Normalized defining polynomial
\( x^{20} + 4x^{16} - 4x^{12} + 4x^{8} - 8x^{4} + 4 \)
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: | \(1618299543010938978304\) \(\medspace = 2^{38}\cdot 277^{4}\) | 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: | \(11.49\) | 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: | \(2\), \(277\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{20}a^{16}+\frac{1}{10}a^{12}-\frac{2}{5}a^{8}-\frac{1}{2}a^{6}-\frac{2}{5}$, $\frac{1}{20}a^{17}+\frac{1}{10}a^{13}-\frac{2}{5}a^{9}-\frac{1}{2}a^{7}-\frac{2}{5}a$, $\frac{1}{20}a^{18}+\frac{1}{10}a^{14}+\frac{1}{10}a^{10}-\frac{1}{2}a^{8}-\frac{2}{5}a^{2}$, $\frac{1}{20}a^{19}+\frac{1}{10}a^{15}+\frac{1}{10}a^{11}-\frac{1}{2}a^{9}-\frac{2}{5}a^{3}$
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: | \( -\frac{13}{10} a^{18} - \frac{61}{10} a^{14} + \frac{9}{10} a^{10} - 5 a^{6} + \frac{32}{5} a^{2} \) (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: | $\frac{3}{5}a^{19}-\frac{2}{5}a^{17}+\frac{59}{20}a^{15}-\frac{9}{5}a^{13}+\frac{1}{5}a^{11}+\frac{7}{10}a^{9}+2a^{7}-\frac{3}{2}a^{5}-\frac{14}{5}a^{3}+\frac{11}{5}a$, $\frac{13}{10}a^{18}+\frac{61}{10}a^{14}-\frac{9}{10}a^{10}+5a^{6}-\frac{37}{5}a^{2}$, $\frac{1}{20}a^{19}-\frac{11}{20}a^{17}+\frac{7}{20}a^{15}-\frac{13}{5}a^{13}+\frac{3}{5}a^{11}+\frac{2}{5}a^{9}+\frac{1}{2}a^{7}-\frac{3}{2}a^{5}-\frac{2}{5}a^{3}+\frac{17}{5}a$, $\frac{1}{10}a^{18}+\frac{9}{10}a^{17}+\frac{1}{10}a^{16}+\frac{9}{20}a^{14}+\frac{43}{10}a^{13}+\frac{9}{20}a^{12}-\frac{1}{20}a^{10}-\frac{1}{5}a^{9}-\frac{3}{10}a^{8}+a^{6}+\frac{7}{2}a^{5}-\frac{3}{10}a^{2}-\frac{26}{5}a-\frac{3}{10}$, $\frac{5}{4}a^{19}+\frac{13}{20}a^{18}+\frac{9}{20}a^{17}+\frac{3}{5}a^{16}+6a^{15}+\frac{61}{20}a^{14}+\frac{43}{20}a^{13}+\frac{59}{20}a^{12}-\frac{1}{4}a^{11}-\frac{9}{20}a^{10}-\frac{1}{10}a^{9}+\frac{1}{5}a^{8}+\frac{9}{2}a^{7}+\frac{5}{2}a^{6}+2a^{5}+2a^{4}-\frac{13}{2}a^{3}-\frac{37}{10}a^{2}-\frac{21}{10}a-\frac{33}{10}$, $\frac{5}{4}a^{19}-\frac{13}{20}a^{18}+\frac{9}{20}a^{17}-\frac{3}{5}a^{16}+6a^{15}-\frac{61}{20}a^{14}+\frac{43}{20}a^{13}-\frac{59}{20}a^{12}-\frac{1}{4}a^{11}+\frac{9}{20}a^{10}-\frac{1}{10}a^{9}-\frac{1}{5}a^{8}+\frac{9}{2}a^{7}-\frac{5}{2}a^{6}+2a^{5}-2a^{4}-\frac{13}{2}a^{3}+\frac{37}{10}a^{2}-\frac{21}{10}a+\frac{33}{10}$, $\frac{5}{4}a^{19}-\frac{1}{10}a^{18}-\frac{9}{20}a^{17}+\frac{9}{20}a^{16}+6a^{15}-\frac{9}{20}a^{14}-\frac{43}{20}a^{13}+\frac{43}{20}a^{12}-\frac{1}{4}a^{11}+\frac{1}{20}a^{10}+\frac{1}{10}a^{9}-\frac{1}{10}a^{8}+\frac{9}{2}a^{7}-a^{6}-2a^{5}+2a^{4}-\frac{13}{2}a^{3}+\frac{3}{10}a^{2}+\frac{21}{10}a-\frac{21}{10}$, $\frac{13}{20}a^{19}+\frac{13}{20}a^{18}-\frac{1}{20}a^{17}-\frac{3}{5}a^{16}+\frac{61}{20}a^{15}+\frac{61}{20}a^{14}-\frac{7}{20}a^{13}-\frac{59}{20}a^{12}-\frac{9}{20}a^{11}-\frac{9}{20}a^{10}-\frac{3}{5}a^{9}-\frac{1}{5}a^{8}+\frac{5}{2}a^{7}+\frac{5}{2}a^{6}-\frac{1}{2}a^{5}-2a^{4}-\frac{37}{10}a^{3}-\frac{37}{10}a^{2}-\frac{1}{10}a+\frac{33}{10}$, $\frac{1}{20}a^{18}+\frac{7}{20}a^{17}+a^{16}+\frac{7}{20}a^{14}+\frac{17}{10}a^{13}+\frac{19}{4}a^{12}+\frac{3}{5}a^{10}+\frac{1}{5}a^{9}-\frac{1}{2}a^{8}+\frac{1}{2}a^{6}+2a^{5}+\frac{7}{2}a^{4}+\frac{1}{10}a^{2}-\frac{9}{5}a-5$ | 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: | \( 476.032179845 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 476.032179845 \cdot 1}{4\cdot\sqrt{1618299543010938978304}}\cr\approx \mathstrut & 0.283690993743 \end{aligned}\]
Galois group
$C_2\wr S_5$ (as 20T279):
A non-solvable group of order 3840 |
The 36 conjugacy class representatives for $C_2\wr S_5$ |
Character table for $C_2\wr S_5$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 5.1.4432.1, 10.0.1257127936.1, 10.2.5028511744.1, 10.0.5028511744.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 10.0.5028511744.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 $20$ | $20$ | $1$ | $38$ | |||
\(277\) | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |