Normalized defining polynomial
\( x^{16} - x^{15} - 75 x^{14} + 29 x^{13} + 2024 x^{12} + 527 x^{11} - 24414 x^{10} - 18647 x^{9} + 132021 x^{8} + 143796 x^{7} - 298501 x^{6} - 431294 x^{5} + 137649 x^{4} + 453033 x^{3} + 232820 x^{2} + 33022 x - 1889 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[16, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(32790240335000876777587890625=5^{12}\cdot 7^{8}\cdot 13^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $60.57$ | 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: | $5, 7, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(455=5\cdot 7\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{455}(64,·)$, $\chi_{455}(1,·)$, $\chi_{455}(118,·)$, $\chi_{455}(8,·)$, $\chi_{455}(272,·)$, $\chi_{455}(274,·)$, $\chi_{455}(148,·)$, $\chi_{455}(216,·)$, $\chi_{455}(27,·)$, $\chi_{455}(34,·)$, $\chi_{455}(356,·)$, $\chi_{455}(363,·)$, $\chi_{455}(174,·)$, $\chi_{455}(372,·)$, $\chi_{455}(246,·)$, $\chi_{455}(57,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{14} a^{12} - \frac{3}{14} a^{11} - \frac{3}{14} a^{10} + \frac{1}{7} a^{9} + \frac{1}{14} a^{7} - \frac{1}{14} a^{6} + \frac{5}{14} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{3} - \frac{1}{14} a^{2} + \frac{1}{14} a - \frac{5}{14}$, $\frac{1}{2954} a^{13} + \frac{10}{1477} a^{12} + \frac{286}{1477} a^{11} + \frac{129}{2954} a^{10} - \frac{488}{1477} a^{9} - \frac{125}{2954} a^{8} - \frac{157}{1477} a^{7} - \frac{618}{1477} a^{6} - \frac{145}{2954} a^{5} - \frac{62}{1477} a^{4} + \frac{85}{422} a^{3} - \frac{354}{1477} a^{2} - \frac{159}{1477} a - \frac{283}{2954}$, $\frac{1}{50218} a^{14} + \frac{4}{25109} a^{13} + \frac{1387}{50218} a^{12} + \frac{9301}{50218} a^{11} + \frac{848}{25109} a^{10} + \frac{4835}{50218} a^{9} + \frac{2070}{25109} a^{8} + \frac{3}{238} a^{7} + \frac{339}{50218} a^{6} - \frac{7632}{25109} a^{5} + \frac{14321}{50218} a^{4} - \frac{628}{1477} a^{3} - \frac{22417}{50218} a^{2} - \frac{11659}{50218} a + \frac{11615}{25109}$, $\frac{1}{63544657034625641378} a^{15} - \frac{190611062590237}{31772328517312820689} a^{14} - \frac{7939872466884695}{63544657034625641378} a^{13} - \frac{914327711043381154}{31772328517312820689} a^{12} + \frac{5302708991194368287}{63544657034625641378} a^{11} + \frac{798481351553135341}{9077808147803663054} a^{10} - \frac{7757679965750400748}{31772328517312820689} a^{9} - \frac{5163656606325484395}{63544657034625641378} a^{8} + \frac{9400171674153215727}{31772328517312820689} a^{7} - \frac{17081331044411037049}{63544657034625641378} a^{6} + \frac{5679807071739789379}{63544657034625641378} a^{5} - \frac{11498143269604121265}{31772328517312820689} a^{4} - \frac{7761039415025831085}{63544657034625641378} a^{3} - \frac{272209189081506658}{1868960501018401217} a^{2} - \frac{4312891080364821471}{9077808147803663054} a - \frac{1570435748469265284}{31772328517312820689}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 956063383.6451516 \) (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 | ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| $7$ | 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13 | Data not computed | ||||||