Normalized defining polynomial
\( x^{16} + 520 x^{14} + 109460 x^{12} + 11965200 x^{10} + 725842325 x^{8} + 24327381000 x^{6} + 426030156500 x^{4} + 3372768490000 x^{2} + 8400432722500 \)
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: | \(10504372386022553663859326976000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 13^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $317.20$ | 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: | $2, 3, 5, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3120=2^{4}\cdot 3\cdot 5\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3120}(1,·)$, $\chi_{3120}(649,·)$, $\chi_{3120}(1867,·)$, $\chi_{3120}(1871,·)$, $\chi_{3120}(1877,·)$, $\chi_{3120}(599,·)$, $\chi_{3120}(1561,·)$, $\chi_{3120}(1373,·)$, $\chi_{3120}(2209,·)$, $\chi_{3120}(1123,·)$, $\chi_{3120}(2159,·)$, $\chi_{3120}(307,·)$, $\chi_{3120}(2933,·)$, $\chi_{3120}(311,·)$, $\chi_{3120}(2683,·)$, $\chi_{3120}(317,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{65} a^{4}$, $\frac{1}{65} a^{5}$, $\frac{1}{65} a^{6}$, $\frac{1}{455} a^{7} + \frac{3}{7} a$, $\frac{1}{29575} a^{8} - \frac{2}{7} a^{2}$, $\frac{1}{29575} a^{9} - \frac{2}{7} a^{3}$, $\frac{1}{1035125} a^{10} + \frac{1}{207025} a^{8} + \frac{1}{2275} a^{6} - \frac{12}{3185} a^{4} + \frac{4}{245} a^{2}$, $\frac{1}{1035125} a^{11} + \frac{1}{207025} a^{9} + \frac{1}{2275} a^{7} - \frac{12}{3185} a^{5} + \frac{4}{245} a^{3}$, $\frac{1}{645918000} a^{12} + \frac{1}{2484300} a^{10} + \frac{3}{1656200} a^{8} - \frac{8}{9555} a^{6} - \frac{101}{50960} a^{4} + \frac{61}{588} a^{2} + \frac{1}{24}$, $\frac{1}{63299964000} a^{13} + \frac{209}{1217307000} a^{11} + \frac{2547}{162307600} a^{9} - \frac{1679}{2340975} a^{7} - \frac{293}{76832} a^{5} - \frac{111439}{288120} a^{3} + \frac{361}{2352} a$, $\frac{1}{13964288558220000} a^{14} + \frac{6337}{9309525705480} a^{12} + \frac{3008009}{107417604294000} a^{10} + \frac{37504699}{2685440107350} a^{8} - \frac{15879558479}{3305157055200} a^{6} - \frac{921330413}{165257852760} a^{4} + \frac{56192051}{172954320} a^{2} + \frac{10045}{37818}$, $\frac{1}{13964288558220000} a^{15} + \frac{433}{310317523516000} a^{13} + \frac{40157891}{107417604294000} a^{11} - \frac{290963203}{21483520858800} a^{9} + \frac{3242835889}{3305157055200} a^{7} + \frac{263793997}{50848570080} a^{5} - \frac{13515445}{242136048} a^{3} + \frac{1397531}{14824656} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{8}\times C_{24}\times C_{83640}$, which has order $128471040$ (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: | \( 5519632.427467192 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2\times C_4$ (as 16T2):
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_4\times C_2^2$ |
| Character table for $C_4\times C_2^2$ |
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 | R | R | ${\href{/LocalNumberField/7.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.24.5 | $x^{8} - 15$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $[2, 3, 4]$ |
| 2.8.24.5 | $x^{8} - 15$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $[2, 3, 4]$ | |
| $3$ | 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 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $13$ | 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |