Normalized defining polynomial
\( x^{20} - 3x^{15} - 31x^{10} + 3x^{5} + 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(439949333667755126953125\) \(\medspace = 3^{10}\cdot 5^{27}\) | 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: | \(15.21\) | 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: | $3^{1/2}5^{27/20}\approx 15.211433151416195$ | ||
Ramified primes: | \(3\), \(5\) | 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: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{8}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{3}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{9}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{10}+\frac{1}{5}a^{5}-\frac{1}{5}$, $\frac{1}{5}a^{11}+\frac{1}{5}a^{6}-\frac{1}{5}a$, $\frac{1}{5}a^{12}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{3}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{25}a^{14}+\frac{2}{25}a^{13}-\frac{1}{25}a^{12}-\frac{2}{25}a^{11}+\frac{1}{25}a^{10}+\frac{1}{25}a^{9}+\frac{2}{25}a^{8}-\frac{1}{25}a^{7}-\frac{2}{25}a^{6}+\frac{1}{25}a^{5}-\frac{6}{25}a^{4}-\frac{12}{25}a^{3}+\frac{6}{25}a^{2}+\frac{12}{25}a-\frac{6}{25}$, $\frac{1}{25}a^{15}-\frac{1}{25}a^{10}-\frac{8}{25}a^{5}+\frac{12}{25}$, $\frac{1}{25}a^{16}-\frac{1}{25}a^{11}-\frac{8}{25}a^{6}+\frac{12}{25}a$, $\frac{1}{25}a^{17}-\frac{1}{25}a^{12}+\frac{2}{25}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{8}{25}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{25}a^{18}-\frac{1}{25}a^{13}+\frac{2}{25}a^{8}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{8}{25}a^{3}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{25}a^{19}+\frac{2}{25}a^{13}-\frac{1}{25}a^{12}-\frac{2}{25}a^{11}+\frac{1}{25}a^{10}-\frac{2}{25}a^{9}+\frac{2}{25}a^{8}-\frac{1}{25}a^{7}-\frac{12}{25}a^{6}-\frac{9}{25}a^{5}-\frac{4}{25}a^{4}-\frac{12}{25}a^{3}+\frac{6}{25}a^{2}+\frac{7}{25}a-\frac{11}{25}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $a$, $\frac{4}{25}a^{17}-\frac{14}{25}a^{12}-\frac{117}{25}a^{7}+\frac{58}{25}a^{2}$, $\frac{1}{5}a^{15}-\frac{3}{5}a^{10}-6a^{5}-\frac{1}{5}$, $\frac{11}{25}a^{19}-\frac{3}{25}a^{18}+\frac{2}{25}a^{17}-\frac{1}{25}a^{16}-\frac{2}{25}a^{15}-\frac{32}{25}a^{14}+\frac{11}{25}a^{13}-\frac{6}{25}a^{12}+\frac{3}{25}a^{11}+\frac{6}{25}a^{10}-\frac{344}{25}a^{9}+\frac{87}{25}a^{8}-\frac{12}{5}a^{7}+\frac{6}{5}a^{6}+\frac{12}{5}a^{5}+\frac{3}{25}a^{4}-\frac{69}{25}a^{3}+\frac{3}{25}a^{2}+\frac{11}{25}a-\frac{3}{25}$, $\frac{9}{25}a^{18}+\frac{1}{5}a^{17}-\frac{1}{5}a^{16}-\frac{29}{25}a^{13}-\frac{3}{5}a^{12}+\frac{3}{5}a^{11}-\frac{272}{25}a^{8}-\frac{31}{5}a^{7}+\frac{31}{5}a^{6}+\frac{88}{25}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{25}a^{19}-\frac{7}{25}a^{18}+\frac{4}{25}a^{17}-\frac{1}{25}a^{16}-\frac{2}{25}a^{14}+\frac{4}{5}a^{13}-\frac{13}{25}a^{12}+\frac{3}{25}a^{11}-\frac{1}{25}a^{10}-\frac{34}{25}a^{9}+\frac{219}{25}a^{8}-\frac{121}{25}a^{7}+\frac{6}{5}a^{6}+\frac{4}{25}a^{5}-\frac{32}{25}a^{4}+\frac{13}{25}a^{3}+\frac{37}{25}a^{2}+\frac{11}{25}a+\frac{21}{25}$, $\frac{2}{5}a^{19}+\frac{1}{5}a^{17}-\frac{3}{25}a^{16}-\frac{6}{5}a^{14}-\frac{3}{5}a^{12}+\frac{8}{25}a^{11}-\frac{62}{5}a^{9}-\frac{31}{5}a^{7}+\frac{94}{25}a^{6}+\frac{7}{5}a^{4}+\frac{1}{5}a^{2}+\frac{29}{25}a$, $\frac{2}{5}a^{19}+\frac{2}{5}a^{18}+\frac{2}{25}a^{17}-\frac{6}{5}a^{14}-\frac{6}{5}a^{13}-\frac{7}{25}a^{12}-\frac{62}{5}a^{9}-\frac{62}{5}a^{8}-\frac{61}{25}a^{7}+\frac{7}{5}a^{4}+\frac{7}{5}a^{3}+\frac{34}{25}a^{2}$, $\frac{6}{25}a^{19}-\frac{1}{25}a^{18}+\frac{4}{25}a^{17}-\frac{2}{25}a^{16}-\frac{4}{25}a^{15}-\frac{18}{25}a^{14}+\frac{2}{25}a^{13}-\frac{12}{25}a^{12}+\frac{6}{25}a^{11}+\frac{12}{25}a^{10}-\frac{37}{5}a^{9}+\frac{34}{25}a^{8}-5a^{7}+\frac{12}{5}a^{6}+5a^{5}+\frac{19}{25}a^{4}+\frac{32}{25}a^{3}+\frac{16}{25}a^{2}-\frac{3}{25}a-\frac{16}{25}$, $\frac{7}{25}a^{19}-\frac{2}{25}a^{18}-\frac{1}{25}a^{17}+\frac{2}{25}a^{16}-\frac{1}{25}a^{15}-\frac{4}{5}a^{14}+\frac{6}{25}a^{13}+\frac{4}{25}a^{12}-\frac{6}{25}a^{11}+\frac{3}{25}a^{10}-\frac{219}{25}a^{9}+\frac{12}{5}a^{8}+\frac{26}{25}a^{7}-\frac{12}{5}a^{6}+\frac{7}{5}a^{5}-\frac{13}{25}a^{4}-\frac{3}{25}a^{3}-\frac{7}{5}a^{2}+\frac{3}{25}a+\frac{1}{25}$, $\frac{3}{25}a^{19}-\frac{4}{25}a^{17}+\frac{2}{25}a^{16}+\frac{2}{25}a^{15}-\frac{11}{25}a^{14}-\frac{1}{25}a^{13}+\frac{12}{25}a^{12}-\frac{6}{25}a^{11}-\frac{1}{5}a^{10}-\frac{87}{25}a^{9}+\frac{4}{25}a^{8}+5a^{7}-\frac{12}{5}a^{6}-\frac{64}{25}a^{5}+\frac{69}{25}a^{4}+\frac{21}{25}a^{3}-\frac{16}{25}a^{2}+\frac{3}{25}a-\frac{18}{25}$ | 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: | \( 7296.61382034 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 7296.61382034 \cdot 1}{2\cdot\sqrt{439949333667755126953125}}\cr\approx \mathstrut & 0.213771085813 \end{aligned}\]
Galois group
$C_4\times D_5$ (as 20T6):
A solvable group of order 40 |
The 16 conjugacy class representatives for $C_4\times D_5$ |
Character table for $C_4\times D_5$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 5.1.140625.1, 10.2.98876953125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 20 sibling: | 20.0.48883259296417236328125.1 |
Minimal sibling: | 20.0.48883259296417236328125.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $20$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.2.0.1}{2} }^{10}$ | ${\href{/padicField/13.4.0.1}{4} }^{5}$ | $20$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | Deg $20$ | $20$ | $1$ | $27$ |