Normalized defining polynomial
\( x^{20} + x^{16} + 5x^{12} + 10x^{10} + 9x^{8} + 7x^{6} + 3x^{4} + 4x^{2} + 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(750661066024355819776\) \(\medspace = 2^{8}\cdot 41381^{4}\) | 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: | \(11.06\) | 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: | $2^{3/2}41381^{1/2}\approx 575.3677085134341$ | ||
Ramified primes: | \(2\), \(41381\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{8326}a^{18}+\frac{1123}{8326}a^{16}-\frac{259}{8326}a^{14}-\frac{1805}{4163}a^{12}-\frac{1}{2}a^{11}+\frac{737}{8326}a^{10}+\frac{3387}{8326}a^{8}-\frac{686}{4163}a^{6}-\frac{1}{2}a^{5}+\frac{1862}{4163}a^{4}-\frac{1}{2}a^{3}-\frac{880}{4163}a^{2}+\frac{949}{8326}$, $\frac{1}{8326}a^{19}+\frac{1123}{8326}a^{17}-\frac{259}{8326}a^{15}-\frac{1805}{4163}a^{13}-\frac{1}{2}a^{12}+\frac{737}{8326}a^{11}+\frac{3387}{8326}a^{9}-\frac{686}{4163}a^{7}-\frac{1}{2}a^{6}+\frac{1862}{4163}a^{5}-\frac{1}{2}a^{4}-\frac{880}{4163}a^{3}+\frac{949}{8326}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $\frac{2032}{4163}a^{19}+\frac{612}{4163}a^{17}+\frac{2413}{4163}a^{15}-\frac{314}{4163}a^{13}+\frac{11393}{4163}a^{11}+\frac{21760}{4163}a^{9}+\frac{26284}{4163}a^{7}+\frac{19649}{4163}a^{5}+\frac{8023}{4163}a^{3}+\frac{9225}{4163}a$, $a$, $\frac{4545}{8326}a^{19}-\frac{611}{8326}a^{18}+\frac{197}{8326}a^{17}+\frac{371}{4163}a^{16}+\frac{5137}{8326}a^{15}+\frac{55}{8326}a^{14}-\frac{1067}{8326}a^{13}-\frac{340}{4163}a^{12}+\frac{11714}{4163}a^{11}-\frac{703}{8326}a^{10}+\frac{22467}{4163}a^{9}-\frac{4609}{8326}a^{8}+\frac{46227}{8326}a^{7}+\frac{2846}{4163}a^{6}+\frac{16063}{4163}a^{5}+\frac{1797}{8326}a^{4}+\frac{5206}{4163}a^{3}+\frac{653}{4163}a^{2}+\frac{6413}{4163}a-\frac{591}{4163}$, $\frac{4545}{8326}a^{19}+\frac{611}{8326}a^{18}+\frac{197}{8326}a^{17}-\frac{371}{4163}a^{16}+\frac{5137}{8326}a^{15}-\frac{55}{8326}a^{14}-\frac{1067}{8326}a^{13}+\frac{340}{4163}a^{12}+\frac{11714}{4163}a^{11}+\frac{703}{8326}a^{10}+\frac{22467}{4163}a^{9}+\frac{4609}{8326}a^{8}+\frac{46227}{8326}a^{7}-\frac{2846}{4163}a^{6}+\frac{16063}{4163}a^{5}-\frac{1797}{8326}a^{4}+\frac{5206}{4163}a^{3}-\frac{653}{4163}a^{2}+\frac{6413}{4163}a+\frac{591}{4163}$, $\frac{1632}{4163}a^{19}+\frac{1827}{8326}a^{18}-\frac{2131}{8326}a^{17}-\frac{319}{4163}a^{16}+\frac{1938}{4163}a^{15}+\frac{1389}{8326}a^{14}-\frac{875}{4163}a^{13}-\frac{639}{4163}a^{12}+\frac{8003}{4163}a^{11}+\frac{5088}{4163}a^{10}+\frac{11609}{4163}a^{9}+\frac{7160}{4163}a^{8}+\frac{4753}{4163}a^{7}+\frac{3904}{4163}a^{6}+\frac{11665}{8326}a^{5}+\frac{703}{4163}a^{4}+\frac{150}{4163}a^{3}-\frac{842}{4163}a^{2}+\frac{12753}{8326}a+\frac{10341}{8326}$, $\frac{1632}{4163}a^{19}-\frac{1827}{8326}a^{18}-\frac{2131}{8326}a^{17}+\frac{319}{4163}a^{16}+\frac{1938}{4163}a^{15}-\frac{1389}{8326}a^{14}-\frac{875}{4163}a^{13}+\frac{639}{4163}a^{12}+\frac{8003}{4163}a^{11}-\frac{5088}{4163}a^{10}+\frac{11609}{4163}a^{9}-\frac{7160}{4163}a^{8}+\frac{4753}{4163}a^{7}-\frac{3904}{4163}a^{6}+\frac{11665}{8326}a^{5}-\frac{703}{4163}a^{4}+\frac{150}{4163}a^{3}+\frac{842}{4163}a^{2}+\frac{12753}{8326}a-\frac{10341}{8326}$, $\frac{1590}{4163}a^{18}-\frac{357}{4163}a^{16}+\frac{327}{4163}a^{14}+\frac{877}{4163}a^{12}+\frac{6190}{4163}a^{10}+\frac{15060}{4163}a^{8}+\frac{4095}{4163}a^{6}+\frac{1374}{4163}a^{4}-\frac{864}{4163}a^{2}+\frac{1904}{4163}$, $\frac{104}{181}a^{19}-\frac{386}{4163}a^{18}-\frac{87}{362}a^{17}-\frac{526}{4163}a^{16}+\frac{247}{362}a^{15}+\frac{62}{4163}a^{14}-\frac{46}{181}a^{13}-\frac{1145}{4163}a^{12}+\frac{1075}{362}a^{11}-\frac{1398}{4163}a^{10}+\frac{1673}{362}a^{9}-\frac{12889}{8326}a^{8}+\frac{1147}{362}a^{7}-\frac{7435}{4163}a^{6}+\frac{499}{181}a^{5}-\frac{5392}{4163}a^{4}+\frac{445}{362}a^{3}-\frac{10907}{8326}a^{2}+\frac{645}{362}a-\frac{4103}{8326}$, $\frac{2018}{4163}a^{19}+\frac{1066}{4163}a^{18}-\frac{1079}{8326}a^{17}+\frac{511}{8326}a^{16}+\frac{1876}{4163}a^{15}+\frac{1491}{8326}a^{14}+\frac{270}{4163}a^{13}+\frac{867}{8326}a^{12}+\frac{9401}{4163}a^{11}+\frac{10159}{8326}a^{10}+\frac{36107}{8326}a^{9}+\frac{23257}{8326}a^{8}+\frac{12188}{4163}a^{7}+\frac{11150}{4163}a^{6}+\frac{22449}{8326}a^{5}+\frac{17379}{8326}a^{4}+\frac{11207}{8326}a^{3}+\frac{6869}{8326}a^{2}+\frac{8428}{4163}a+\frac{4213}{8326}$ | 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: | \( 91.2702526158 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 91.2702526158 \cdot 1}{2\cdot\sqrt{750661066024355819776}}\cr\approx \mathstrut & 0.159726139411 \end{aligned}\]
Galois group
$C_2\times C_4^4:S_5$ (as 20T673):
A non-solvable group of order 61440 |
The 126 conjugacy class representatives for $C_2\times C_4^4:S_5$ are not computed |
Character table for $C_2\times C_4^4:S_5$ is not computed |
Intermediate fields
5.1.41381.1, 10.2.1712387161.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 20 sibling: | data not computed |
Degree 40 siblings: | data not computed |
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 | $16{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.8.0.1}{8} }$ | $16{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.8.0.1}{8} }$ | $16{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.5 | $x^{8} + 8 x^{7} + 56 x^{6} + 184 x^{5} + 576 x^{4} + 960 x^{3} + 1632 x^{2} + 1120 x + 304$ | $2$ | $4$ | $8$ | $C_8:C_2$ | $[2, 2]^{4}$ |
2.12.0.1 | $x^{12} + x^{7} + x^{6} + x^{5} + x^{3} + x + 1$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(41381\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |