Normalized defining polynomial
\( x^{16} + 820 x^{14} + 243950 x^{12} + 33333000 x^{10} + 2178022500 x^{8} + 67912400000 x^{6} + 921188000000 x^{4} + 3765440000000 x^{2} + 2689600000000 \)
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: | \(260647664583545828058954286602649600000000=2^{44}\cdot 5^{8}\cdot 41^{14}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $387.71$ | 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$$\chi_{3280}(1,·)$, $\chi_{3280}(2949,·)$, $\chi_{3280}(1069,·)$, $\chi_{3280}(81,·)$, $\chi_{3280}(2709,·)$, $\chi_{3280}(1721,·)$, $\chi_{3280}(2269,·)$, $\chi_{3280}(2961,·)$, $\chi_{3280}(401,·)$, $\chi_{3280}(1321,·)$, $\chi_{3280}(109,·)$, $\chi_{3280}(1309,·)$, $\chi_{3280}(629,·)$, $\chi_{3280}(1641,·)$, $\chi_{3280}(2041,·)$, $\chi_{3280}(1749,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{50} a^{4}$, $\frac{1}{50} a^{5}$, $\frac{1}{250} a^{6}$, $\frac{1}{250} a^{7}$, $\frac{1}{102500} a^{8}$, $\frac{1}{205000} a^{9} - \frac{1}{100} a^{5} - \frac{1}{2} a$, $\frac{1}{47150000} a^{10} + \frac{2}{589375} a^{8} - \frac{29}{23000} a^{6} + \frac{1}{230} a^{4} - \frac{19}{460} a^{2} + \frac{10}{23}$, $\frac{1}{94300000} a^{11} + \frac{1}{589375} a^{9} + \frac{63}{46000} a^{7} - \frac{9}{1150} a^{5} + \frac{73}{920} a^{3} - \frac{13}{46} a$, $\frac{1}{75440000000} a^{12} + \frac{29}{3772000000} a^{10} + \frac{799}{1508800000} a^{8} + \frac{201}{1840000} a^{6} - \frac{5711}{736000} a^{4} + \frac{771}{9200} a^{2} - \frac{79}{920}$, $\frac{1}{150880000000} a^{13} + \frac{29}{7544000000} a^{11} + \frac{799}{3017600000} a^{9} - \frac{7159}{3680000} a^{7} + \frac{9009}{1472000} a^{5} + \frac{771}{18400} a^{3} + \frac{841}{1840} a$, $\frac{1}{381893876800000000} a^{14} + \frac{96489}{19094693840000000} a^{12} - \frac{30056961}{7637877536000000} a^{10} + \frac{952911141}{381893876800000} a^{8} + \frac{1758883729}{3725793920000} a^{6} + \frac{74683363}{23286212000} a^{4} + \frac{183761921}{4657242400} a^{2} - \frac{2596773}{11643106}$, $\frac{1}{763787753600000000} a^{15} + \frac{96489}{38189387680000000} a^{13} - \frac{30056961}{15275755072000000} a^{11} + \frac{952911141}{763787753600000} a^{9} + \frac{1758883729}{7451587840000} a^{7} + \frac{74683363}{46572424000} a^{5} + \frac{183761921}{9314484800} a^{3} + \frac{9046333}{23286212} a$
Class group and class number
$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{8}\times C_{112712}$, which has order $230834176$ (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: | \( 533954.940478744 \) (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
| \(\Q(\sqrt{82}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{41})\), 4.4.4410944.2, 4.4.68921.1, 8.8.19456426971136.3, 8.0.510536643722608640000.2, 8.0.510536643722608640000.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.22.5 | $x^{8} + 10 x^{4} + 16 x + 36$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ |
| 2.8.22.5 | $x^{8} + 10 x^{4} + 16 x + 36$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ | |
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $41$ | 41.8.7.1 | $x^{8} - 41$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 41.8.7.1 | $x^{8} - 41$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |