Normalized defining polynomial
\( x^{20} - 2x^{18} + x^{16} + 2x^{14} - 4x^{10} - x^{8} + 6x^{6} - x^{4} - 2x^{2} + 1 \)
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: | \(684008168396450430976\) \(\medspace = 2^{30}\cdot 798143^{2}\) | 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: | \(11.01\) | 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: | not computed | ||
Ramified primes: | \(2\), \(798143\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{1}{2} a^{19} - \frac{3}{2} a^{15} + \frac{5}{2} a^{13} + a^{11} - a^{9} - \frac{9}{2} a^{7} + 3 a^{5} + 4 a^{3} - 2 a \) (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: | $a$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}+\frac{3}{2}a^{13}+a^{11}-2a^{9}-2a^{7}+2a^{5}+\frac{5}{2}a^{3}-a$, $\frac{1}{2}a^{18}+\frac{1}{4}a^{17}-a^{16}-\frac{3}{4}a^{15}+\frac{1}{2}a^{14}+a^{13}+\frac{3}{4}a^{12}-\frac{1}{4}a^{11}+\frac{1}{2}a^{10}+\frac{1}{4}a^{9}-\frac{9}{4}a^{8}-\frac{5}{4}a^{7}-\frac{3}{4}a^{6}+a^{5}+3a^{4}+\frac{3}{2}a^{3}-\frac{3}{2}a-\frac{1}{4}$, $\frac{1}{2}a^{19}+\frac{3}{4}a^{18}-\frac{1}{2}a^{17}-\frac{3}{2}a^{16}-\frac{1}{2}a^{15}+\frac{3}{2}a^{14}+\frac{3}{2}a^{13}+\frac{3}{4}a^{11}+\frac{5}{4}a^{10}-\frac{7}{4}a^{9}-\frac{5}{2}a^{8}-\frac{5}{2}a^{7}+\frac{7}{4}a^{5}+2a^{4}+\frac{9}{4}a^{3}-\frac{5}{4}a^{2}-\frac{5}{4}a+\frac{3}{4}$, $a^{19}-\frac{3}{2}a^{17}-\frac{3}{4}a^{16}+\frac{3}{2}a^{14}+\frac{9}{4}a^{13}-\frac{5}{4}a^{12}+\frac{5}{4}a^{11}-\frac{1}{2}a^{10}-4a^{9}-\frac{3}{4}a^{8}-\frac{13}{4}a^{7}+\frac{5}{2}a^{6}+\frac{19}{4}a^{5}+\frac{1}{2}a^{4}+\frac{9}{4}a^{3}-\frac{9}{4}a^{2}-\frac{3}{2}a+\frac{3}{4}$, $\frac{5}{4}a^{19}-2a^{17}-\frac{1}{4}a^{16}+\frac{1}{4}a^{15}+\frac{3}{4}a^{14}+3a^{13}-a^{12}+\frac{3}{4}a^{11}+\frac{1}{4}a^{10}-\frac{9}{2}a^{9}-\frac{1}{4}a^{8}-\frac{7}{2}a^{7}+\frac{5}{4}a^{6}+\frac{27}{4}a^{5}-a^{4}+\frac{7}{4}a^{3}-\frac{3}{2}a^{2}-\frac{5}{2}a+\frac{3}{2}$, $\frac{1}{4}a^{19}-\frac{3}{4}a^{18}-\frac{1}{4}a^{17}+\frac{3}{2}a^{16}-\frac{1}{4}a^{15}-a^{14}+a^{13}-a^{12}+\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-a^{9}+\frac{11}{4}a^{8}-\frac{3}{4}a^{7}+\frac{1}{2}a^{6}+\frac{7}{4}a^{5}-\frac{15}{4}a^{4}+\frac{1}{2}a^{3}+a^{2}-\frac{3}{2}a+\frac{1}{2}$, $\frac{1}{4}a^{19}+\frac{3}{4}a^{18}-\frac{1}{4}a^{17}-\frac{3}{2}a^{16}-\frac{1}{4}a^{15}+a^{14}+a^{13}+a^{12}+\frac{1}{4}a^{11}+\frac{1}{2}a^{10}-a^{9}-\frac{11}{4}a^{8}-\frac{3}{4}a^{7}-\frac{1}{2}a^{6}+\frac{7}{4}a^{5}+\frac{15}{4}a^{4}+\frac{1}{2}a^{3}-a^{2}-\frac{3}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{4}a^{16}+\frac{1}{2}a^{15}+a^{14}+\frac{1}{4}a^{13}-2a^{12}-\frac{3}{4}a^{10}-\frac{1}{2}a^{9}+\frac{5}{4}a^{8}+\frac{1}{4}a^{7}+\frac{15}{4}a^{6}+\frac{3}{4}a^{5}-\frac{9}{4}a^{4}-a^{3}-3a^{2}-\frac{1}{4}a+\frac{5}{4}$ | 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: | \( 171.360926564 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 171.360926564 \cdot 1}{4\cdot\sqrt{684008168396450430976}}\cr\approx \mathstrut & 0.157079622895 \end{aligned}\]
Galois group
$C_2\times S_{10}$ (as 20T1021):
A non-solvable group of order 7257600 |
The 84 conjugacy class representatives for $C_2\times S_{10}$ |
Character table for $C_2\times S_{10}$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 10.2.817298432.1 |
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 sibling: | 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 | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 $20$ | $4$ | $5$ | $30$ | |||
\(798143\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |