Normalized defining polynomial
\( x^{16} + 6x^{12} + 39x^{8} + 30x^{4} + 25 \)
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: | \(671088640000000000\) \(\medspace = 2^{36}\cdot 5^{10}\) | 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: | \(13.01\) | 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: | $2^{9/4}5^{3/4}\approx 15.905414575341013$ | ||
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{820}a^{12}-\frac{1}{4}a^{10}-\frac{26}{205}a^{8}-\frac{1}{4}a^{6}+\frac{51}{205}a^{4}-\frac{1}{4}a^{2}-\frac{13}{164}$, $\frac{1}{820}a^{13}-\frac{1}{4}a^{11}-\frac{26}{205}a^{9}-\frac{1}{4}a^{7}+\frac{51}{205}a^{5}-\frac{1}{4}a^{3}-\frac{13}{164}a$, $\frac{1}{4100}a^{14}+\frac{921}{4100}a^{10}-\frac{1}{4}a^{8}-\frac{821}{4100}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{7}{205}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4100}a^{15}+\frac{921}{4100}a^{11}-\frac{1}{4}a^{9}-\frac{821}{4100}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{191}{410}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{3}{82} a^{12} - \frac{8}{41} a^{8} - \frac{60}{41} a^{4} - \frac{51}{82} \) (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: | $\frac{33}{1025}a^{14}+\frac{311}{2050}a^{10}+\frac{2189}{2050}a^{6}+\frac{3}{410}a^{2}$, $\frac{103}{4100}a^{14}+\frac{8}{205}a^{12}+\frac{563}{4100}a^{10}+\frac{157}{820}a^{8}+\frac{3587}{4100}a^{6}+\frac{993}{820}a^{4}+\frac{106}{205}a^{2}-\frac{47}{164}$, $\frac{57}{4100}a^{14}+\frac{17}{820}a^{12}+\frac{111}{2050}a^{10}+\frac{77}{820}a^{8}+\frac{689}{2050}a^{6}+\frac{393}{820}a^{4}-\frac{659}{820}a^{2}-\frac{4}{41}$, $\frac{23}{2050}a^{15}+\frac{33}{2050}a^{14}-\frac{19}{820}a^{13}+\frac{3}{164}a^{12}+\frac{341}{4100}a^{11}+\frac{311}{4100}a^{10}-\frac{37}{410}a^{9}+\frac{4}{41}a^{8}+\frac{2209}{4100}a^{7}+\frac{2189}{4100}a^{6}-\frac{149}{205}a^{5}+\frac{30}{41}a^{4}+\frac{673}{820}a^{3}-\frac{407}{820}a^{2}+\frac{83}{164}a+\frac{51}{164}$, $\frac{28}{1025}a^{15}-\frac{47}{2050}a^{14}-\frac{1}{205}a^{12}+\frac{163}{1025}a^{11}-\frac{237}{2050}a^{10}+\frac{3}{410}a^{8}+\frac{2199}{2050}a^{7}-\frac{694}{1025}a^{6}+\frac{1}{205}a^{4}+\frac{169}{205}a^{3}+\frac{119}{410}a^{2}-\frac{1}{2}a+\frac{67}{82}$, $\frac{103}{4100}a^{15}+\frac{33}{2050}a^{14}-\frac{7}{205}a^{13}-\frac{3}{164}a^{12}+\frac{563}{4100}a^{11}+\frac{311}{4100}a^{10}-\frac{163}{820}a^{9}-\frac{4}{41}a^{8}+\frac{3587}{4100}a^{7}+\frac{2189}{4100}a^{6}-\frac{997}{820}a^{5}-\frac{30}{41}a^{4}+\frac{7}{410}a^{3}-\frac{407}{820}a^{2}-\frac{169}{164}a-\frac{51}{164}$, $\frac{33}{2050}a^{15}-\frac{47}{2050}a^{14}+\frac{3}{164}a^{13}+\frac{1}{205}a^{12}+\frac{311}{4100}a^{11}-\frac{237}{2050}a^{10}+\frac{4}{41}a^{9}-\frac{3}{410}a^{8}+\frac{2189}{4100}a^{7}-\frac{694}{1025}a^{6}+\frac{30}{41}a^{5}-\frac{1}{205}a^{4}-\frac{407}{820}a^{3}+\frac{119}{410}a^{2}+\frac{51}{164}a-\frac{67}{82}$ | 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: | \( 342.10379149 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 342.10379149 \cdot 1}{4\cdot\sqrt{671088640000000000}}\cr\approx \mathstrut & 0.25359862783 \end{aligned}\]
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
\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), 4.2.400.1 x2, 4.0.320.1 x2, \(\Q(i, \sqrt{5})\), 8.0.32768000.1, 8.0.819200000.1, 8.0.2560000.1 |
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: | 8.0.819200000.1, 8.0.32768000.1 |
Degree 16 sibling: | 16.4.16777216000000000000.2 |
Minimal sibling: | 8.0.32768000.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $8$ | $2$ | $36$ | |||
\(5\) | 5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |