899629941080982350350977} a^{5} - \frac{1476709188808168979727176}{4766214372432311761540083} a^{3} - \frac{6666328756204800823471}{16047859839839433540539} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{28}\times C_{169456}$, which has order $1214660608$ (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: | \( 55505673.28953015 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_8$ (as 16T5):
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_8\times C_2$ |
| Character table for $C_8\times C_2$ |
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 | R | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 | ||||||
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 41 | Data not computed | ||||||