Normalized defining polynomial
\( x^{16} - 15x^{12} + 200x^{8} - 375x^{4} + 625 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(6879707136000000000000\) \(\medspace = 2^{32}\cdot 3^{8}\cdot 5^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(23.17\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | $2^{2}3^{1/2}5^{3/4}\approx 23.165843705765383$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(120=2^{3}\cdot 3\cdot 5\) | ||
Dirichlet character group: | $\lbrace$$\chi_{120}(1,·)$, $\chi_{120}(67,·)$, $\chi_{120}(71,·)$, $\chi_{120}(13,·)$, $\chi_{120}(77,·)$, $\chi_{120}(43,·)$, $\chi_{120}(83,·)$, $\chi_{120}(89,·)$, $\chi_{120}(79,·)$, $\chi_{120}(31,·)$, $\chi_{120}(37,·)$, $\chi_{120}(41,·)$, $\chi_{120}(107,·)$, $\chi_{120}(49,·)$, $\chi_{120}(53,·)$, $\chi_{120}(119,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{25}a^{9}$, $\frac{1}{125}a^{10}+\frac{1}{25}a^{6}+\frac{1}{5}a^{2}$, $\frac{1}{125}a^{11}+\frac{1}{25}a^{7}+\frac{1}{5}a^{3}$, $\frac{1}{1000}a^{12}-\frac{3}{8}$, $\frac{1}{1000}a^{13}-\frac{3}{8}a$, $\frac{1}{5000}a^{14}-\frac{11}{40}a^{2}$, $\frac{1}{5000}a^{15}-\frac{11}{40}a^{3}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{11}{5000} a^{14} - \frac{4}{125} a^{10} + \frac{11}{25} a^{6} - \frac{33}{40} a^{2} \) (order $12$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{3}{1000}a^{12}-\frac{1}{25}a^{8}+\frac{2}{5}a^{4}-\frac{1}{8}$, $\frac{3}{1250}a^{14}-\frac{3}{1000}a^{12}-\frac{4}{125}a^{10}+\frac{1}{25}a^{8}+\frac{11}{25}a^{6}-\frac{3}{5}a^{4}-\frac{1}{10}a^{2}+\frac{9}{8}$, $\frac{1}{250}a^{14}+\frac{1}{500}a^{12}-\frac{7}{125}a^{10}-\frac{1}{25}a^{8}+\frac{18}{25}a^{6}+\frac{3}{5}a^{4}-\frac{9}{10}a^{2}-\frac{7}{4}$, $\frac{9}{5000}a^{14}-\frac{3}{1000}a^{13}-\frac{1}{1000}a^{12}-\frac{3}{125}a^{10}+\frac{1}{25}a^{9}+\frac{7}{25}a^{6}-\frac{3}{5}a^{5}-\frac{3}{40}a^{2}+\frac{9}{8}a-\frac{5}{8}$, $\frac{3}{1250}a^{14}+\frac{3}{1000}a^{13}-\frac{4}{125}a^{10}-\frac{1}{25}a^{9}+\frac{11}{25}a^{6}+\frac{3}{5}a^{5}-\frac{1}{10}a^{2}-\frac{9}{8}a-1$, $\frac{3}{5000}a^{15}+\frac{11}{5000}a^{14}-\frac{3}{1000}a^{12}-\frac{4}{125}a^{10}+\frac{1}{25}a^{8}+\frac{11}{25}a^{6}-\frac{3}{5}a^{4}+\frac{47}{40}a^{3}-\frac{33}{40}a^{2}+\frac{9}{8}$, $\frac{1}{1000}a^{15}+\frac{1}{2500}a^{14}+\frac{1}{500}a^{13}+\frac{3}{1000}a^{12}-\frac{1}{125}a^{11}-\frac{1}{125}a^{10}-\frac{1}{25}a^{9}-\frac{1}{25}a^{8}+\frac{4}{25}a^{7}+\frac{4}{25}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{17}{40}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{8}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 27342.8963735 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 27342.8963735 \cdot 4}{12\cdot\sqrt{6879707136000000000000}}\cr\approx \mathstrut & 0.266917613487 \end{aligned}\] (assuming GRH)
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 | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.2 | $x^{8} + 4 x^{6} + 8 x^{5} + 16 x^{4} + 16 x^{3} + 40 x^{2} + 48 x + 84$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ |
2.8.16.2 | $x^{8} + 4 x^{6} + 8 x^{5} + 16 x^{4} + 16 x^{3} + 40 x^{2} + 48 x + 84$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ | |
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |