Normalized defining polynomial
\( x^{16} - 5x^{14} - 16x^{12} + 25x^{10} + 71x^{8} + 25x^{6} - 16x^{4} - 5x^{2} + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1949444636015869140625\) \(\medspace = 5^{12}\cdot 41^{8}\) | 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: | \(21.41\) | 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: | $5^{3/4}41^{1/2}\approx 21.410136276713814$ | ||
Ramified primes: | \(5\), \(41\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{22}a^{12}-\frac{2}{11}a^{10}+\frac{1}{22}a^{8}+\frac{4}{11}a^{6}-\frac{1}{2}a^{5}+\frac{1}{22}a^{4}-\frac{1}{2}a^{3}+\frac{7}{22}a^{2}-\frac{1}{2}a+\frac{1}{22}$, $\frac{1}{22}a^{13}-\frac{2}{11}a^{11}+\frac{1}{22}a^{9}+\frac{4}{11}a^{7}-\frac{1}{2}a^{6}+\frac{1}{22}a^{5}-\frac{1}{2}a^{4}+\frac{7}{22}a^{3}-\frac{1}{2}a^{2}+\frac{1}{22}a$, $\frac{1}{198}a^{14}-\frac{2}{99}a^{12}+\frac{17}{99}a^{10}+\frac{41}{198}a^{8}+\frac{67}{198}a^{6}-\frac{1}{2}a^{5}-\frac{35}{99}a^{4}+\frac{67}{198}a^{2}-\frac{1}{2}a+\frac{2}{9}$, $\frac{1}{198}a^{15}-\frac{2}{99}a^{13}+\frac{17}{99}a^{11}+\frac{41}{198}a^{9}+\frac{67}{198}a^{7}-\frac{1}{2}a^{6}-\frac{35}{99}a^{5}+\frac{67}{198}a^{3}-\frac{1}{2}a^{2}+\frac{2}{9}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | 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: | $\frac{28}{99}a^{14}-\frac{148}{99}a^{12}-\frac{389}{99}a^{10}+\frac{716}{99}a^{8}+\frac{1588}{99}a^{6}+\frac{776}{99}a^{4}+\frac{40}{99}a^{2}-\frac{91}{99}$, $a$, $\frac{127}{198}a^{15}-\frac{19}{66}a^{14}-\frac{643}{198}a^{13}+\frac{59}{33}a^{12}-\frac{1973}{198}a^{11}+\frac{8}{3}a^{10}+\frac{3191}{198}a^{9}-\frac{35}{3}a^{8}+\frac{8617}{198}a^{7}-\frac{607}{66}a^{6}+\frac{3251}{198}a^{5}+\frac{356}{33}a^{4}-\frac{772}{99}a^{3}+\frac{8}{3}a^{2}-\frac{392}{99}a-\frac{67}{33}$, $\frac{119}{198}a^{15}+\frac{71}{198}a^{14}-\frac{575}{198}a^{13}-\frac{223}{99}a^{12}-\frac{1993}{198}a^{11}-\frac{601}{198}a^{10}+\frac{1301}{99}a^{9}+\frac{2749}{198}a^{8}+\frac{8765}{198}a^{7}+\frac{988}{99}a^{6}+\frac{4639}{198}a^{5}-\frac{1865}{198}a^{4}-\frac{320}{99}a^{3}-\frac{535}{198}a^{2}-\frac{55}{18}a+\frac{289}{198}$, $\frac{119}{198}a^{15}-\frac{1}{9}a^{14}-\frac{575}{198}a^{13}+\frac{62}{99}a^{12}-\frac{1993}{198}a^{11}+\frac{148}{99}a^{10}+\frac{1301}{99}a^{9}-\frac{433}{99}a^{8}+\frac{8765}{198}a^{7}-\frac{1285}{198}a^{6}+\frac{4639}{198}a^{5}+\frac{883}{198}a^{4}-\frac{320}{99}a^{3}+\frac{478}{99}a^{2}-\frac{23}{9}a-\frac{70}{99}$, $\frac{119}{198}a^{15}-\frac{13}{33}a^{14}-\frac{575}{198}a^{13}+\frac{70}{33}a^{12}-\frac{1993}{198}a^{11}+\frac{179}{33}a^{10}+\frac{1301}{99}a^{9}-\frac{383}{33}a^{8}+\frac{8765}{198}a^{7}-\frac{1487}{66}a^{6}+\frac{4639}{198}a^{5}-\frac{223}{66}a^{4}-\frac{320}{99}a^{3}+\frac{146}{33}a^{2}-\frac{23}{9}a+\frac{40}{33}$, $\frac{47}{33}a^{15}+\frac{14}{99}a^{14}-\frac{269}{33}a^{13}-\frac{74}{99}a^{12}-\frac{553}{33}a^{11}-\frac{389}{198}a^{10}+\frac{1549}{33}a^{9}+\frac{358}{99}a^{8}+\frac{2171}{33}a^{7}+\frac{794}{99}a^{6}-\frac{505}{66}a^{5}+\frac{388}{99}a^{4}-\frac{388}{33}a^{3}+\frac{20}{99}a^{2}+\frac{7}{33}a-\frac{91}{198}$, $\frac{49}{99}a^{15}-\frac{259}{99}a^{13}-\frac{1411}{198}a^{11}+\frac{2803}{198}a^{9}+\frac{6053}{198}a^{7}+\frac{467}{99}a^{5}-\frac{1345}{198}a^{3}-\frac{1}{2}a^{2}+\frac{127}{198}a+\frac{1}{2}$, $\frac{1}{2}a^{15}+\frac{49}{99}a^{14}-\frac{5}{2}a^{13}-\frac{259}{99}a^{12}-8a^{11}-\frac{1411}{198}a^{10}+\frac{25}{2}a^{9}+\frac{2803}{198}a^{8}+\frac{71}{2}a^{7}+\frac{6053}{198}a^{6}+\frac{25}{2}a^{5}+\frac{467}{99}a^{4}-8a^{3}-\frac{1345}{198}a^{2}-2a-\frac{71}{198}$, $\frac{155}{198}a^{15}-\frac{29}{99}a^{14}-\frac{881}{198}a^{13}+\frac{331}{198}a^{12}-\frac{173}{18}a^{11}+\frac{701}{198}a^{10}+\frac{241}{9}a^{9}-\frac{991}{99}a^{8}+\frac{3901}{99}a^{7}-\frac{2797}{198}a^{6}-\frac{1051}{99}a^{5}+\frac{595}{198}a^{4}-\frac{133}{9}a^{3}+\frac{767}{198}a^{2}+\frac{260}{99}a-\frac{7}{18}$, $\frac{152}{99}a^{15}-\frac{85}{198}a^{14}-\frac{79}{9}a^{13}+\frac{511}{198}a^{12}-\frac{1807}{99}a^{11}+\frac{881}{198}a^{10}+\frac{5080}{99}a^{9}-\frac{3215}{198}a^{8}+\frac{1301}{18}a^{7}-\frac{3337}{198}a^{6}-\frac{109}{9}a^{5}+\frac{1130}{99}a^{4}-\frac{1642}{99}a^{3}+\frac{721}{99}a^{2}+\frac{281}{198}a-\frac{203}{198}$ (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)]
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Regulator: | \( 62312.6670751 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 62312.6670751 \cdot 1}{2\cdot\sqrt{1949444636015869140625}}\cr\approx \mathstrut & 0.281546627998 \end{aligned}\] (assuming GRH)
Galois group
$C_2^3:C_4$ (as 16T33):
A solvable group of order 32 |
The 11 conjugacy class representatives for $C_2^3:C_4$ |
Character table for $C_2^3:C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{205}) \), 4.4.210125.1 x2, \(\Q(\sqrt{5}, \sqrt{41})\), 4.4.5125.1 x2, 8.4.44152515625.1 x2, 8.4.8830503125.1 x2, 8.8.44152515625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 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 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(41\) | 41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |