Normalized defining polynomial
\( x^{16} + 36 x^{14} - 12 x^{13} + 564 x^{12} - 120 x^{11} + 4590 x^{10} + 900 x^{9} + 19573 x^{8} + 18072 x^{7} + 48258 x^{6} + 83100 x^{5} + 110460 x^{4} + 160680 x^{3} + 209970 x^{2} + 175032 x + 168121 \)
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: | \(11703378935126425600000000=2^{32}\cdot 5^{8}\cdot 17^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $36.88$ | 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: | $2, 5, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(680=2^{3}\cdot 5\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{680}(1,·)$, $\chi_{680}(579,·)$, $\chi_{680}(69,·)$, $\chi_{680}(271,·)$, $\chi_{680}(339,·)$, $\chi_{680}(341,·)$, $\chi_{680}(409,·)$, $\chi_{680}(611,·)$, $\chi_{680}(101,·)$, $\chi_{680}(679,·)$, $\chi_{680}(169,·)$, $\chi_{680}(171,·)$, $\chi_{680}(239,·)$, $\chi_{680}(441,·)$, $\chi_{680}(509,·)$, $\chi_{680}(511,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{4} - \frac{1}{6}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{6} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{6} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{7} - \frac{1}{6} a^{3}$, $\frac{1}{42} a^{12} + \frac{1}{21} a^{11} + \frac{1}{21} a^{10} - \frac{1}{42} a^{9} - \frac{1}{42} a^{8} - \frac{1}{7} a^{7} - \frac{1}{14} a^{5} + \frac{17}{42} a^{4} + \frac{2}{21} a^{3} + \frac{8}{21} a^{2} + \frac{19}{42} a - \frac{10}{21}$, $\frac{1}{1218} a^{13} - \frac{1}{203} a^{12} + \frac{1}{174} a^{11} - \frac{31}{1218} a^{10} - \frac{2}{29} a^{9} - \frac{9}{203} a^{8} - \frac{19}{406} a^{7} - \frac{1}{406} a^{6} + \frac{10}{609} a^{5} - \frac{43}{203} a^{4} - \frac{331}{1218} a^{3} - \frac{53}{1218} a^{2} - \frac{69}{406} a - \frac{62}{203}$, $\frac{1}{10517430} a^{14} + \frac{1556}{5258715} a^{13} - \frac{12323}{5258715} a^{12} - \frac{9373}{1502490} a^{11} - \frac{43151}{5258715} a^{10} + \frac{482087}{10517430} a^{9} - \frac{13207}{1502490} a^{8} + \frac{156421}{701162} a^{7} + \frac{2377741}{5258715} a^{6} + \frac{4127351}{10517430} a^{5} + \frac{2557543}{10517430} a^{4} - \frac{398263}{2103486} a^{3} + \frac{960667}{10517430} a^{2} + \frac{2948839}{10517430} a + \frac{2206801}{10517430}$, $\frac{1}{59582451833679529212150} a^{15} + \frac{118116294503761}{2837259611127596629150} a^{14} + \frac{527416814350210946}{1418629805563798314575} a^{13} - \frac{6253743261239890}{5158653838413812053} a^{12} + \frac{101335460778915875161}{4255889416691394943725} a^{11} + \frac{192599861954796058577}{2708293265167251327825} a^{10} + \frac{3012703864626082062109}{59582451833679529212150} a^{9} + \frac{1611118138396844081437}{29791225916839764606075} a^{8} + \frac{11255987132823207083251}{29791225916839764606075} a^{7} + \frac{4650796254735073321089}{9930408638946588202025} a^{6} + \frac{191502606216609442451}{1045306172520693494950} a^{5} - \frac{2965611522187110888033}{9930408638946588202025} a^{4} - \frac{22341013792221406685873}{59582451833679529212150} a^{3} - \frac{1629714163407414402833}{59582451833679529212150} a^{2} + \frac{4896326314436113531}{15447874470749164950} a - \frac{166824383360422930592}{334732875470109714675}$
Class group and class number
$C_{4}\times C_{12}$, which has order $48$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{253785592338616}{9035858634164320475} a^{15} - \frac{12599264662073713}{379506062634901459950} a^{14} + \frac{109318271796367663}{126502020878300486650} a^{13} - \frac{55097723516055883}{37950606263490145995} a^{12} + \frac{209373410413100973}{18071717268328640950} a^{11} - \frac{3291135341367623971}{189753031317450729975} a^{10} + \frac{26258577983465153833}{379506062634901459950} a^{9} - \frac{1914091580495030053}{27107575902492961425} a^{8} + \frac{13269451097423595373}{126502020878300486650} a^{7} + \frac{7613615730394936379}{189753031317450729975} a^{6} - \frac{190638455598928233}{605272827168901850} a^{5} + \frac{14988741694668195467}{189753031317450729975} a^{4} - \frac{57751087375407061077}{126502020878300486650} a^{3} - \frac{531438612475646588791}{379506062634901459950} a^{2} - \frac{10962498515472088862}{9987001648286880525} a - \frac{3948252338674724314}{2132056531656749775} \) (order $8$) | 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: | \( 202451.134101 \) (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 $C_2^4$ |
| Character table for $C_2^4$ |
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 | ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
| 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| $5$ | 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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $17$ | 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |