Normalized defining polynomial
\( x^{20} + 5x^{18} + 14x^{16} + 27x^{14} + 41x^{12} + 53x^{10} + 55x^{8} + 43x^{6} + 22x^{4} + 7x^{2} + 1 \)
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: | \(1267791677353905767424\) \(\medspace = 2^{10}\cdot 3^{4}\cdot 11119^{4}\) | 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: | \(11.35\) | 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: | $2^{15/8}3^{1/2}11119^{1/2}\approx 669.9227623971964$ | ||
Ramified primes: | \(2\), \(3\), \(11119\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{22}a^{18}-\frac{1}{22}a^{16}+\frac{9}{22}a^{14}-\frac{1}{2}a^{13}-\frac{5}{22}a^{12}-\frac{1}{2}a^{11}+\frac{5}{22}a^{10}-\frac{1}{2}a^{9}-\frac{5}{11}a^{8}-\frac{1}{2}a^{7}+\frac{5}{22}a^{6}-\frac{9}{22}a^{4}-\frac{1}{2}a^{3}+\frac{5}{11}a^{2}-\frac{9}{22}$, $\frac{1}{22}a^{19}-\frac{1}{22}a^{17}-\frac{1}{11}a^{15}-\frac{1}{2}a^{14}-\frac{5}{22}a^{13}-\frac{3}{11}a^{11}-\frac{1}{2}a^{10}+\frac{1}{22}a^{9}-\frac{1}{2}a^{8}+\frac{5}{22}a^{7}-\frac{9}{22}a^{5}-\frac{1}{22}a^{3}-\frac{1}{2}a^{2}+\frac{1}{11}a-\frac{1}{2}$
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: | \( -1 \) (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: | $a$, $\frac{4}{11}a^{18}+\frac{29}{11}a^{16}+\frac{91}{11}a^{14}+\frac{189}{11}a^{12}+\frac{295}{11}a^{10}+\frac{389}{11}a^{8}+\frac{427}{11}a^{6}+\frac{338}{11}a^{4}+\frac{172}{11}a^{2}+\frac{41}{11}$, $\frac{37}{11}a^{19}+\frac{172}{11}a^{17}+\frac{454}{11}a^{15}+\frac{827}{11}a^{13}+\frac{1197}{11}a^{11}+\frac{1489}{11}a^{9}+\frac{1439}{11}a^{7}+\frac{998}{11}a^{5}+\frac{392}{11}a^{3}+\frac{74}{11}a$, $\frac{1}{11}a^{19}-\frac{7}{22}a^{18}+\frac{10}{11}a^{17}-\frac{13}{11}a^{16}+\frac{31}{11}a^{15}-\frac{63}{22}a^{14}+\frac{61}{11}a^{13}-\frac{54}{11}a^{12}+\frac{175}{22}a^{11}-\frac{78}{11}a^{10}+\frac{211}{22}a^{9}-\frac{97}{11}a^{8}+\frac{219}{22}a^{7}-\frac{167}{22}a^{6}+\frac{125}{22}a^{5}-\frac{62}{11}a^{4}+\frac{10}{11}a^{3}-\frac{59}{22}a^{2}-\frac{29}{22}a-\frac{25}{22}$, $\frac{3}{11}a^{18}+\frac{19}{11}a^{16}+\frac{60}{11}a^{14}+\frac{128}{11}a^{12}+\frac{202}{11}a^{10}+\frac{267}{11}a^{8}+\frac{290}{11}a^{6}+\frac{237}{11}a^{4}+\frac{118}{11}a^{2}+\frac{17}{11}$, $\frac{25}{22}a^{19}+\frac{37}{22}a^{18}+\frac{107}{22}a^{17}+\frac{86}{11}a^{16}+\frac{269}{22}a^{15}+\frac{227}{11}a^{14}+\frac{469}{22}a^{13}+\frac{827}{22}a^{12}+\frac{332}{11}a^{11}+\frac{1197}{22}a^{10}+\frac{817}{22}a^{9}+\frac{1489}{22}a^{8}+\frac{376}{11}a^{7}+\frac{1439}{22}a^{6}+\frac{245}{11}a^{5}+\frac{987}{22}a^{4}+\frac{92}{11}a^{3}+\frac{381}{22}a^{2}+\frac{25}{11}a+\frac{63}{22}$, $\frac{39}{22}a^{19}+\frac{26}{11}a^{18}+\frac{181}{22}a^{17}+\frac{245}{22}a^{16}+\frac{236}{11}a^{15}+\frac{655}{22}a^{14}+\frac{425}{11}a^{13}+\frac{1203}{22}a^{12}+\frac{609}{11}a^{11}+\frac{878}{11}a^{10}+\frac{1513}{22}a^{9}+\frac{2197}{22}a^{8}+\frac{1449}{22}a^{7}+\frac{1076}{11}a^{6}+\frac{490}{11}a^{5}+\frac{1523}{22}a^{4}+\frac{184}{11}a^{3}+\frac{315}{11}a^{2}+\frac{67}{22}a+\frac{137}{22}$, $\frac{39}{22}a^{19}-\frac{26}{11}a^{18}+\frac{181}{22}a^{17}-\frac{245}{22}a^{16}+\frac{236}{11}a^{15}-\frac{655}{22}a^{14}+\frac{425}{11}a^{13}-\frac{1203}{22}a^{12}+\frac{609}{11}a^{11}-\frac{878}{11}a^{10}+\frac{1513}{22}a^{9}-\frac{2197}{22}a^{8}+\frac{1449}{22}a^{7}-\frac{1076}{11}a^{6}+\frac{490}{11}a^{5}-\frac{1523}{22}a^{4}+\frac{184}{11}a^{3}-\frac{315}{11}a^{2}+\frac{67}{22}a-\frac{137}{22}$, $\frac{1}{11}a^{19}-\frac{21}{22}a^{18}+\frac{10}{11}a^{17}-\frac{39}{11}a^{16}+\frac{31}{11}a^{15}-\frac{167}{22}a^{14}+\frac{61}{11}a^{13}-\frac{118}{11}a^{12}+\frac{175}{22}a^{11}-\frac{135}{11}a^{10}+\frac{211}{22}a^{9}-\frac{137}{11}a^{8}+\frac{219}{22}a^{7}-\frac{127}{22}a^{6}+\frac{125}{22}a^{5}+\frac{34}{11}a^{4}+\frac{10}{11}a^{3}+\frac{153}{22}a^{2}-\frac{29}{22}a+\frac{57}{22}$ | 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: | \( 117.777592282 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 117.777592282 \cdot 1}{2\cdot\sqrt{1267791677353905767424}}\cr\approx \mathstrut & 0.158601544229 \end{aligned}\]
Galois group
$C_2^9.C_2^4:S_5$ (as 20T964):
A non-solvable group of order 983040 |
The 155 conjugacy class representatives for $C_2^9.C_2^4:S_5$ are not computed |
Character table for $C_2^9.C_2^4:S_5$ is not computed |
Intermediate fields
5.3.11119.1, 10.2.1112689449.1 |
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 | R | R | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{4}$ | $16{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |
2.10.10.6 | $x^{10} - 6 x^{9} + 42 x^{8} - 104 x^{7} - 256 x^{6} - 112 x^{5} - 1568 x^{4} - 2016 x^{3} - 2832 x^{2} - 4960 x - 3616$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2]^{10}$ | |
\(3\) | 3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(11119\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |