Normalized defining polynomial
\( x^{16} + 820 x^{14} + 192290 x^{12} + 19384800 x^{10} + 899179200 x^{8} + 17643776000 x^{6} + 133834496000 x^{4} + 240988160000 x^{2} + 68853760000 \)
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: | \(162904790364716142536846429126656000000000000=2^{44}\cdot 5^{12}\cdot 41^{14}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $579.76$ | 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$$\chi_{3280}(1,·)$, $\chi_{3280}(2627,·)$, $\chi_{3280}(2369,·)$, $\chi_{3280}(9,·)$, $\chi_{3280}(1227,·)$, $\chi_{3280}(81,·)$, $\chi_{3280}(2323,·)$, $\chi_{3280}(729,·)$, $\chi_{3280}(987,·)$, $\chi_{3280}(2843,·)$, $\chi_{3280}(1203,·)$, $\chi_{3280}(1641,·)$, $\chi_{3280}(683,·)$, $\chi_{3280}(1649,·)$, $\chi_{3280}(2867,·)$, $\chi_{3280}(1721,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{10} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{2} a$, $\frac{1}{40} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{80} a^{7} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{1049600} a^{8} - \frac{7}{1280} a^{6} + \frac{117}{2560} a^{4} + \frac{13}{32} a^{2} + \frac{1}{16}$, $\frac{1}{2099200} a^{9} - \frac{7}{2560} a^{7} + \frac{117}{5120} a^{5} - \frac{19}{64} a^{3} - \frac{15}{32} a$, $\frac{1}{20992000} a^{10} + \frac{397}{51200} a^{6} - \frac{33}{2560} a^{4} + \frac{159}{320} a^{2} - \frac{1}{16}$, $\frac{1}{41984000} a^{11} + \frac{397}{102400} a^{7} - \frac{33}{5120} a^{5} + \frac{159}{640} a^{3} - \frac{1}{32} a$, $\frac{1}{166592512000} a^{12} + \frac{873}{41648128000} a^{10} + \frac{7597}{16659251200} a^{8} - \frac{154883}{12697600} a^{6} - \frac{6283}{1269760} a^{4} + \frac{36319}{79360} a^{2} - \frac{2979}{15872}$, $\frac{1}{333185024000} a^{13} + \frac{873}{83296256000} a^{11} + \frac{7597}{33318502400} a^{9} - \frac{154883}{25395200} a^{7} - \frac{6283}{2539520} a^{5} + \frac{36319}{158720} a^{3} - \frac{2979}{31744} a$, $\frac{1}{2102397501440000} a^{14} - \frac{9}{10511987507200} a^{12} + \frac{3575157}{210239750144000} a^{10} - \frac{3837691}{10511987507200} a^{8} - \frac{104454673}{16024371200} a^{6} + \frac{20554913}{801218560} a^{4} + \frac{5616261}{40060928} a^{2} - \frac{997555}{10015232}$, $\frac{1}{4204795002880000} a^{15} - \frac{9}{21023975014400} a^{13} + \frac{3575157}{420479500288000} a^{11} - \frac{3837691}{21023975014400} a^{9} - \frac{104454673}{32048742400} a^{7} + \frac{20554913}{1602437120} a^{5} + \frac{5616261}{80121856} a^{3} + \frac{9017677}{20030464} a$
Class group and class number
$C_{2}\times C_{6}\times C_{6}\times C_{6}\times C_{6}\times C_{12}\times C_{39576}$, which has order $1230971904$ (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: | \( 42893219.69984098 \) (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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.11.2 | $x^{4} + 8 x + 14$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ |
| 2.4.11.2 | $x^{4} + 8 x + 14$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ | |
| 2.4.11.2 | $x^{4} + 8 x + 14$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ | |
| 2.4.11.2 | $x^{4} + 8 x + 14$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $41$ | 41.8.7.1 | $x^{8} - 41$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 41.8.7.1 | $x^{8} - 41$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |