Normalized defining polynomial
\( x^{16} - 5x^{14} + 8x^{12} + 4x^{10} - 17x^{8} - 4x^{6} + 32x^{4} - x^{2} + 1 \)
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: | \(217678233600000000\) \(\medspace = 2^{20}\cdot 3^{12}\cdot 5^{8}\) | 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: | \(12.12\) | 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: | $2^{3/2}3^{3/4}5^{1/2}\approx 14.416868484808525$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{572}a^{14}-\frac{1}{4}a^{13}+\frac{11}{52}a^{12}+\frac{239}{572}a^{10}+\frac{1}{4}a^{9}-\frac{9}{26}a^{8}+\frac{1}{4}a^{7}-\frac{113}{286}a^{6}+\frac{1}{4}a^{5}+\frac{263}{572}a^{4}+\frac{137}{572}a^{2}-\frac{1}{4}a+\frac{101}{572}$, $\frac{1}{572}a^{15}-\frac{1}{26}a^{13}-\frac{1}{4}a^{12}+\frac{239}{572}a^{11}-\frac{5}{52}a^{9}+\frac{1}{4}a^{8}-\frac{83}{572}a^{7}+\frac{1}{4}a^{6}-\frac{83}{286}a^{5}+\frac{1}{4}a^{4}+\frac{137}{572}a^{3}-\frac{21}{286}a-\frac{1}{4}$
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: | \( -\frac{2}{11} a^{14} + a^{12} - \frac{16}{11} a^{10} - a^{8} + \frac{34}{11} a^{6} + \frac{24}{11} a^{4} - \frac{54}{11} a^{2} + \frac{7}{11} \) (order $6$) | 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: | $\frac{9}{286}a^{15}-\frac{5}{26}a^{13}+\frac{149}{286}a^{11}-\frac{3}{13}a^{9}-\frac{159}{143}a^{7}+\frac{365}{286}a^{5}+\frac{661}{286}a^{3}-\frac{521}{286}a$, $\frac{15}{286}a^{14}-\frac{2}{13}a^{12}-\frac{133}{286}a^{10}+\frac{29}{26}a^{8}+\frac{185}{286}a^{6}-\frac{387}{143}a^{4}-\frac{805}{286}a^{2}-\frac{29}{143}$, $\frac{3}{286}a^{14}-\frac{3}{13}a^{12}+\frac{145}{286}a^{10}+\frac{11}{26}a^{8}-\frac{535}{286}a^{6}-\frac{106}{143}a^{4}+\frac{983}{286}a^{2}+\frac{80}{143}$, $\frac{137}{572}a^{15}-\frac{61}{286}a^{14}-\frac{33}{26}a^{13}+\frac{49}{52}a^{12}+\frac{1283}{572}a^{11}-\frac{279}{286}a^{10}+\frac{43}{52}a^{9}-\frac{79}{52}a^{8}-\frac{2791}{572}a^{7}+\frac{1403}{572}a^{6}-\frac{217}{286}a^{5}+\frac{1233}{572}a^{4}+\frac{5613}{572}a^{3}-\frac{921}{286}a^{2}-\frac{17}{286}a+\frac{119}{572}$, $\frac{15}{52}a^{15}-\frac{2}{13}a^{14}-\frac{35}{26}a^{13}+\frac{33}{52}a^{12}+\frac{101}{52}a^{11}-\frac{10}{13}a^{10}+\frac{59}{52}a^{9}-\frac{41}{52}a^{8}-\frac{205}{52}a^{7}+\frac{79}{52}a^{6}-\frac{23}{26}a^{5}+\frac{67}{52}a^{4}+\frac{339}{52}a^{3}-\frac{40}{13}a^{2}-\frac{55}{26}a-\frac{15}{52}$, $\frac{49}{572}a^{15}-\frac{23}{143}a^{14}-\frac{5}{13}a^{13}+\frac{41}{52}a^{12}+\frac{271}{572}a^{11}-\frac{206}{143}a^{10}+\frac{41}{52}a^{9}-\frac{21}{52}a^{8}-\frac{921}{572}a^{7}+\frac{1773}{572}a^{6}-\frac{103}{143}a^{5}-\frac{315}{572}a^{4}+\frac{1565}{572}a^{3}-\frac{863}{143}a^{2}+\frac{129}{143}a+\frac{3}{572}$, $\frac{17}{286}a^{15}+\frac{131}{572}a^{14}-\frac{29}{52}a^{13}-\frac{41}{52}a^{12}+\frac{345}{286}a^{11}+\frac{421}{572}a^{10}-\frac{1}{52}a^{9}+\frac{15}{13}a^{8}-\frac{1535}{572}a^{7}-\frac{180}{143}a^{6}+\frac{219}{572}a^{5}-\frac{725}{572}a^{4}+\frac{899}{286}a^{3}+\frac{1931}{572}a^{2}-\frac{999}{572}a+\frac{361}{572}$ | 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: | \( 302.605971735 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 302.605971735 \cdot 1}{6\cdot\sqrt{217678233600000000}}\cr\approx \mathstrut & 0.262577608455 \end{aligned}\]
Galois group
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_4:D_4$ |
Character table for $C_4:D_4$ |
Intermediate fields
\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), 4.2.400.1, 4.2.3600.1, 4.0.10800.2, \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.432.1, 8.0.116640000.1, 8.0.7290000.1, 8.0.12960000.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 16 siblings: | 16.0.3482851737600000000.1, 16.0.5572562780160000.1, 16.4.3482851737600000000.1 |
Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.15 | $x^{8} + 4 x^{7} + 4 x^{6} + 12$ | $4$ | $2$ | $12$ | $C_2^2:C_4$ | $[2, 2]^{4}$ |
2.8.8.2 | $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ | |
\(3\) | 3.16.12.1 | $x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 36 x^{12} + 120 x^{11} + 312 x^{10} + 352 x^{9} - 522 x^{8} - 2664 x^{7} - 3672 x^{6} - 1440 x^{5} + 4292 x^{4} + 7720 x^{3} + 6408 x^{2} + 2592 x + 433$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $[\ ]_{4}^{4}$ |
\(5\) | 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |