Normalized defining polynomial
\( x^{18} + x^{16} - 6x^{14} - 5x^{12} + 2x^{10} + 7x^{8} + 18x^{6} + 7x^{4} - 3x^{2} + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-8173509334225000000\) \(\medspace = -\,2^{6}\cdot 5^{8}\cdot 83^{6}\) | 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: | \(11.24\) | 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: | $2\cdot 5^{2/3}83^{1/2}\approx 53.27813877645612$ | ||
Ramified primes: | \(2\), \(5\), \(83\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6}a^{12}+\frac{1}{6}a^{10}-\frac{1}{3}a^{8}-\frac{1}{2}a^{7}-\frac{1}{3}a^{6}-\frac{1}{2}a^{5}-\frac{1}{6}a^{4}-\frac{1}{3}a^{2}-\frac{1}{2}a+\frac{1}{6}$, $\frac{1}{6}a^{13}+\frac{1}{6}a^{11}-\frac{1}{3}a^{9}-\frac{1}{2}a^{8}-\frac{1}{3}a^{7}-\frac{1}{2}a^{6}-\frac{1}{6}a^{5}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a$, $\frac{1}{6}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{1}{6}a^{6}-\frac{1}{2}a^{5}-\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{6}$, $\frac{1}{6}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}+\frac{1}{6}a^{7}-\frac{1}{2}a^{6}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{6}a$, $\frac{1}{582}a^{16}+\frac{7}{194}a^{14}+\frac{13}{291}a^{12}-\frac{1}{2}a^{11}+\frac{112}{291}a^{10}-\frac{1}{2}a^{9}+\frac{39}{194}a^{8}-\frac{136}{291}a^{6}-\frac{1}{2}a^{5}-\frac{281}{582}a^{4}-\frac{139}{291}a^{2}-\frac{17}{291}$, $\frac{1}{582}a^{17}+\frac{7}{194}a^{15}+\frac{13}{291}a^{13}+\frac{112}{291}a^{11}+\frac{39}{194}a^{9}+\frac{19}{582}a^{7}-\frac{1}{2}a^{6}+\frac{5}{291}a^{5}-\frac{1}{2}a^{4}-\frac{139}{291}a^{3}+\frac{257}{582}a-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{81}{194}a^{17}-\frac{91}{582}a^{16}+\frac{253}{582}a^{15}-\frac{34}{291}a^{14}-\frac{721}{291}a^{13}+\frac{272}{291}a^{12}-\frac{1343}{582}a^{11}+\frac{277}{582}a^{10}+\frac{301}{582}a^{9}-\frac{57}{194}a^{8}+\frac{1901}{582}a^{7}-\frac{78}{97}a^{6}+\frac{2282}{291}a^{5}-\frac{1589}{582}a^{4}+\frac{2383}{582}a^{3}-\frac{19}{582}a^{2}-\frac{19}{97}a+\frac{63}{97}$, $\frac{49}{291}a^{17}-\frac{41}{194}a^{16}+\frac{59}{291}a^{15}-\frac{61}{582}a^{14}-\frac{278}{291}a^{13}+\frac{341}{291}a^{12}-\frac{649}{582}a^{11}+\frac{95}{291}a^{10}-\frac{77}{582}a^{9}-\frac{35}{582}a^{8}+\frac{446}{291}a^{7}-\frac{147}{97}a^{6}+\frac{747}{194}a^{5}-\frac{507}{194}a^{4}+\frac{180}{97}a^{3}+\frac{122}{291}a^{2}+\frac{80}{291}a+\frac{151}{291}$, $\frac{45}{194}a^{17}+\frac{11}{291}a^{16}+\frac{36}{97}a^{15}+\frac{57}{194}a^{14}-\frac{379}{291}a^{13}-\frac{5}{291}a^{12}-\frac{1091}{582}a^{11}-\frac{446}{291}a^{10}+\frac{89}{291}a^{9}-\frac{56}{97}a^{8}+\frac{458}{291}a^{7}+\frac{127}{582}a^{6}+\frac{1354}{291}a^{5}+\frac{1093}{582}a^{4}+\frac{926}{291}a^{3}+\frac{1016}{291}a^{2}-\frac{419}{582}a-\frac{83}{291}$, $\frac{55}{291}a^{17}-\frac{24}{97}a^{16}+\frac{88}{291}a^{15}-\frac{211}{582}a^{14}-\frac{535}{582}a^{13}+\frac{815}{582}a^{12}-\frac{871}{582}a^{11}+\frac{556}{291}a^{10}-\frac{419}{582}a^{9}-\frac{67}{582}a^{8}+\frac{635}{582}a^{7}-\frac{592}{291}a^{6}+\frac{2555}{582}a^{5}-\frac{1496}{291}a^{4}+\frac{703}{194}a^{3}-\frac{839}{291}a^{2}+\frac{273}{194}a+\frac{40}{97}$, $\frac{45}{194}a^{17}-\frac{11}{291}a^{16}+\frac{36}{97}a^{15}-\frac{57}{194}a^{14}-\frac{379}{291}a^{13}+\frac{5}{291}a^{12}-\frac{1091}{582}a^{11}+\frac{446}{291}a^{10}+\frac{89}{291}a^{9}+\frac{56}{97}a^{8}+\frac{458}{291}a^{7}-\frac{127}{582}a^{6}+\frac{1354}{291}a^{5}-\frac{1093}{582}a^{4}+\frac{926}{291}a^{3}-\frac{1016}{291}a^{2}-\frac{419}{582}a+\frac{83}{291}$, $\frac{91}{291}a^{17}+\frac{68}{291}a^{15}-\frac{544}{291}a^{13}-\frac{277}{291}a^{11}+\frac{57}{97}a^{9}+\frac{156}{97}a^{7}+\frac{1589}{291}a^{5}+\frac{310}{291}a^{3}-\frac{126}{97}a$, $\frac{43}{291}a^{17}+\frac{10}{97}a^{15}-\frac{80}{97}a^{13}-\frac{55}{97}a^{11}-\frac{110}{291}a^{9}+\frac{332}{291}a^{7}+\frac{305}{97}a^{5}+\frac{656}{291}a^{3}+\frac{127}{97}a$, $a$ | 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: | \( 72.3431657313 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 72.3431657313 \cdot 1}{2\cdot\sqrt{8173509334225000000}}\cr\approx \mathstrut & 0.193099899447 \end{aligned}\]
Galois group
$C_6^2:D_6$ (as 18T156):
A solvable group of order 432 |
The 38 conjugacy class representatives for $C_6^2:D_6$ |
Character table for $C_6^2:D_6$ is not computed |
Intermediate fields
3.1.83.1, 6.0.27556.1, 9.3.357366875.1 |
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 | R | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
2.12.0.1 | $x^{12} + x^{7} + x^{6} + x^{5} + x^{3} + x + 1$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(5\) | 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
5.12.8.2 | $x^{12} + 100 x^{6} - 500 x^{3} + 1250$ | $3$ | $4$ | $8$ | $C_3\times (C_3 : C_4)$ | $[\ ]_{3}^{12}$ | |
\(83\) | 83.6.0.1 | $x^{6} + x^{4} + 76 x^{3} + 32 x^{2} + 17 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
83.6.3.2 | $x^{6} + 255 x^{4} + 162 x^{3} + 20676 x^{2} - 39852 x + 537761$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
83.6.3.2 | $x^{6} + 255 x^{4} + 162 x^{3} + 20676 x^{2} - 39852 x + 537761$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |