Normalized defining polynomial
\( x^{16} - 28 x^{14} + 392 x^{12} - 1818 x^{10} - 2632 x^{8} + 31416 x^{6} - 9127 x^{4} - 105196 x^{2} + 132496 \)
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: | \(12052057944074269250390625=3^{8}\cdot 5^{8}\cdot 7^{8}\cdot 13^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $36.95$ | 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: | $3, 5, 7, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1365=3\cdot 5\cdot 7\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1365}(64,·)$, $\chi_{1365}(1,·)$, $\chi_{1365}(1091,·)$, $\chi_{1365}(1156,·)$, $\chi_{1365}(454,·)$, $\chi_{1365}(391,·)$, $\chi_{1365}(974,·)$, $\chi_{1365}(911,·)$, $\chi_{1365}(209,·)$, $\chi_{1365}(274,·)$, $\chi_{1365}(1364,·)$, $\chi_{1365}(1301,·)$, $\chi_{1365}(664,·)$, $\chi_{1365}(1184,·)$, $\chi_{1365}(181,·)$, $\chi_{1365}(701,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{24} a^{8} - \frac{1}{4} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{7}{24} a^{2} + \frac{1}{4} a + \frac{1}{6}$, $\frac{1}{24} a^{9} + \frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{7}{24} a^{3} - \frac{1}{3} a$, $\frac{1}{24} a^{10} + \frac{1}{6} a^{6} - \frac{1}{24} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{10} - \frac{1}{48} a^{9} - \frac{1}{12} a^{6} + \frac{1}{16} a^{5} + \frac{1}{48} a^{4} - \frac{19}{48} a^{3} - \frac{1}{4} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{48} a^{12} - \frac{1}{48} a^{9} - \frac{1}{12} a^{7} + \frac{7}{48} a^{6} + \frac{1}{12} a^{5} + \frac{1}{12} a^{4} + \frac{17}{48} a^{3} - \frac{5}{12} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{48} a^{13} - \frac{1}{48} a^{10} + \frac{7}{48} a^{7} - \frac{1}{12} a^{6} + \frac{1}{12} a^{5} + \frac{1}{48} a^{4} + \frac{1}{12} a^{3} + \frac{1}{4} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{196043407835184} a^{14} - \frac{63179032303}{9335400373104} a^{12} - \frac{513416857261}{28006201119312} a^{10} - \frac{3886637811691}{196043407835184} a^{8} - \frac{1}{4} a^{7} + \frac{118866926413}{28006201119312} a^{6} - \frac{2035726116707}{28006201119312} a^{4} - \frac{1}{2} a^{3} + \frac{3637493469745}{10891300435288} a^{2} + \frac{1}{4} a + \frac{36746083471}{269290395378}$, $\frac{1}{35679900226003488} a^{15} - \frac{1}{392086815670368} a^{14} + \frac{10050171371893}{1699042867904928} a^{13} + \frac{63179032303}{18670800746208} a^{12} - \frac{3574120405237}{728161229102112} a^{11} - \frac{653508189377}{56012402238624} a^{10} - \frac{690038565234835}{35679900226003488} a^{9} - \frac{4281837514775}{392086815670368} a^{8} + \frac{1164710063471137}{5097128603714784} a^{7} - \frac{9454267299517}{56012402238624} a^{6} + \frac{451898117025475}{5097128603714784} a^{5} + \frac{7870351349897}{56012402238624} a^{4} + \frac{2603041984007779}{5946650037667248} a^{3} - \frac{1026354409295}{16336950652932} a^{2} + \frac{3447757758259}{49010851958796} a + \frac{48949557109}{269290395378}$
Class group and class number
$C_{2}\times C_{20}$, which has order $40$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{375}{50940464} a^{14} + \frac{26659}{152821392} a^{12} - \frac{286663}{152821392} a^{10} - \frac{104881}{50940464} a^{8} + \frac{5980655}{50940464} a^{6} - \frac{46178833}{152821392} a^{4} - \frac{17265157}{19102674} a^{2} + \frac{6705137}{3183779} \) (order $6$) | 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: | \( 593389.965104 \) (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 | ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ | R | R | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $13$ | 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |