Normalized defining polynomial
\( x^{18} - 95x^{12} + 3667x^{6} + 27 \)
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: | \(-998220592474975337089609728\) \(\medspace = -\,2^{12}\cdot 3^{21}\cdot 13^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(31.62\) | 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: | $2^{2/3}3^{7/6}13^{2/3}\approx 31.619647871494593$ | ||
Ramified primes: | \(2\), \(3\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{6}a^{8}-\frac{1}{6}a^{2}$, $\frac{1}{12}a^{9}-\frac{1}{4}a^{6}-\frac{1}{12}a^{3}+\frac{1}{4}$, $\frac{1}{36}a^{10}-\frac{1}{4}a^{7}+\frac{11}{36}a^{4}+\frac{1}{4}a$, $\frac{1}{36}a^{11}-\frac{1}{12}a^{8}+\frac{11}{36}a^{5}+\frac{1}{12}a^{2}$, $\frac{1}{6120}a^{12}+\frac{613}{3060}a^{6}-\frac{1}{2}a^{3}-\frac{183}{680}$, $\frac{1}{6120}a^{13}+\frac{613}{3060}a^{7}-\frac{1}{2}a^{4}-\frac{183}{680}a$, $\frac{1}{6120}a^{14}+\frac{103}{3060}a^{8}-\frac{1}{2}a^{5}-\frac{209}{2040}a^{2}$, $\frac{1}{18360}a^{15}+\frac{1}{18360}a^{13}-\frac{1}{108}a^{11}+\frac{103}{9180}a^{9}-\frac{1}{12}a^{8}-\frac{917}{9180}a^{7}-\frac{47}{108}a^{5}-\frac{1}{2}a^{4}-\frac{209}{6120}a^{3}+\frac{1}{12}a^{2}-\frac{523}{2040}a-\frac{1}{2}$, $\frac{1}{18360}a^{16}+\frac{1}{18360}a^{14}+\frac{1}{18360}a^{12}+\frac{103}{9180}a^{10}+\frac{613}{9180}a^{8}+\frac{2143}{9180}a^{6}-\frac{1}{2}a^{5}-\frac{209}{6120}a^{4}-\frac{1}{2}a^{3}-\frac{863}{2040}a^{2}-\frac{1}{2}a+\frac{279}{680}$, $\frac{1}{18360}a^{17}-\frac{67}{9180}a^{11}-\frac{1}{36}a^{9}-\frac{1}{6}a^{7}-\frac{1}{4}a^{6}+\frac{1753}{18360}a^{5}-\frac{11}{36}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a+\frac{1}{4}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{1}{360} a^{15} - \frac{47}{180} a^{9} + \frac{1211}{120} a^{3} + \frac{1}{2} \) (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)
| |
Fundamental units: | $\frac{109}{9180}a^{17}+\frac{7}{1224}a^{16}+\frac{1}{540}a^{15}+\frac{7}{18360}a^{13}-\frac{863}{765}a^{11}-\frac{37}{68}a^{10}-\frac{47}{270}a^{9}-\frac{299}{9180}a^{7}+\frac{400007}{9180}a^{5}+\frac{25667}{1224}a^{4}+\frac{1211}{180}a^{3}+\frac{2119}{2040}a$, $\frac{109}{18360}a^{17}-\frac{7}{1224}a^{16}+\frac{1}{1080}a^{15}+\frac{1}{1224}a^{14}-\frac{7}{18360}a^{13}-\frac{863}{1530}a^{11}+\frac{37}{68}a^{10}-\frac{47}{540}a^{9}-\frac{25}{306}a^{8}+\frac{299}{9180}a^{7}+\frac{400007}{18360}a^{5}-\frac{25667}{1224}a^{4}+\frac{1211}{360}a^{3}+\frac{1321}{408}a^{2}-\frac{2119}{2040}a-\frac{1}{2}$, $\frac{19}{1080}a^{17}-\frac{61}{9180}a^{16}-\frac{1}{270}a^{15}+\frac{37}{18360}a^{14}+\frac{7}{18360}a^{13}-\frac{1}{9180}a^{12}-\frac{301}{180}a^{11}+\frac{2897}{4590}a^{10}+\frac{47}{135}a^{9}-\frac{1799}{9180}a^{8}-\frac{299}{9180}a^{7}+\frac{76}{2295}a^{6}+\frac{69707}{1080}a^{5}-\frac{74461}{3060}a^{4}-\frac{1211}{90}a^{3}+\frac{15329}{2040}a^{2}+\frac{1099}{2040}a-\frac{109}{340}$, $\frac{7}{1224}a^{17}-\frac{61}{9180}a^{16}+\frac{7}{18360}a^{14}-\frac{7}{6120}a^{13}-\frac{1}{9180}a^{12}-\frac{37}{68}a^{11}+\frac{2897}{4590}a^{10}-\frac{299}{9180}a^{8}+\frac{299}{3060}a^{7}+\frac{76}{2295}a^{6}+\frac{25667}{1224}a^{5}-\frac{74461}{3060}a^{4}+\frac{2119}{2040}a^{2}-\frac{2459}{680}a-\frac{789}{340}$, $\frac{1}{4590}a^{17}+\frac{131}{3060}a^{16}+\frac{1}{918}a^{15}-\frac{43}{6120}a^{14}+\frac{41}{18360}a^{13}-\frac{5}{1224}a^{12}-\frac{61}{3060}a^{11}-\frac{4123}{1020}a^{10}-\frac{251}{1836}a^{9}+\frac{973}{1530}a^{8}-\frac{28}{2295}a^{7}+\frac{37}{153}a^{6}+\frac{2911}{9180}a^{5}+\frac{119482}{765}a^{4}+\frac{4189}{612}a^{3}-\frac{18821}{680}a^{2}-\frac{1213}{2040}a+\frac{745}{136}$, $\frac{317}{18360}a^{17}-\frac{1}{1530}a^{16}-\frac{71}{18360}a^{15}+\frac{1}{765}a^{14}-\frac{1}{9180}a^{13}-\frac{1}{306}a^{12}-\frac{1661}{1020}a^{11}+\frac{61}{1020}a^{10}+\frac{2887}{9180}a^{9}-\frac{49}{765}a^{8}+\frac{1069}{9180}a^{7}-\frac{1}{153}a^{6}+\frac{1146961}{18360}a^{5}-\frac{4441}{3060}a^{4}-\frac{75601}{6120}a^{3}+\frac{7}{510}a^{2}+\frac{662}{255}a-\frac{2}{17}$, $\frac{157}{18360}a^{17}-\frac{1}{1530}a^{16}-\frac{13}{4590}a^{15}-\frac{7}{3060}a^{14}+\frac{1}{9180}a^{13}+\frac{4}{765}a^{12}-\frac{1229}{1530}a^{11}+\frac{61}{1020}a^{10}+\frac{2039}{9180}a^{9}+\frac{343}{3060}a^{8}-\frac{1069}{9180}a^{7}-\frac{1039}{3060}a^{6}+\frac{570851}{18360}a^{5}-\frac{4441}{3060}a^{4}-\frac{22531}{3060}a^{3}-\frac{2371}{510}a^{2}-\frac{662}{255}a-\frac{293}{340}$, $\frac{1}{1020}a^{15}-\frac{7}{1224}a^{12}-\frac{49}{1020}a^{9}+\frac{73}{306}a^{6}+\frac{193}{170}a^{3}+\frac{23}{136}$ (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)]
| |
Regulator: | \( 6280321.20776 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 6280321.20776 \cdot 3}{6\cdot\sqrt{998220592474975337089609728}}\cr\approx \mathstrut & 1.51690161434 \end{aligned}\] (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
\(\Q(\sqrt{-3}) \), 3.1.2028.1 x3, 3.1.18252.1 x3, 3.1.108.1 x3, 3.1.4563.1 x3, 6.0.12338352.2, 6.0.999406512.1, 6.0.34992.1, 6.0.62462907.1, 9.1.18241167657024.4 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 | ${\href{/padicField/5.2.0.1}{2} }^{9}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(3\) | 3.6.7.4 | $x^{6} + 3 x^{2} + 3$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ |
3.6.7.4 | $x^{6} + 3 x^{2} + 3$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ | |
3.6.7.4 | $x^{6} + 3 x^{2} + 3$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ | |
\(13\) | 13.9.6.1 | $x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
13.9.6.1 | $x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |