Normalized defining polynomial
\( x^{18} - 2 x^{17} + x^{16} + 5 x^{15} - 10 x^{14} + 5 x^{13} + 13 x^{12} - 29 x^{11} + 24 x^{10} + \cdots + 1 \)
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: | \(-1678237502892756571\) \(\medspace = -\,139\cdot 367^{2}\cdot 299401^{2}\) | 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: | \(10.29\) | 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: | $139^{1/2}367^{1/2}299401^{1/2}\approx 123585.36811856006$ | ||
Ramified primes: | \(139\), \(367\), \(299401\) | 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: | \(\Q(\sqrt{-139}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{15073}a^{17}+\frac{1035}{15073}a^{16}+\frac{3113}{15073}a^{15}+\frac{2564}{15073}a^{14}+\frac{6010}{15073}a^{13}+\frac{7226}{15073}a^{12}+\frac{2094}{15073}a^{11}+\frac{937}{15073}a^{10}+\frac{7021}{15073}a^{9}+\frac{527}{15073}a^{8}+\frac{3823}{15073}a^{7}+\frac{308}{15073}a^{6}+\frac{2847}{15073}a^{5}-\frac{2013}{15073}a^{4}-\frac{7333}{15073}a^{3}+\frac{7484}{15073}a^{2}-\frac{1658}{15073}a-\frac{1032}{15073}$
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)
| |
Fundamental units: | $\frac{28906}{15073}a^{17}-\frac{47414}{15073}a^{16}+\frac{28714}{15073}a^{15}+\frac{136700}{15073}a^{14}-\frac{247506}{15073}a^{13}+\frac{143852}{15073}a^{12}+\frac{342675}{15073}a^{11}-\frac{754909}{15073}a^{10}+\frac{654293}{15073}a^{9}+\frac{235827}{15073}a^{8}-\frac{1258657}{15073}a^{7}+\frac{1502205}{15073}a^{6}-\frac{455388}{15073}a^{5}-\frac{1151546}{15073}a^{4}+\frac{2008610}{15073}a^{3}-\frac{1668295}{15073}a^{2}+\frac{744569}{15073}a-\frac{152255}{15073}$, $\frac{110167}{15073}a^{17}-\frac{155130}{15073}a^{16}+\frac{8975}{15073}a^{15}+\frac{572942}{15073}a^{14}-\frac{761651}{15073}a^{13}+\frac{46539}{15073}a^{12}+\frac{1534879}{15073}a^{11}-\frac{2284521}{15073}a^{10}+\frac{1141987}{15073}a^{9}+\frac{1880938}{15073}a^{8}-\frac{4266984}{15073}a^{7}+\frac{3453830}{15073}a^{6}+\frac{684750}{15073}a^{5}-\frac{4745117}{15073}a^{4}+\frac{5243301}{15073}a^{3}-\frac{3048018}{15073}a^{2}+\frac{917181}{15073}a-\frac{102216}{15073}$, $a$, $\frac{62425}{15073}a^{17}-\frac{68368}{15073}a^{16}-\frac{7164}{15073}a^{15}+\frac{314046}{15073}a^{14}-\frac{339399}{15073}a^{13}-\frac{36767}{15073}a^{12}+\frac{818836}{15073}a^{11}-\frac{1061198}{15073}a^{10}+\frac{430348}{15073}a^{9}+\frac{1078872}{15073}a^{8}-\frac{2065035}{15073}a^{7}+\frac{1485979}{15073}a^{6}+\frac{601152}{15073}a^{5}-\frac{2379458}{15073}a^{4}+\frac{2370946}{15073}a^{3}-\frac{1295213}{15073}a^{2}+\frac{352320}{15073}a-\frac{30744}{15073}$, $\frac{109774}{15073}a^{17}-\frac{124768}{15073}a^{16}-\frac{23667}{15073}a^{15}+\frac{560108}{15073}a^{14}-\frac{606390}{15073}a^{13}-\frac{110285}{15073}a^{12}+\frac{1480660}{15073}a^{11}-\frac{1868966}{15073}a^{10}+\frac{658757}{15073}a^{9}+\frac{2005433}{15073}a^{8}-\frac{3674276}{15073}a^{7}+\frac{2473625}{15073}a^{6}+\frac{1269128}{15073}a^{5}-\frac{4300687}{15073}a^{4}+\frac{4055460}{15073}a^{3}-\frac{2025031}{15073}a^{2}+\frac{483519}{15073}a-\frac{28246}{15073}$, $\frac{109533}{15073}a^{17}-\frac{117962}{15073}a^{16}-\frac{5177}{15073}a^{15}+\frac{545104}{15073}a^{14}-\frac{592719}{15073}a^{13}-\frac{27918}{15073}a^{12}+\frac{1413123}{15073}a^{11}-\frac{1868688}{15073}a^{10}+\frac{835748}{15073}a^{9}+\frac{1818134}{15073}a^{8}-\frac{3600801}{15073}a^{7}+\frac{2715930}{15073}a^{6}+\frac{884461}{15073}a^{5}-\frac{4086868}{15073}a^{4}+\frac{4270194}{15073}a^{3}-\frac{2472105}{15073}a^{2}+\frac{762513}{15073}a-\frac{96067}{15073}$, $\frac{43873}{15073}a^{17}-\frac{96832}{15073}a^{16}+\frac{30342}{15073}a^{15}+\frac{241741}{15073}a^{14}-\frac{477595}{15073}a^{13}+\frac{146619}{15073}a^{12}+\frac{678412}{15073}a^{11}-\frac{1351640}{15073}a^{10}+\frac{889812}{15073}a^{9}+\frac{662301}{15073}a^{8}-\frac{2342180}{15073}a^{7}+\frac{2298572}{15073}a^{6}-\frac{229615}{15073}a^{5}-\frac{2415322}{15073}a^{4}+\frac{3223025}{15073}a^{3}-\frac{2129993}{15073}a^{2}+\frac{769587}{15073}a-\frac{118228}{15073}$, $\frac{25055}{15073}a^{17}-\frac{38854}{15073}a^{16}-\frac{6560}{15073}a^{15}+\frac{135551}{15073}a^{14}-\frac{179596}{15073}a^{13}-\frac{24519}{15073}a^{12}+\frac{372882}{15073}a^{11}-\frac{519681}{15073}a^{10}+\frac{190121}{15073}a^{9}+\frac{497446}{15073}a^{8}-\frac{998468}{15073}a^{7}+\frac{692922}{15073}a^{6}+\frac{307609}{15073}a^{5}-\frac{1147005}{15073}a^{4}+\frac{1111884}{15073}a^{3}-\frac{524055}{15073}a^{2}+\frac{120582}{15073}a+\frac{8508}{15073}$ | 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: | \( 27.3003078572 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 27.3003078572 \cdot 1}{2\cdot\sqrt{1678237502892756571}}\cr\approx \mathstrut & 0.160816147175 \end{aligned}\]
Galois group
$C_2^9.S_9$ (as 18T968):
A non-solvable group of order 185794560 |
The 300 conjugacy class representatives for $C_2^9.S_9$ are not computed |
Character table for $C_2^9.S_9$ is not computed |
Intermediate fields
9.3.109880167.1 |
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 | $18$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.9.0.1}{9} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | $18$ | ${\href{/padicField/23.7.0.1}{7} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $18$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(139\) | 139.2.1.1 | $x^{2} + 278$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
139.2.0.1 | $x^{2} + 138 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
139.2.0.1 | $x^{2} + 138 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
139.12.0.1 | $x^{12} + 120 x^{7} + 75 x^{6} + 41 x^{5} + 77 x^{4} + 106 x^{3} + 8 x^{2} + 10 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(367\) | 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 $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(299401\) | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |