Normalized defining polynomial
\( x^{18} - 87x^{12} + 8991x^{6} + 19683 \)
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: | \(-315164892649262158461997559808\) \(\medspace = -\,2^{12}\cdot 3^{33}\cdot 7^{12}\) | 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: | \(43.53\) | 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: | $2^{2/3}3^{11/6}7^{2/3}\approx 43.531903466499294$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | 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: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{9}a^{8}+\frac{1}{3}a^{2}$, $\frac{1}{54}a^{9}-\frac{1}{6}a^{6}+\frac{7}{18}a^{3}-\frac{1}{2}$, $\frac{1}{54}a^{10}-\frac{1}{6}a^{7}+\frac{7}{18}a^{4}-\frac{1}{2}a$, $\frac{1}{162}a^{11}-\frac{1}{18}a^{8}+\frac{25}{54}a^{5}-\frac{1}{6}a^{2}$, $\frac{1}{43740}a^{12}-\frac{227}{1458}a^{6}-\frac{1}{2}a^{3}+\frac{77}{180}$, $\frac{1}{131220}a^{13}+\frac{259}{4374}a^{7}-\frac{1}{2}a^{4}-\frac{103}{540}a$, $\frac{1}{131220}a^{14}-\frac{227}{4374}a^{8}-\frac{1}{2}a^{5}+\frac{257}{540}a^{2}$, $\frac{1}{393660}a^{15}+\frac{8}{6561}a^{9}+\frac{347}{1620}a^{3}-\frac{1}{2}$, $\frac{1}{1180980}a^{16}+\frac{259}{39366}a^{10}-\frac{1}{6}a^{7}+\frac{977}{4860}a^{4}$, $\frac{1}{1180980}a^{17}+\frac{8}{19683}a^{11}-\frac{1273}{4860}a^{5}-\frac{1}{2}a^{2}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (assuming GRH)
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{5}{78732} a^{15} + \frac{43}{6561} a^{9} - \frac{187}{324} a^{3} + \frac{1}{2} \) (order $6$) | 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: | $\frac{1}{393660}a^{15}-\frac{13}{43740}a^{12}+\frac{8}{6561}a^{9}+\frac{35}{1458}a^{6}-\frac{463}{1620}a^{3}-\frac{11}{180}$, $\frac{1}{21870}a^{12}+\frac{16}{729}a^{6}-\frac{13}{90}$, $\frac{19}{196830}a^{17}-\frac{5}{39366}a^{16}-\frac{1}{10935}a^{15}+\frac{19}{65610}a^{14}+\frac{11}{65610}a^{13}-\frac{1}{1620}a^{12}-\frac{121}{13122}a^{11}+\frac{86}{6561}a^{10}+\frac{17}{1458}a^{9}-\frac{121}{4374}a^{8}-\frac{67}{2187}a^{7}+\frac{1}{27}a^{6}+\frac{394}{405}a^{5}-\frac{187}{162}a^{4}-\frac{47}{45}a^{3}+\frac{259}{135}a^{2}+\frac{487}{270}a-\frac{41}{20}$, $\frac{19}{196830}a^{17}-\frac{1}{14580}a^{16}-\frac{7}{98415}a^{15}+\frac{19}{65610}a^{14}-\frac{11}{131220}a^{13}+\frac{1}{2916}a^{12}-\frac{121}{13122}a^{11}+\frac{1}{243}a^{10}+\frac{19}{6561}a^{9}-\frac{121}{4374}a^{8}+\frac{67}{4374}a^{7}-\frac{1}{486}a^{6}+\frac{394}{405}a^{5}-\frac{91}{180}a^{4}-\frac{583}{810}a^{3}+\frac{259}{135}a^{2}-\frac{1297}{540}a+\frac{29}{12}$, $\frac{1}{295245}a^{17}-\frac{83}{590490}a^{16}+\frac{1}{21870}a^{15}+\frac{47}{65610}a^{14}+\frac{127}{131220}a^{13}-\frac{2}{10935}a^{12}-\frac{179}{39366}a^{11}+\frac{17}{39366}a^{10}-\frac{11}{729}a^{9}-\frac{197}{4374}a^{8}+\frac{44}{2187}a^{7}+\frac{179}{729}a^{6}+\frac{1189}{2430}a^{5}-\frac{2251}{2430}a^{4}-\frac{83}{90}a^{3}+\frac{167}{135}a^{2}+\frac{689}{540}a-\frac{64}{45}$, $\frac{16}{295245}a^{17}-\frac{23}{295245}a^{16}-\frac{1}{131220}a^{14}-\frac{1}{3645}a^{13}+\frac{1}{21870}a^{12}-\frac{191}{39366}a^{11}+\frac{229}{39366}a^{10}-\frac{8}{2187}a^{8}+\frac{17}{486}a^{7}+\frac{16}{729}a^{6}+\frac{512}{1215}a^{5}-\frac{1607}{2430}a^{4}+\frac{193}{540}a^{2}-\frac{49}{30}a-\frac{283}{90}$, $\frac{103}{295245}a^{17}-\frac{19}{65610}a^{16}+\frac{5}{19683}a^{15}-\frac{7}{21870}a^{13}-\frac{592}{19683}a^{11}+\frac{121}{4374}a^{10}-\frac{172}{6561}a^{9}+\frac{19}{1458}a^{7}+\frac{3971}{1215}a^{5}-\frac{394}{135}a^{4}+\frac{187}{81}a^{3}-\frac{112}{45}a+5$, $\frac{103}{590490}a^{17}+\frac{19}{65610}a^{16}+\frac{35}{78732}a^{15}+\frac{13}{21870}a^{14}+\frac{7}{21870}a^{13}-\frac{296}{19683}a^{11}-\frac{121}{4374}a^{10}-\frac{301}{6561}a^{9}-\frac{35}{729}a^{8}-\frac{19}{1458}a^{7}+\frac{3971}{2430}a^{5}+\frac{394}{135}a^{4}+\frac{1309}{324}a^{3}+\frac{371}{90}a^{2}+\frac{112}{45}a-\frac{1}{2}$ (assuming GRH) | 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: | \( 30334220.019 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 30334220.019 \cdot 27}{6\cdot\sqrt{315164892649262158461997559808}}\cr\approx \mathstrut & 3.7110366591 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 18 |
The 6 conjugacy class representatives for $C_3^2 : C_2$ |
Character table for $C_3^2 : C_2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 3.1.972.2 x3, 3.1.47628.3 x3, 3.1.588.1 x3, 3.1.11907.2 x3, 6.0.2834352.2, 6.0.6805279152.2, 6.0.1037232.1, 6.0.425329947.5, 9.1.324121835451456.10 x9 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | ${\href{/padicField/5.2.0.1}{2} }^{9}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(3\) | 3.6.11.7 | $x^{6} + 9 x^{2} + 12$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ |
3.6.11.7 | $x^{6} + 9 x^{2} + 12$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ | |
3.6.11.7 | $x^{6} + 9 x^{2} + 12$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ | |
\(7\) | 7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |