Normalized defining polynomial
\( x^{20} - 12x^{15} + 31x^{10} + 30x^{5} - 1 \)
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: | \(200608841562366962432861328125\) \(\medspace = 5^{21}\cdot 29^{10}\) | 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: | \(29.18\) | 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: | $5^{23/20}29^{1/2}\approx 34.27792339979557$ | ||
Ramified primes: | \(5\), \(29\) | 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: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{4}$, $\frac{1}{10}a^{15}+\frac{1}{10}a^{10}-\frac{1}{10}a^{5}+\frac{1}{5}$, $\frac{1}{10}a^{16}+\frac{1}{10}a^{11}-\frac{1}{10}a^{6}+\frac{1}{5}a$, $\frac{1}{10}a^{17}+\frac{1}{10}a^{12}-\frac{1}{10}a^{7}+\frac{1}{5}a^{2}$, $\frac{1}{50}a^{18}+\frac{1}{50}a^{17}+\frac{1}{25}a^{16}-\frac{1}{25}a^{15}-\frac{9}{50}a^{13}-\frac{9}{50}a^{12}+\frac{7}{50}a^{11}-\frac{7}{50}a^{10}-\frac{21}{50}a^{8}-\frac{21}{50}a^{7}-\frac{17}{50}a^{6}+\frac{17}{50}a^{5}-\frac{4}{25}a^{3}-\frac{4}{25}a^{2}+\frac{9}{50}a-\frac{9}{50}$, $\frac{1}{50}a^{19}+\frac{1}{50}a^{17}+\frac{1}{50}a^{16}+\frac{1}{25}a^{15}-\frac{9}{50}a^{14}-\frac{9}{50}a^{12}-\frac{9}{50}a^{11}+\frac{7}{50}a^{10}-\frac{21}{50}a^{9}-\frac{21}{50}a^{7}-\frac{21}{50}a^{6}-\frac{17}{50}a^{5}-\frac{4}{25}a^{4}-\frac{4}{25}a^{2}-\frac{4}{25}a+\frac{9}{50}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $\frac{1}{10}a^{17}-\frac{7}{5}a^{12}+\frac{22}{5}a^{7}+\frac{27}{10}a^{2}$, $a$, $\frac{13}{25}a^{19}-\frac{4}{25}a^{18}+\frac{4}{25}a^{17}+\frac{3}{50}a^{15}-\frac{309}{50}a^{14}+\frac{97}{50}a^{13}-\frac{97}{50}a^{12}-\frac{27}{50}a^{10}+\frac{779}{50}a^{9}-\frac{257}{50}a^{8}+\frac{257}{50}a^{7}+\frac{37}{50}a^{5}+\frac{817}{50}a^{4}-\frac{211}{50}a^{3}+\frac{261}{50}a^{2}+\frac{13}{25}$, $\frac{9}{50}a^{19}-\frac{1}{50}a^{17}+\frac{2}{25}a^{16}-\frac{1}{25}a^{15}-\frac{53}{25}a^{14}+\frac{9}{50}a^{12}-\frac{18}{25}a^{11}-\frac{7}{50}a^{10}+\frac{143}{25}a^{9}-\frac{29}{50}a^{7}+\frac{33}{25}a^{6}+\frac{117}{50}a^{5}+\frac{153}{50}a^{4}+\frac{29}{25}a^{2}+\frac{34}{25}a-\frac{9}{50}$, $\frac{1}{2}a^{11}-\frac{5}{2}a^{6}-\frac{5}{2}a$, $\frac{4}{25}a^{19}+\frac{4}{25}a^{18}-\frac{9}{50}a^{17}-\frac{1}{50}a^{16}-\frac{97}{50}a^{14}-\frac{97}{50}a^{13}+\frac{53}{25}a^{12}+\frac{9}{50}a^{11}+\frac{257}{50}a^{9}+\frac{257}{50}a^{8}-\frac{143}{25}a^{7}-\frac{29}{50}a^{6}+\frac{211}{50}a^{4}+\frac{211}{50}a^{3}-\frac{153}{50}a^{2}+\frac{29}{25}a-1$, $\frac{1}{2}a^{19}-\frac{3}{50}a^{18}+\frac{7}{50}a^{17}+\frac{2}{25}a^{16}-\frac{2}{25}a^{15}-6a^{14}+\frac{27}{50}a^{13}-\frac{63}{50}a^{12}-\frac{18}{25}a^{11}+\frac{18}{25}a^{10}+15a^{9}-\frac{37}{50}a^{8}+\frac{103}{50}a^{7}+\frac{33}{25}a^{6}-\frac{33}{25}a^{5}+\frac{35}{2}a^{4}-\frac{38}{25}a^{3}+\frac{47}{25}a^{2}+\frac{34}{25}a-\frac{9}{25}$, $\frac{12}{25}a^{19}+\frac{1}{5}a^{18}+\frac{9}{50}a^{17}-\frac{1}{50}a^{16}+\frac{3}{50}a^{15}-\frac{291}{50}a^{14}-\frac{23}{10}a^{13}-\frac{53}{25}a^{12}+\frac{9}{50}a^{11}-\frac{26}{25}a^{10}+\frac{771}{50}a^{9}+\frac{53}{10}a^{8}+\frac{143}{25}a^{7}-\frac{29}{50}a^{6}+\frac{106}{25}a^{5}+\frac{683}{50}a^{4}+\frac{79}{10}a^{3}+\frac{153}{50}a^{2}+\frac{54}{25}a+\frac{51}{50}$, $\frac{61}{50}a^{19}+\frac{19}{25}a^{18}+\frac{19}{50}a^{17}+\frac{7}{50}a^{16}+\frac{3}{25}a^{15}-\frac{362}{25}a^{14}-\frac{221}{25}a^{13}-\frac{221}{50}a^{12}-\frac{44}{25}a^{11}-\frac{27}{25}a^{10}+\frac{922}{25}a^{9}+\frac{551}{25}a^{8}+\frac{551}{50}a^{7}+\frac{114}{25}a^{6}+\frac{37}{25}a^{5}+\frac{1787}{50}a^{4}+\frac{523}{25}a^{3}+\frac{249}{25}a^{2}+\frac{219}{50}a+\frac{51}{25}$, $\frac{39}{50}a^{19}+\frac{3}{5}a^{18}+\frac{9}{50}a^{17}+\frac{9}{50}a^{16}+\frac{3}{50}a^{15}-\frac{238}{25}a^{14}-\frac{69}{10}a^{13}-\frac{131}{50}a^{12}-\frac{81}{50}a^{11}-\frac{27}{50}a^{10}+\frac{628}{25}a^{9}+\frac{169}{10}a^{8}+\frac{411}{50}a^{7}+\frac{111}{50}a^{6}+\frac{37}{50}a^{5}+\frac{1213}{50}a^{4}+\frac{157}{10}a^{3}+\frac{214}{25}a^{2}+\frac{89}{25}a+\frac{38}{25}$, $\frac{7}{25}a^{19}+\frac{9}{50}a^{18}-\frac{6}{25}a^{17}+\frac{1}{25}a^{16}-\frac{1}{5}a^{15}-\frac{88}{25}a^{14}-\frac{53}{25}a^{13}+\frac{133}{50}a^{12}-\frac{9}{25}a^{11}+\frac{23}{10}a^{10}+\frac{253}{25}a^{9}+\frac{143}{25}a^{8}-\frac{323}{50}a^{7}+\frac{4}{25}a^{6}-\frac{63}{10}a^{5}+\frac{144}{25}a^{4}+\frac{153}{50}a^{3}-\frac{229}{50}a^{2}+\frac{92}{25}a-\frac{9}{10}$ (assuming GRH) | 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: | \( 10784558.4809 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 10784558.4809 \cdot 1}{2\cdot\sqrt{200608841562366962432861328125}}\cr\approx \mathstrut & 0.467903472900 \end{aligned}\] (assuming GRH)
Galois group
$D_{10}:C_4$ (as 20T19):
A solvable group of order 80 |
The 14 conjugacy class representatives for $D_{10}:C_4$ |
Character table for $D_{10}:C_4$ |
Intermediate fields
\(\Q(\sqrt{29}) \), 4.4.4205.1, 5.1.2628125.1, 10.2.200304189453125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 20 siblings: | 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 | ${\href{/padicField/2.4.0.1}{4} }^{5}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{10}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{10}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.10.11.7 | $x^{10} + 5 x^{2} + 10$ | $10$ | $1$ | $11$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ |
5.10.10.7 | $x^{10} - 50 x^{7} + 10 x^{6} + 10 x^{5} + 175 x^{4} - 250 x^{3} - 225 x^{2} + 50 x + 25$ | $5$ | $2$ | $10$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ | |
\(29\) | 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |