Normalized defining polynomial
\( x^{18} + 546 x^{16} + 97461 x^{14} + 7758660 x^{12} + 321369048 x^{10} + 7362272736 x^{8} + 93766723596 x^{6} + 635072713821 x^{4} + 2051773383114 x^{2} + 2393735613633 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-4129014843306746341732347206220114431311872=-\,2^{18}\cdot 3^{27}\cdot 7^{9}\cdot 13^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $233.10$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 7, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3276=2^{2}\cdot 3^{2}\cdot 7\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3276}(1,·)$, $\chi_{3276}(1091,·)$, $\chi_{3276}(2435,·)$, $\chi_{3276}(1093,·)$, $\chi_{3276}(2183,·)$, $\chi_{3276}(2185,·)$, $\chi_{3276}(3275,·)$, $\chi_{3276}(1933,·)$, $\chi_{3276}(335,·)$, $\chi_{3276}(3025,·)$, $\chi_{3276}(1427,·)$, $\chi_{3276}(2519,·)$, $\chi_{3276}(757,·)$, $\chi_{3276}(841,·)$, $\chi_{3276}(1849,·)$, $\chi_{3276}(251,·)$, $\chi_{3276}(2941,·)$, $\chi_{3276}(1343,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{49} a^{4}$, $\frac{1}{49} a^{5}$, $\frac{1}{13377} a^{6}$, $\frac{1}{13377} a^{7}$, $\frac{1}{93639} a^{8}$, $\frac{1}{93639} a^{9}$, $\frac{1}{655473} a^{10}$, $\frac{1}{655473} a^{11}$, $\frac{1}{894720645} a^{12} + \frac{2}{468195} a^{8} - \frac{1}{5}$, $\frac{1}{894720645} a^{13} + \frac{2}{468195} a^{9} - \frac{1}{5} a$, $\frac{1}{50104356120} a^{14} + \frac{1}{2184910} a^{10} - \frac{1}{187278} a^{8} - \frac{1}{26754} a^{6} - \frac{1}{98} a^{4} - \frac{2}{35} a^{2} + \frac{3}{8}$, $\frac{1}{50104356120} a^{15} + \frac{1}{2184910} a^{11} - \frac{1}{187278} a^{9} - \frac{1}{26754} a^{7} - \frac{1}{98} a^{5} - \frac{2}{35} a^{3} + \frac{3}{8} a$, $\frac{1}{148709728964160} a^{16} + \frac{151}{21244246994880} a^{14} - \frac{139}{758723106960} a^{12} - \frac{131}{231600460} a^{10} - \frac{327}{66171560} a^{8} + \frac{5}{810264} a^{6} + \frac{1221}{207760} a^{4} + \frac{879}{118720} a^{2} - \frac{6831}{16960}$, $\frac{1}{148709728964160} a^{17} + \frac{151}{21244246994880} a^{15} - \frac{139}{758723106960} a^{13} - \frac{131}{231600460} a^{11} - \frac{327}{66171560} a^{9} + \frac{5}{810264} a^{7} + \frac{1221}{207760} a^{5} + \frac{879}{118720} a^{3} - \frac{6831}{16960} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{532}\times C_{6916}$, which has order $117737984$ (assuming GRH)
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 400417.1364448253 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times C_6$ (as 18T2):
| An abelian group of order 18 |
| The 18 conjugacy class representatives for $C_6 \times C_3$ |
| Character table for $C_6 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{-273}) \), \(\Q(\zeta_{9})^+\), 3.3.169.1, 3.3.13689.2, 3.3.13689.1, 6.0.949282431552.8, 6.0.220066846272.2, 6.0.160428730932288.1, 6.0.160428730932288.2, 9.9.2565164201769.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.6.3 | $x^{6} + 2 x^{4} + x^{2} - 7$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.6.6.3 | $x^{6} + 2 x^{4} + x^{2} - 7$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.3 | $x^{6} + 2 x^{4} + x^{2} - 7$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 3 | Data not computed | ||||||
| $7$ | 7.6.3.2 | $x^{6} - 49 x^{2} + 686$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 7.6.3.2 | $x^{6} - 49 x^{2} + 686$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 7.6.3.2 | $x^{6} - 49 x^{2} + 686$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 13 | Data not computed | ||||||