Normalized defining polynomial
\( x^{18} - x^{17} + 3 x^{16} - 5 x^{15} + 4 x^{14} - 6 x^{13} - 5 x^{12} - 2 x^{11} - 16 x^{10} + x^{9} + \cdots + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(10913148516977234837\) \(\medspace = 53\cdot 453771377^{2}\) | 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: | \(11.42\) | 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: | $53^{1/2}453771377^{1/2}\approx 155080.2469078509$ | ||
Ramified primes: | \(53\), \(453771377\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{53}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11}a^{14}-\frac{2}{11}a^{13}+\frac{4}{11}a^{11}+\frac{1}{11}a^{10}-\frac{3}{11}a^{9}-\frac{3}{11}a^{7}-\frac{3}{11}a^{5}+\frac{1}{11}a^{4}+\frac{4}{11}a^{3}-\frac{2}{11}a+\frac{1}{11}$, $\frac{1}{11}a^{15}-\frac{4}{11}a^{13}+\frac{4}{11}a^{12}-\frac{2}{11}a^{11}-\frac{1}{11}a^{10}+\frac{5}{11}a^{9}-\frac{3}{11}a^{8}+\frac{5}{11}a^{7}-\frac{3}{11}a^{6}-\frac{5}{11}a^{5}-\frac{5}{11}a^{4}-\frac{3}{11}a^{3}-\frac{2}{11}a^{2}-\frac{3}{11}a+\frac{2}{11}$, $\frac{1}{121}a^{16}+\frac{4}{121}a^{15}+\frac{24}{121}a^{13}+\frac{3}{121}a^{12}-\frac{59}{121}a^{11}-\frac{6}{121}a^{10}-\frac{6}{121}a^{9}-\frac{29}{121}a^{8}-\frac{6}{121}a^{7}+\frac{16}{121}a^{6}-\frac{26}{121}a^{5}-\frac{19}{121}a^{4}+\frac{13}{121}a^{3}+\frac{2}{11}a^{2}+\frac{48}{121}a-\frac{21}{121}$, $\frac{1}{121}a^{17}-\frac{5}{121}a^{15}+\frac{2}{121}a^{14}+\frac{28}{121}a^{13}-\frac{27}{121}a^{12}-\frac{1}{121}a^{11}-\frac{15}{121}a^{10}-\frac{5}{121}a^{9}-\frac{4}{11}a^{8}+\frac{40}{121}a^{7}-\frac{2}{121}a^{6}-\frac{25}{121}a^{5}+\frac{12}{121}a^{4}-\frac{30}{121}a^{3}+\frac{59}{121}a^{2}+\frac{40}{121}a-\frac{37}{121}$
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: | $a$, $\frac{41}{121}a^{17}+\frac{42}{121}a^{16}+\frac{51}{121}a^{15}+\frac{5}{121}a^{14}-\frac{20}{11}a^{13}-\frac{24}{121}a^{12}-\frac{53}{11}a^{11}-\frac{548}{121}a^{10}-\frac{754}{121}a^{9}-\frac{1108}{121}a^{8}-\frac{482}{121}a^{7}-\frac{1005}{121}a^{6}-\frac{269}{121}a^{5}-\frac{218}{121}a^{4}-\frac{288}{121}a^{3}+\frac{263}{121}a^{2}-\frac{205}{121}a+\frac{120}{121}$, $\frac{53}{121}a^{17}-\frac{19}{121}a^{16}+\frac{6}{11}a^{15}-\frac{147}{121}a^{14}-\frac{94}{121}a^{13}+\frac{19}{121}a^{12}-\frac{516}{121}a^{11}-\frac{10}{121}a^{10}-\frac{503}{121}a^{9}-\frac{98}{121}a^{8}+\frac{67}{121}a^{7}-\frac{179}{121}a^{6}+\frac{434}{121}a^{5}+\frac{40}{121}a^{4}+\frac{26}{11}a^{3}+\frac{201}{121}a^{2}+\frac{9}{121}a+\frac{8}{11}$, $\frac{25}{121}a^{17}+\frac{27}{121}a^{16}+\frac{49}{121}a^{15}+\frac{50}{121}a^{14}-\frac{126}{121}a^{13}+\frac{3}{11}a^{12}-\frac{540}{121}a^{11}-\frac{361}{121}a^{10}-\frac{804}{121}a^{9}-\frac{1113}{121}a^{8}-\frac{768}{121}a^{7}-\frac{1268}{121}a^{6}-\frac{568}{121}a^{5}-\frac{664}{121}a^{4}-\frac{355}{121}a^{3}+\frac{1}{121}a^{2}-\frac{80}{121}a+\frac{92}{121}$, $\frac{49}{121}a^{17}-\frac{28}{121}a^{16}+\frac{83}{121}a^{15}-\frac{188}{121}a^{14}-\frac{4}{121}a^{13}-\frac{131}{121}a^{12}-\frac{300}{121}a^{11}-\frac{83}{121}a^{10}-\frac{26}{11}a^{9}-\frac{2}{121}a^{8}+\frac{104}{121}a^{7}+\frac{191}{121}a^{6}+\frac{339}{121}a^{5}+\frac{207}{121}a^{4}+\frac{300}{121}a^{3}+\frac{185}{121}a^{2}-\frac{1}{11}a+\frac{95}{121}$, $\frac{23}{121}a^{17}+\frac{1}{11}a^{16}+\frac{72}{121}a^{15}-\frac{97}{121}a^{14}+\frac{17}{121}a^{13}-\frac{258}{121}a^{12}-\frac{78}{121}a^{11}-\frac{334}{121}a^{10}-\frac{489}{121}a^{9}-\frac{17}{11}a^{8}-\frac{422}{121}a^{7}+\frac{64}{121}a^{6}-\frac{179}{121}a^{5}+\frac{56}{121}a^{4}+\frac{25}{121}a^{3}-\frac{18}{121}a^{2}+\frac{95}{121}a-\frac{92}{121}$, $\frac{74}{121}a^{17}-\frac{81}{121}a^{16}+\frac{208}{121}a^{15}-\frac{336}{121}a^{14}+\frac{271}{121}a^{13}-\frac{327}{121}a^{12}-\frac{487}{121}a^{11}-\frac{74}{121}a^{10}-\frac{1061}{121}a^{9}-\frac{104}{121}a^{8}-\frac{998}{121}a^{7}-\frac{520}{121}a^{6}-\frac{261}{121}a^{5}-\frac{510}{121}a^{4}+\frac{71}{121}a^{3}-\frac{188}{121}a^{2}-\frac{4}{121}a+\frac{41}{121}$, $\frac{9}{121}a^{17}+\frac{6}{121}a^{16}+\frac{45}{121}a^{15}+\frac{7}{121}a^{14}+\frac{3}{11}a^{13}-\frac{82}{121}a^{12}-\frac{16}{11}a^{11}-\frac{248}{121}a^{10}-\frac{444}{121}a^{9}-\frac{526}{121}a^{8}-\frac{644}{121}a^{7}-\frac{725}{121}a^{6}-\frac{436}{121}a^{5}-\frac{468}{121}a^{4}-\frac{192}{121}a^{3}-\frac{74}{121}a^{2}-\frac{133}{121}a+\frac{25}{121}$, $\frac{30}{121}a^{17}+\frac{6}{121}a^{16}+\frac{72}{121}a^{15}-\frac{72}{121}a^{14}-\frac{28}{121}a^{13}-a^{12}-\frac{340}{121}a^{11}-\frac{211}{121}a^{10}-\frac{736}{121}a^{9}-\frac{273}{121}a^{8}-\frac{596}{121}a^{7}-\frac{316}{121}a^{6}-\frac{169}{121}a^{5}-\frac{271}{121}a^{4}+\frac{234}{121}a^{3}-\frac{188}{121}a^{2}+\frac{190}{121}a-\frac{125}{121}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 106.041791279 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 106.041791279 \cdot 1}{2\cdot\sqrt{10913148516977234837}}\cr\approx \mathstrut & 0.155944824179 \end{aligned}\]
Galois group
$C_2^9.S_9$ (as 18T968):
A non-solvable group of order 185794560 |
The 300 conjugacy class representatives for $C_2^9.S_9$ |
Character table for $C_2^9.S_9$ |
Intermediate fields
9.5.453771377.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 18 sibling: | data not computed |
Degree 36 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 | $18$ | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | ${\href{/padicField/5.7.0.1}{7} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.7.0.1}{7} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(53\) | 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
53.6.0.1 | $x^{6} + x^{4} + 7 x^{3} + 4 x^{2} + 45 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
53.6.0.1 | $x^{6} + x^{4} + 7 x^{3} + 4 x^{2} + 45 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(453771377\) | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |