Normalized defining polynomial
\( x^{18} - 6 x^{17} - 38 x^{16} + 280 x^{15} + 350 x^{14} - 4460 x^{13} + 932 x^{12} + 30696 x^{11} - 23739 x^{10} - 93216 x^{9} + 86432 x^{8} + 120652 x^{7} - 101484 x^{6} - 59266 x^{5} + 34798 x^{4} + 12148 x^{3} - 3288 x^{2} - 564 x + 109 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[18, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(13620176948980906425236472453660672=2^{18}\cdot 3^{9}\cdot 1129^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $78.77$ | 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: | $2, 3, 1129$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4199} a^{16} - \frac{639}{4199} a^{15} - \frac{30}{323} a^{14} - \frac{462}{4199} a^{13} - \frac{523}{4199} a^{12} - \frac{1362}{4199} a^{11} - \frac{115}{323} a^{10} + \frac{1896}{4199} a^{9} + \frac{1044}{4199} a^{8} + \frac{9}{4199} a^{7} - \frac{617}{4199} a^{6} + \frac{116}{4199} a^{5} + \frac{92}{323} a^{4} - \frac{1436}{4199} a^{3} - \frac{1743}{4199} a^{2} - \frac{14}{4199} a - \frac{1999}{4199}$, $\frac{1}{495592318845721493905131846119} a^{17} + \frac{52622567534171960477313678}{495592318845721493905131846119} a^{16} - \frac{86845420254241814164496318677}{495592318845721493905131846119} a^{15} - \frac{5311756428627284634882457923}{17089390305024879100176960211} a^{14} + \frac{177529547592763400170761799217}{495592318845721493905131846119} a^{13} + \frac{8098657668409352965639563021}{29152489343865970229713638007} a^{12} + \frac{11472131985860705740674980584}{29152489343865970229713638007} a^{11} - \frac{178821181857095500771481582866}{495592318845721493905131846119} a^{10} + \frac{107024398583711988959469830284}{495592318845721493905131846119} a^{9} - \frac{53703031825800881969451675014}{495592318845721493905131846119} a^{8} - \frac{169091363465271992028364386113}{495592318845721493905131846119} a^{7} + \frac{45681986622596379034022456795}{495592318845721493905131846119} a^{6} + \frac{233215486358487906847071043417}{495592318845721493905131846119} a^{5} + \frac{76387488063041928686492961090}{495592318845721493905131846119} a^{4} + \frac{192201465852386345675738666695}{495592318845721493905131846119} a^{3} - \frac{6999416075642892324762477954}{38122486065055499531163988163} a^{2} - \frac{9747462986942533398347937563}{29152489343865970229713638007} a + \frac{165597214091235437877460185439}{495592318845721493905131846119}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $17$ | 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: | \( 236007763005 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 36 |
| The 12 conjugacy class representatives for $D_{18}$ |
| Character table for $D_{18}$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), 3.3.1129.1, 6.6.2202579648.2, 9.9.1624709678881.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 18 sibling: | data not computed |
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 | $18$ | $18$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ | $18$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 | Data not computed | ||||||
| $3$ | 3.6.3.1 | $x^{6} - 6 x^{4} + 9 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.1 | $x^{6} - 6 x^{4} + 9 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.1 | $x^{6} - 6 x^{4} + 9 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 1129 | Data not computed | ||||||