Normalized defining polynomial
\( x^{16} + 424 x^{14} + 69404 x^{12} + 5652768 x^{10} + 251569339 x^{8} + 6316872536 x^{6} + 90249360616 x^{4} + 723888668752 x^{2} + 2944652568001 \)
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: | \(803207007062310661390336000000000000=2^{48}\cdot 5^{12}\cdot 43^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $175.41$ | 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: | $2, 5, 43$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3440=2^{4}\cdot 5\cdot 43\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3440}(1,·)$, $\chi_{3440}(3267,·)$, $\chi_{3440}(3269,·)$, $\chi_{3440}(1547,·)$, $\chi_{3440}(1549,·)$, $\chi_{3440}(2063,·)$, $\chi_{3440}(2581,·)$, $\chi_{3440}(343,·)$, $\chi_{3440}(861,·)$, $\chi_{3440}(2407,·)$, $\chi_{3440}(2409,·)$, $\chi_{3440}(2923,·)$, $\chi_{3440}(687,·)$, $\chi_{3440}(689,·)$, $\chi_{3440}(1203,·)$, $\chi_{3440}(1721,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{1715999} a^{11} - \frac{158528}{1715999} a^{9} + \frac{840487}{1715999} a^{7} - \frac{343583}{1715999} a^{5} + \frac{280430}{1715999} a^{3} + \frac{460128}{1715999} a$, $\frac{1}{4151001581} a^{12} - \frac{489218243}{4151001581} a^{10} + \frac{1740863473}{4151001581} a^{8} + \frac{2038263229}{4151001581} a^{6} - \frac{1133994909}{4151001581} a^{4} - \frac{1154407199}{4151001581} a^{2} + \frac{685}{2419}$, $\frac{1}{4151001581} a^{13} + \frac{317}{4151001581} a^{11} + \frac{63521616}{4151001581} a^{9} - \frac{1885507168}{4151001581} a^{7} - \frac{1807475956}{4151001581} a^{5} - \frac{195878449}{4151001581} a^{3} - \frac{331701054}{4151001581} a$, $\frac{1}{112498311904673869160663635859} a^{14} - \frac{13367451685650883783}{112498311904673869160663635859} a^{12} + \frac{7865823855013807800718713054}{112498311904673869160663635859} a^{10} + \frac{14824778139450052794118263883}{112498311904673869160663635859} a^{8} + \frac{29429123350622074768917107952}{112498311904673869160663635859} a^{6} - \frac{41375080475054947538119781122}{112498311904673869160663635859} a^{4} - \frac{18147374814408356492214633198}{112498311904673869160663635859} a^{2} - \frac{15815719974539426}{38204273443723859}$, $\frac{1}{112498311904673869160663635859} a^{15} - \frac{13367451685650883783}{112498311904673869160663635859} a^{13} - \frac{15495094543177128444408}{112498311904673869160663635859} a^{11} + \frac{39316120710948092152556538663}{112498311904673869160663635859} a^{9} - \frac{30491341102371147495691766048}{112498311904673869160663635859} a^{7} - \frac{19640538177859772500741755533}{112498311904673869160663635859} a^{5} + \frac{31423501145953428121990221414}{112498311904673869160663635859} a^{3} - \frac{25530137211858706849186}{65558495025156698320141} a$
Class group and class number
$C_{3}\times C_{30}\times C_{120}\times C_{1200}$, which has order $12960000$ (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: | \( 12198.951274811623 \) (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 $C_4^2$ |
| Character table for $C_4^2$ |
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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $43$ | 43.8.4.1 | $x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 43.8.4.1 | $x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |