Normalized defining polynomial
\( x^{16} - 10x^{12} + 95x^{8} - 250x^{4} + 625 \)
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: | \(419430400000000000000\) \(\medspace = 2^{36}\cdot 5^{14}\) | 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: | \(19.45\) | 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: | $2^{9/4}5^{7/8}\approx 19.449849449321462$ | ||
Ramified primes: | \(2\), \(5\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{10}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{10}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{10}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{50}a^{11}-\frac{1}{5}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{2}{5}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2100}a^{12}-\frac{1}{20}a^{10}-\frac{1}{42}a^{8}-\frac{1}{4}a^{6}-\frac{53}{210}a^{4}-\frac{1}{4}a^{2}-\frac{23}{84}$, $\frac{1}{2100}a^{13}-\frac{1}{100}a^{11}-\frac{1}{42}a^{9}-\frac{3}{20}a^{7}+\frac{26}{105}a^{5}-\frac{1}{2}a^{4}+\frac{1}{20}a^{3}-\frac{1}{2}a^{2}-\frac{23}{84}a-\frac{1}{2}$, $\frac{1}{10500}a^{14}+\frac{53}{2100}a^{10}-\frac{1}{20}a^{8}+\frac{419}{2100}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{2}{21}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{10500}a^{15}+\frac{11}{2100}a^{11}-\frac{1}{20}a^{9}-\frac{211}{2100}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{32}{105}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{1}{210} a^{12} + \frac{4}{105} a^{8} - \frac{10}{21} a^{4} + \frac{31}{42} \) (order $4$) | 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: | $\frac{4}{525}a^{12}-\frac{17}{210}a^{8}+\frac{97}{210}a^{4}-\frac{37}{42}$, $\frac{4}{875}a^{14}-\frac{1}{420}a^{12}-\frac{27}{700}a^{10}+\frac{2}{105}a^{8}+\frac{229}{700}a^{6}-\frac{5}{21}a^{4}-\frac{19}{28}a^{2}-\frac{11}{84}$, $\frac{1}{750}a^{14}-\frac{1}{420}a^{12}+\frac{1}{300}a^{10}+\frac{2}{105}a^{8}+\frac{13}{300}a^{6}-\frac{5}{21}a^{4}+\frac{7}{12}a^{2}-\frac{11}{84}$, $\frac{1}{2100}a^{15}-\frac{17}{10500}a^{14}-\frac{13}{2100}a^{13}-\frac{17}{2100}a^{12}-\frac{2}{525}a^{11}+\frac{11}{525}a^{10}+\frac{5}{84}a^{9}+\frac{23}{420}a^{8}+\frac{1}{21}a^{7}-\frac{149}{1050}a^{6}-\frac{197}{420}a^{5}-\frac{193}{420}a^{4}+\frac{53}{420}a^{3}+\frac{53}{84}a^{2}+\frac{17}{21}a+\frac{17}{42}$, $\frac{1}{1050}a^{15}-\frac{17}{10500}a^{14}-\frac{1}{525}a^{13}+\frac{17}{2100}a^{12}-\frac{4}{525}a^{11}+\frac{11}{525}a^{10}-\frac{1}{210}a^{9}-\frac{23}{420}a^{8}+\frac{2}{21}a^{7}-\frac{149}{1050}a^{6}+\frac{1}{105}a^{5}+\frac{193}{420}a^{4}-\frac{26}{105}a^{3}+\frac{53}{84}a^{2}-\frac{19}{21}a-\frac{17}{42}$, $\frac{1}{2625}a^{15}-\frac{1}{750}a^{14}+\frac{1}{700}a^{13}-\frac{19}{2100}a^{12}+\frac{23}{2100}a^{11}-\frac{1}{300}a^{10}+\frac{1}{35}a^{9}+\frac{11}{210}a^{8}-\frac{109}{2100}a^{7}-\frac{13}{300}a^{6}-\frac{9}{35}a^{5}-\frac{43}{210}a^{4}+\frac{139}{420}a^{3}+\frac{5}{12}a^{2}+\frac{19}{28}a+\frac{17}{84}$, $\frac{1}{2625}a^{15}-\frac{17}{10500}a^{14}-\frac{1}{2100}a^{13}+\frac{1}{84}a^{12}+\frac{23}{2100}a^{11}+\frac{11}{525}a^{10}+\frac{1}{42}a^{9}-\frac{19}{420}a^{8}-\frac{109}{2100}a^{7}-\frac{149}{1050}a^{6}-\frac{26}{105}a^{5}+\frac{37}{84}a^{4}+\frac{349}{420}a^{3}+\frac{53}{84}a^{2}+\frac{149}{84}a+\frac{59}{42}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 8166.23968032 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 8166.23968032 \cdot 2}{4\cdot\sqrt{419430400000000000000}}\cr\approx \mathstrut & 0.484285115571 \end{aligned}\]
Galois group
$C_4\wr C_2$ (as 16T42):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_4\wr C_2$ |
Character table for $C_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), 4.0.8000.1 x2, 4.2.2000.1 x2, \(\Q(i, \sqrt{5})\), 8.0.20480000000.2, 8.0.20480000000.3, 8.0.64000000.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 8 siblings: | 8.0.20480000000.2, 8.0.20480000000.3 |
Degree 16 sibling: | 16.4.419430400000000000000.4 |
Minimal sibling: | 8.0.20480000000.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $8$ | $2$ | $36$ | |||
\(5\) | 5.8.7.1 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
5.8.7.1 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |