Normalized defining polynomial
\( x^{16} - 52 x^{14} + 1112 x^{12} - 12702 x^{10} + 84248 x^{8} - 327816 x^{6} + 743633 x^{4} - 790564 x^{2} + 364816 \)
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: | \(13153238385373209600000000=2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 23^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $37.15$ | 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: | $2, 3, 5, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1380=2^{2}\cdot 3\cdot 5\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1380}(1,·)$, $\chi_{1380}(1289,·)$, $\chi_{1380}(599,·)$, $\chi_{1380}(139,·)$, $\chi_{1380}(781,·)$, $\chi_{1380}(461,·)$, $\chi_{1380}(919,·)$, $\chi_{1380}(1241,·)$, $\chi_{1380}(91,·)$, $\chi_{1380}(1379,·)$, $\chi_{1380}(229,·)$, $\chi_{1380}(551,·)$, $\chi_{1380}(689,·)$, $\chi_{1380}(691,·)$, $\chi_{1380}(829,·)$, $\chi_{1380}(1151,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{2} a^{6} - \frac{3}{8} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{3}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{24} a^{12} + \frac{1}{24} a^{10} - \frac{7}{24} a^{6} + \frac{3}{8} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{24} a^{13} + \frac{1}{24} a^{11} + \frac{5}{24} a^{7} + \frac{3}{8} a^{5} - \frac{1}{3} a^{3} + \frac{1}{6} a$, $\frac{1}{24339618089514624} a^{14} + \frac{20984649219917}{8113206029838208} a^{12} + \frac{514959578320037}{24339618089514624} a^{10} - \frac{536621817790471}{24339618089514624} a^{8} - \frac{5258149870174373}{24339618089514624} a^{6} + \frac{7796850367998721}{24339618089514624} a^{4} + \frac{547108507101485}{2028301507459552} a^{2} - \frac{540663927666595}{1521226130594664}$, $\frac{1}{3675282331516708224} a^{15} + \frac{16247396708896333}{1225094110505569408} a^{13} + \frac{79618718369242565}{3675282331516708224} a^{11} - \frac{40088501213251735}{3675282331516708224} a^{9} + \frac{128609749622156059}{3675282331516708224} a^{7} - \frac{1020552013913994143}{3675282331516708224} a^{5} + \frac{63170917549915153}{306273527626392352} a^{3} + \frac{78563094863255933}{229705145719794264} a$
Class group and class number
$C_{2}\times C_{12}$, which has order $24$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{6333766631}{2790647176550272} a^{15} + \frac{1044882293773}{8371941529650816} a^{13} - \frac{23935887073577}{8371941529650816} a^{11} + \frac{99253106765185}{2790647176550272} a^{9} - \frac{2191284577843063}{8371941529650816} a^{7} + \frac{3179970662639065}{2790647176550272} a^{5} - \frac{5667730224543071}{2092985382412704} a^{3} + \frac{1334662852651915}{523246345603176} a \) (order $12$) | 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: | \( 612241.341839 \) (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 | R | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $5$ | 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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $23$ | 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |