Normalized defining polynomial
\( x^{20} - x^{19} + x^{18} - 2 x^{17} + 2 x^{16} - 5 x^{15} + 5 x^{14} - 4 x^{13} + 5 x^{12} - 3 x^{11} + \cdots + 1 \)
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: | \(1245300904083251953125\) \(\medspace = 3^{4}\cdot 5^{17}\cdot 67^{4}\) | 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: | \(11.34\) | 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: | $3^{1/2}5^{11/12}67^{1/2}\approx 61.989842465858416$ | ||
Ramified primes: | \(3\), \(5\), \(67\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{11}+\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{11}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{2}+\frac{2}{5}$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{11}+\frac{2}{5}a^{10}+\frac{1}{5}a^{9}+\frac{2}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{15}+\frac{2}{5}a^{10}-\frac{2}{5}a^{5}+\frac{1}{5}$, $\frac{1}{25}a^{16}+\frac{1}{25}a^{15}-\frac{1}{25}a^{14}-\frac{2}{25}a^{12}-\frac{8}{25}a^{11}-\frac{12}{25}a^{10}-\frac{12}{25}a^{9}-\frac{3}{25}a^{8}-\frac{4}{25}a^{7}+\frac{12}{25}a^{6}-\frac{1}{5}a^{4}-\frac{6}{25}a^{3}-\frac{2}{5}a^{2}+\frac{4}{25}a+\frac{11}{25}$, $\frac{1}{25}a^{17}-\frac{2}{25}a^{15}+\frac{1}{25}a^{14}-\frac{2}{25}a^{13}-\frac{1}{25}a^{12}+\frac{11}{25}a^{11}+\frac{1}{5}a^{10}-\frac{1}{25}a^{9}-\frac{11}{25}a^{8}-\frac{4}{25}a^{7}-\frac{7}{25}a^{6}+\frac{1}{5}a^{5}+\frac{9}{25}a^{4}-\frac{9}{25}a^{3}-\frac{1}{25}a^{2}-\frac{8}{25}a-\frac{6}{25}$, $\frac{1}{25}a^{18}-\frac{2}{25}a^{15}+\frac{1}{25}a^{14}-\frac{1}{25}a^{13}+\frac{2}{25}a^{12}-\frac{6}{25}a^{11}-\frac{1}{5}a^{10}+\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}-\frac{6}{25}a^{6}+\frac{4}{25}a^{5}+\frac{1}{25}a^{4}+\frac{7}{25}a^{3}-\frac{8}{25}a^{2}+\frac{7}{25}a+\frac{2}{25}$, $\frac{1}{25}a^{19}-\frac{2}{25}a^{15}+\frac{2}{25}a^{14}+\frac{2}{25}a^{13}+\frac{4}{25}a^{11}-\frac{9}{25}a^{10}-\frac{4}{25}a^{9}-\frac{1}{25}a^{8}+\frac{6}{25}a^{7}+\frac{8}{25}a^{6}+\frac{1}{25}a^{5}-\frac{3}{25}a^{4}+\frac{2}{5}a^{3}+\frac{12}{25}a^{2}-\frac{1}{5}a-\frac{8}{25}$
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: | \( -\frac{8}{25} a^{19} + \frac{17}{25} a^{18} - \frac{16}{25} a^{17} + \frac{24}{25} a^{16} - \frac{32}{25} a^{15} + \frac{56}{25} a^{14} - \frac{81}{25} a^{13} + \frac{72}{25} a^{12} - \frac{72}{25} a^{11} + \frac{64}{25} a^{10} - \frac{24}{5} a^{9} + \frac{172}{25} a^{8} - \frac{16}{5} a^{7} + \frac{64}{25} a^{6} - \frac{8}{5} a^{5} + \frac{72}{25} a^{4} - \frac{121}{25} a^{3} + \frac{24}{25} a^{2} + \frac{8}{25} a + \frac{8}{25} \) (order $10$) | 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: | $\frac{1}{5}a^{19}-\frac{1}{5}a^{15}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{4}{5}a^{12}-\frac{4}{5}a^{11}+\frac{6}{5}a^{10}+\frac{4}{5}a^{9}+\frac{7}{5}a^{8}-\frac{8}{5}a^{7}+\frac{4}{5}a^{6}-\frac{9}{5}a^{5}-a^{4}-\frac{3}{5}a^{3}+\frac{8}{5}a^{2}-\frac{1}{5}a$, $\frac{4}{25}a^{18}-\frac{8}{25}a^{17}+\frac{8}{25}a^{16}-\frac{9}{25}a^{15}+\frac{13}{25}a^{14}-\frac{28}{25}a^{13}+\frac{8}{5}a^{12}-\frac{36}{25}a^{11}+\frac{34}{25}a^{10}-\frac{33}{25}a^{9}+\frac{69}{25}a^{8}-\frac{19}{5}a^{7}+\frac{58}{25}a^{6}-\frac{34}{25}a^{5}+\frac{27}{25}a^{4}-\frac{63}{25}a^{3}+\frac{86}{25}a^{2}-\frac{21}{25}a+\frac{4}{25}$, $\frac{4}{25}a^{19}-\frac{3}{25}a^{18}-\frac{1}{25}a^{17}-\frac{7}{25}a^{16}+\frac{3}{25}a^{15}-\frac{14}{25}a^{14}+\frac{13}{25}a^{13}+\frac{4}{25}a^{12}+\frac{14}{25}a^{11}-\frac{2}{25}a^{10}+\frac{39}{25}a^{9}-\frac{17}{25}a^{8}-\frac{44}{25}a^{7}-\frac{32}{25}a^{6}-\frac{18}{25}a^{5}-\frac{24}{25}a^{4}+a^{3}+\frac{83}{25}a^{2}+\frac{29}{25}a-\frac{4}{25}$, $\frac{13}{25}a^{19}-\frac{17}{25}a^{18}+\frac{13}{25}a^{17}-\frac{21}{25}a^{16}+\frac{26}{25}a^{15}-\frac{57}{25}a^{14}+\frac{67}{25}a^{13}-\frac{8}{5}a^{12}+\frac{7}{5}a^{11}-\frac{4}{5}a^{10}+\frac{122}{25}a^{9}-\frac{118}{25}a^{8}-\frac{1}{5}a^{7}+\frac{3}{25}a^{6}-\frac{3}{5}a^{5}-\frac{79}{25}a^{4}+\frac{21}{5}a^{3}+\frac{44}{25}a^{2}-\frac{42}{25}a-\frac{17}{25}$, $\frac{1}{5}a^{19}-\frac{1}{5}a^{18}+\frac{1}{5}a^{17}-\frac{2}{5}a^{16}+\frac{3}{5}a^{15}-a^{14}+a^{13}-\frac{4}{5}a^{12}+\frac{4}{5}a^{11}-\frac{4}{5}a^{10}+\frac{11}{5}a^{9}-\frac{6}{5}a^{8}-\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{6}{5}a^{5}-\frac{7}{5}a^{4}+\frac{4}{5}a^{3}+\frac{4}{5}a^{2}-a-1$, $\frac{16}{25}a^{19}-\frac{12}{25}a^{18}+\frac{12}{25}a^{17}-\frac{29}{25}a^{16}+\frac{24}{25}a^{15}-\frac{69}{25}a^{14}+\frac{12}{5}a^{13}-\frac{43}{25}a^{12}+\frac{13}{5}a^{11}-\frac{31}{25}a^{10}+\frac{172}{25}a^{9}-\frac{81}{25}a^{8}+\frac{4}{25}a^{7}-\frac{107}{25}a^{6}-\frac{27}{25}a^{5}-\frac{147}{25}a^{4}+\frac{67}{25}a^{3}+\frac{71}{25}a^{2}+\frac{54}{25}a-\frac{8}{25}$, $\frac{8}{25}a^{19}-\frac{12}{25}a^{18}+\frac{8}{25}a^{17}-\frac{4}{5}a^{16}+\frac{22}{25}a^{15}-\frac{48}{25}a^{14}+\frac{62}{25}a^{13}-\frac{42}{25}a^{12}+\frac{57}{25}a^{11}-\frac{42}{25}a^{10}+\frac{22}{5}a^{9}-\frac{116}{25}a^{8}+\frac{6}{25}a^{7}-\frac{11}{5}a^{6}+\frac{1}{5}a^{5}-\frac{69}{25}a^{4}+\frac{104}{25}a^{3}+\frac{54}{25}a^{2}-\frac{8}{25}a+\frac{4}{25}$, $\frac{8}{25}a^{19}-\frac{12}{25}a^{18}+\frac{8}{25}a^{17}-\frac{17}{25}a^{16}+\frac{4}{5}a^{15}-\frac{41}{25}a^{14}+\frac{57}{25}a^{13}-\frac{38}{25}a^{12}+\frac{43}{25}a^{11}-\frac{33}{25}a^{10}+\frac{94}{25}a^{9}-\frac{22}{5}a^{8}+\frac{9}{25}a^{7}-\frac{39}{25}a^{6}-\frac{1}{5}a^{5}-\frac{44}{25}a^{4}+\frac{106}{25}a^{3}+\frac{24}{25}a^{2}-\frac{21}{25}a+\frac{12}{25}$, $\frac{4}{25}a^{19}-\frac{8}{25}a^{18}+\frac{16}{25}a^{17}-\frac{21}{25}a^{16}+a^{15}-\frac{38}{25}a^{14}+\frac{49}{25}a^{13}-\frac{13}{5}a^{12}+\frac{63}{25}a^{11}-\frac{49}{25}a^{10}+\frac{13}{5}a^{9}-\frac{67}{25}a^{8}+\frac{94}{25}a^{7}-\frac{74}{25}a^{6}+\frac{27}{25}a^{5}-\frac{26}{25}a^{4}+\frac{16}{25}a^{3}-\frac{54}{25}a^{2}+\frac{37}{25}a$ | 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: | \( 676.304774 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 676.304774 \cdot 1}{10\cdot\sqrt{1245300904083251953125}}\cr\approx \mathstrut & 0.183782428 \end{aligned}\]
Galois group
$C_4\times S_5$ (as 20T123):
A non-solvable group of order 480 |
The 28 conjugacy class representatives for $C_4\times S_5$ |
Character table for $C_4\times S_5$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.5025.1, 10.2.3156328125.1 |
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 24 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 | $20$ | R | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{5}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | $20$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.12.0.1 | $x^{12} + x^{6} + x^{5} + x^{4} + x^{2} + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
5.12.11.2 | $x^{12} + 5$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $[\ ]_{12}^{2}$ | |
\(67\) | 67.4.0.1 | $x^{4} + 8 x^{2} + 54 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
67.4.0.1 | $x^{4} + 8 x^{2} + 54 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
67.4.0.1 | $x^{4} + 8 x^{2} + 54 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
67.8.4.1 | $x^{8} + 284 x^{6} + 108 x^{5} + 28074 x^{4} - 13608 x^{3} + 1141144 x^{2} - 1396332 x + 15837397$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |