Normalized defining polynomial
\( x^{18} - 6 x^{17} + 15 x^{16} - 23 x^{15} + 37 x^{14} - 78 x^{13} + 152 x^{12} - 240 x^{11} + 302 x^{10} + \cdots + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-13415474159898041623\) \(\medspace = -\,7^{15}\cdot 41^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.55\) | 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))
| |
Galois root discriminant: | $7^{5/6}41^{2/3}\approx 60.17797984027825$ | ||
Ramified primes: | \(7\), \(41\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}-\frac{2}{7}a^{11}-\frac{2}{7}a^{10}+\frac{2}{7}a^{9}+\frac{1}{7}a^{8}+\frac{1}{7}a^{6}+\frac{1}{7}a^{4}-\frac{2}{7}a^{3}-\frac{2}{7}a^{2}+\frac{2}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{13}+\frac{1}{7}a^{11}-\frac{2}{7}a^{10}-\frac{2}{7}a^{9}+\frac{2}{7}a^{8}+\frac{1}{7}a^{7}+\frac{2}{7}a^{6}+\frac{1}{7}a^{5}+\frac{1}{7}a^{3}-\frac{2}{7}a^{2}-\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{14}+\frac{2}{7}a^{7}-\frac{1}{7}$, $\frac{1}{7}a^{15}+\frac{2}{7}a^{8}-\frac{1}{7}a$, $\frac{1}{91}a^{16}-\frac{2}{91}a^{15}-\frac{4}{91}a^{14}-\frac{4}{91}a^{13}+\frac{31}{91}a^{11}+\frac{29}{91}a^{10}+\frac{3}{91}a^{9}-\frac{5}{91}a^{8}+\frac{16}{91}a^{7}-\frac{36}{91}a^{6}+\frac{45}{91}a^{5}+\frac{6}{13}a^{4}+\frac{17}{91}a^{3}+\frac{3}{13}a^{2}-\frac{25}{91}a+\frac{45}{91}$, $\frac{1}{294931}a^{17}-\frac{899}{294931}a^{16}-\frac{17151}{294931}a^{15}+\frac{18300}{294931}a^{14}-\frac{57}{22687}a^{13}-\frac{18975}{294931}a^{12}-\frac{119038}{294931}a^{11}-\frac{140228}{294931}a^{10}-\frac{298}{637}a^{9}-\frac{129022}{294931}a^{8}-\frac{3845}{294931}a^{7}+\frac{88575}{294931}a^{6}+\frac{54447}{294931}a^{5}-\frac{116450}{294931}a^{4}-\frac{10574}{294931}a^{3}+\frac{11246}{294931}a^{2}+\frac{40085}{294931}a-\frac{6659}{22687}$
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)
| |
Torsion generator: | \( \frac{741}{3241} a^{17} - \frac{39470}{42133} a^{16} + \frac{39303}{42133} a^{15} + \frac{2564}{6019} a^{14} + \frac{31039}{42133} a^{13} - \frac{2328}{463} a^{12} + \frac{240360}{42133} a^{11} + \frac{274}{6019} a^{10} - \frac{1200}{91} a^{9} + \frac{1119011}{42133} a^{8} - \frac{932756}{42133} a^{7} + \frac{836362}{42133} a^{6} - \frac{1636543}{42133} a^{5} + \frac{194663}{6019} a^{4} - \frac{7777}{6019} a^{3} - \frac{76417}{42133} a^{2} - \frac{26186}{6019} a - \frac{13298}{42133} \) (order $14$) | 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: | $\frac{156918}{294931}a^{17}-\frac{876586}{294931}a^{16}+\frac{1934731}{294931}a^{15}-\frac{2652281}{294931}a^{14}+\frac{4664918}{294931}a^{13}-\frac{10386791}{294931}a^{12}+\frac{19164477}{294931}a^{11}-\frac{28939879}{294931}a^{10}+\frac{75149}{637}a^{9}-\frac{34123126}{294931}a^{8}+\frac{35055988}{294931}a^{7}-\frac{34996763}{294931}a^{6}+\frac{21316868}{294931}a^{5}-\frac{7807123}{294931}a^{4}+\frac{5741516}{294931}a^{3}-\frac{3281877}{294931}a^{2}+\frac{80798}{294931}a-\frac{649672}{294931}$, $\frac{115119}{294931}a^{17}-\frac{690702}{294931}a^{16}+\frac{1680346}{294931}a^{15}-\frac{188091}{22687}a^{14}+\frac{3947479}{294931}a^{13}-\frac{8634905}{294931}a^{12}+\frac{16638393}{294931}a^{11}-\frac{1979864}{22687}a^{10}+\frac{68493}{637}a^{9}-\frac{31236407}{294931}a^{8}+\frac{2337301}{22687}a^{7}-\frac{30750851}{294931}a^{6}+\frac{20973369}{294931}a^{5}-\frac{6396347}{294931}a^{4}+\frac{2300548}{294931}a^{3}-\frac{2251394}{294931}a^{2}+\frac{430645}{294931}a-\frac{60227}{294931}$, $\frac{28540}{294931}a^{17}-\frac{27632}{294931}a^{16}-\frac{20544}{22687}a^{15}+\frac{581471}{294931}a^{14}-\frac{302028}{294931}a^{13}+\frac{620944}{294931}a^{12}-\frac{2497491}{294931}a^{11}+\frac{4822205}{294931}a^{10}-\frac{15810}{637}a^{9}+\frac{572675}{22687}a^{8}-\frac{4701972}{294931}a^{7}+\frac{514503}{22687}a^{6}-\frac{8085811}{294931}a^{5}+\frac{723193}{294931}a^{4}+\frac{177048}{22687}a^{3}+\frac{1485747}{294931}a^{2}-\frac{925411}{294931}a-\frac{370872}{294931}$, $\frac{15674}{294931}a^{17}-\frac{177313}{294931}a^{16}+\frac{596315}{294931}a^{15}-\frac{888816}{294931}a^{14}+\frac{1028707}{294931}a^{13}-\frac{2398884}{294931}a^{12}+\frac{5373533}{294931}a^{11}-\frac{8547424}{294931}a^{10}+\frac{22536}{637}a^{9}-\frac{9014038}{294931}a^{8}+\frac{6585577}{294931}a^{7}-\frac{7554704}{294931}a^{6}+\frac{5618257}{294931}a^{5}+\frac{2057946}{294931}a^{4}-\frac{3528067}{294931}a^{3}-\frac{400347}{294931}a^{2}+\frac{730978}{294931}a+\frac{479642}{294931}$, $\frac{12927}{42133}a^{17}-\frac{57485}{42133}a^{16}+\frac{98433}{42133}a^{15}-\frac{16365}{6019}a^{14}+\frac{33218}{6019}a^{13}-\frac{527190}{42133}a^{12}+\frac{923109}{42133}a^{11}-\frac{1244949}{42133}a^{10}+\frac{366}{13}a^{9}-\frac{864137}{42133}a^{8}+\frac{115900}{6019}a^{7}-\frac{583078}{42133}a^{6}-\frac{26527}{6019}a^{5}+\frac{596779}{42133}a^{4}-\frac{49739}{6019}a^{3}+\frac{173760}{42133}a^{2}-\frac{91830}{42133}a+\frac{3186}{42133}$, $\frac{136770}{294931}a^{17}-\frac{582552}{294931}a^{16}+\frac{1026938}{294931}a^{15}-\frac{1445905}{294931}a^{14}+\frac{2981220}{294931}a^{13}-\frac{6095767}{294931}a^{12}+\frac{10752814}{294931}a^{11}-\frac{15699231}{294931}a^{10}+\frac{38148}{637}a^{9}-\frac{17525507}{294931}a^{8}+\frac{18578181}{294931}a^{7}-\frac{15418947}{294931}a^{6}+\frac{8546647}{294931}a^{5}-\frac{3989068}{294931}a^{4}+\frac{1519551}{294931}a^{3}+\frac{79424}{294931}a^{2}+\frac{436000}{294931}a-\frac{152515}{294931}$, $\frac{159195}{294931}a^{17}-\frac{810477}{294931}a^{16}+\frac{1462768}{294931}a^{15}-\frac{1414656}{294931}a^{14}+\frac{2825334}{294931}a^{13}-\frac{7541497}{294931}a^{12}+\frac{12953266}{294931}a^{11}-\frac{16675012}{294931}a^{10}+\frac{33281}{637}a^{9}-\frac{9708081}{294931}a^{8}+\frac{10646893}{294931}a^{7}-\frac{11598898}{294931}a^{6}-\frac{2597384}{294931}a^{5}+\frac{9304640}{294931}a^{4}+\frac{439631}{294931}a^{3}-\frac{145499}{22687}a^{2}-\frac{1921637}{294931}a-\frac{372867}{294931}$, $\frac{77771}{294931}a^{17}-\frac{351305}{294931}a^{16}+\frac{621606}{294931}a^{15}-\frac{814798}{294931}a^{14}+\frac{1766255}{294931}a^{13}-\frac{3788370}{294931}a^{12}+\frac{6473814}{294931}a^{11}-\frac{9436270}{294931}a^{10}+\frac{1796}{49}a^{9}-\frac{10956132}{294931}a^{8}+\frac{12722295}{294931}a^{7}-\frac{11587495}{294931}a^{6}+\frac{503005}{22687}a^{5}-\frac{4358842}{294931}a^{4}+\frac{3511343}{294931}a^{3}-\frac{1011545}{294931}a^{2}+\frac{496569}{294931}a-\frac{203439}{294931}$ | 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: | \( 765.05787444 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 765.05787444 \cdot 1}{14\cdot\sqrt{13415474159898041623}}\cr\approx \mathstrut & 0.22770997496 \end{aligned}\]
Galois group
$C_3^3:C_6$ (as 18T85):
A solvable group of order 162 |
The 13 conjugacy class representatives for $C_3^3:C_6$ |
Character table for $C_3^3:C_6$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 9.3.1384375783.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 siblings: | data not computed |
Minimal sibling: | 9.3.1384375783.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{9}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(41\) | 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.6.4.1 | $x^{6} + 114 x^{5} + 4350 x^{4} + 56322 x^{3} + 30774 x^{2} + 180240 x + 2223605$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |