Normalized defining polynomial
\( x^{16} + 1160 x^{14} + 521420 x^{12} + 117067200 x^{10} + 13863001950 x^{8} + 820543516000 x^{6} + 19044955487000 x^{4} + 410222980000 x^{2} + 1768202500 \)
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: | \(398360031450580799831535291203584000000000000=2^{62}\cdot 5^{12}\cdot 29^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $613.08$ | 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: | $2, 5, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4640=2^{5}\cdot 5\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4640}(1,·)$, $\chi_{4640}(1027,·)$, $\chi_{4640}(521,·)$, $\chi_{4640}(3787,·)$, $\chi_{4640}(2321,·)$, $\chi_{4640}(3347,·)$, $\chi_{4640}(2841,·)$, $\chi_{4640}(929,·)$, $\chi_{4640}(1467,·)$, $\chi_{4640}(1449,·)$, $\chi_{4640}(1003,·)$, $\chi_{4640}(3249,·)$, $\chi_{4640}(563,·)$, $\chi_{4640}(3769,·)$, $\chi_{4640}(2883,·)$, $\chi_{4640}(3323,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{145} a^{4}$, $\frac{1}{145} a^{5}$, $\frac{1}{145} a^{6}$, $\frac{1}{145} a^{7}$, $\frac{1}{1724050} a^{8} + \frac{2}{5945} a^{6} - \frac{9}{5945} a^{4} - \frac{7}{41} a^{2} + \frac{1}{41}$, $\frac{1}{1724050} a^{9} + \frac{2}{5945} a^{7} - \frac{9}{5945} a^{5} - \frac{7}{41} a^{3} + \frac{1}{41} a$, $\frac{1}{1724050} a^{10} + \frac{4}{1189} a^{6} - \frac{18}{5945} a^{4} + \frac{2}{41} a^{2} - \frac{6}{41}$, $\frac{1}{1724050} a^{11} + \frac{4}{1189} a^{7} - \frac{18}{5945} a^{5} + \frac{2}{41} a^{3} - \frac{6}{41} a$, $\frac{1}{249987250} a^{12} - \frac{2}{1189} a^{6} - \frac{7}{5945} a^{4} - \frac{11}{41} a^{2} + \frac{1}{41}$, $\frac{1}{249987250} a^{13} - \frac{2}{1189} a^{7} - \frac{7}{5945} a^{5} - \frac{11}{41} a^{3} + \frac{1}{41} a$, $\frac{1}{303256778981210846750} a^{14} - \frac{267316952584}{151628389490605423375} a^{12} - \frac{16561935492}{149387575852813225} a^{10} + \frac{39388366529}{2091426061939385150} a^{8} - \frac{935189262437}{313557130725545} a^{6} - \frac{2976991242670}{1442362801337507} a^{4} - \frac{10792082519217}{49736648321983} a^{2} + \frac{17798506351534}{49736648321983}$, $\frac{1}{303256778981210846750} a^{15} - \frac{267316952584}{151628389490605423375} a^{13} - \frac{16561935492}{149387575852813225} a^{11} + \frac{39388366529}{2091426061939385150} a^{9} - \frac{935189262437}{313557130725545} a^{7} - \frac{2976991242670}{1442362801337507} a^{5} - \frac{10792082519217}{49736648321983} a^{3} + \frac{17798506351534}{49736648321983} a$
Class group and class number
$C_{2}\times C_{36}\times C_{24789924}$, which has order $1784874528$ (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: | \( 2227936.3231945336 \) (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 | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 | ||||||
| 5 | Data not computed | ||||||
| $29$ | 29.8.6.4 | $x^{8} - 29 x^{4} + 2523$ | $4$ | $2$ | $6$ | $C_8$ | $[\ ]_{4}^{2}$ |
| 29.8.6.4 | $x^{8} - 29 x^{4} + 2523$ | $4$ | $2$ | $6$ | $C_8$ | $[\ ]_{4}^{2}$ | |