Normalized defining polynomial
\( x^{18} - 2 x^{17} + 501 x^{16} + 14648 x^{15} + 272582 x^{14} + 3144591 x^{13} + 27063010 x^{12} + 177529456 x^{11} + 942236178 x^{10} + 4085550909 x^{9} + 15020023463 x^{8} + 47124023126 x^{7} + 130555515789 x^{6} + 316870835959 x^{5} + 684237290359 x^{4} + 1237214238291 x^{3} + 1844496811058 x^{2} + 1914036738569 x + 1310138919769 \)
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: | \(-262152748150496553757816558021167153326662254632938777325568=-\,2^{12}\cdot 3^{9}\cdot 29^{6}\cdot 1129^{14}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $2000.00$ | 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: | $2, 3, 29, 1129$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{1073} a^{12} - \frac{446}{1073} a^{11} + \frac{427}{1073} a^{10} + \frac{462}{1073} a^{9} + \frac{440}{1073} a^{8} - \frac{228}{1073} a^{7} - \frac{110}{1073} a^{6} - \frac{156}{1073} a^{5} + \frac{352}{1073} a^{4} + \frac{23}{1073} a^{3} - \frac{117}{1073} a^{2} + \frac{489}{1073} a - \frac{405}{1073}$, $\frac{1}{1073} a^{13} + \frac{16}{1073} a^{11} - \frac{90}{1073} a^{10} + \frac{476}{1073} a^{9} - \frac{347}{1073} a^{8} + \frac{137}{1073} a^{7} + \frac{142}{1073} a^{6} + \frac{521}{1073} a^{5} + \frac{357}{1073} a^{4} + \frac{484}{1073} a^{3} - \frac{189}{1073} a^{2} - \frac{130}{1073} a - \frac{366}{1073}$, $\frac{1}{1073} a^{14} - \frac{465}{1073} a^{11} + \frac{82}{1073} a^{10} - \frac{228}{1073} a^{9} - \frac{465}{1073} a^{8} - \frac{502}{1073} a^{7} + \frac{135}{1073} a^{6} - \frac{366}{1073} a^{5} + \frac{217}{1073} a^{4} + \frac{516}{1073} a^{3} - \frac{404}{1073} a^{2} + \frac{394}{1073} a + \frac{42}{1073}$, $\frac{1}{33263} a^{15} + \frac{9}{33263} a^{14} + \frac{1}{33263} a^{13} - \frac{1}{33263} a^{12} + \frac{350}{33263} a^{11} - \frac{4249}{33263} a^{10} - \frac{16222}{33263} a^{9} + \frac{5986}{33263} a^{8} - \frac{277}{899} a^{7} - \frac{1764}{33263} a^{6} + \frac{1243}{33263} a^{5} - \frac{4453}{33263} a^{4} + \frac{11104}{33263} a^{3} - \frac{13726}{33263} a^{2} + \frac{7170}{33263} a + \frac{238}{1073}$, $\frac{1}{33263} a^{16} + \frac{13}{33263} a^{14} - \frac{10}{33263} a^{13} - \frac{13}{33263} a^{12} + \frac{15479}{33263} a^{11} + \frac{3853}{33263} a^{10} - \frac{7821}{33263} a^{9} - \frac{4944}{33263} a^{8} - \frac{4445}{33263} a^{7} + \frac{4068}{33263} a^{6} + \frac{8354}{33263} a^{5} + \frac{224}{1073} a^{4} - \frac{7704}{33263} a^{3} + \frac{3604}{33263} a^{2} - \frac{2840}{33263} a - \frac{375}{1073}$, $\frac{1}{2466586928874577805181282359921918162895855008408740499703778337} a^{17} - \frac{11224461886879445830344994090955794994948323600316771641907}{2466586928874577805181282359921918162895855008408740499703778337} a^{16} - \frac{19958685787225900665665373819962321798416261182548592313444}{2466586928874577805181282359921918162895855008408740499703778337} a^{15} + \frac{8815985702125737028812652974356466292162509028903942221052}{2466586928874577805181282359921918162895855008408740499703778337} a^{14} - \frac{493736488329843227543574662532672147956030736887588214088884}{2466586928874577805181282359921918162895855008408740499703778337} a^{13} - \frac{226884686641578697357188430404852029981445266097597028623393}{2466586928874577805181282359921918162895855008408740499703778337} a^{12} + \frac{249346739249836824040785234881505126267965537103185505181450660}{2466586928874577805181282359921918162895855008408740499703778337} a^{11} + \frac{836863240213302068644256465627145739974957581408082098583074804}{2466586928874577805181282359921918162895855008408740499703778337} a^{10} + \frac{622547319224250035163632156314496502691737767393344165738062174}{2466586928874577805181282359921918162895855008408740499703778337} a^{9} - \frac{810822474247530404601931183703233123660928430883406493810911995}{2466586928874577805181282359921918162895855008408740499703778337} a^{8} + \frac{446542398395861970809761083496773748135467756457848068725872900}{2466586928874577805181282359921918162895855008408740499703778337} a^{7} - \frac{831549807834631259047966530500890400606852525969346572939925355}{2466586928874577805181282359921918162895855008408740499703778337} a^{6} + \frac{80616256951738842192007300207852906798906828218252112482375846}{2466586928874577805181282359921918162895855008408740499703778337} a^{5} - \frac{1019915211695583278598045731705105850349855516056741748736119312}{2466586928874577805181282359921918162895855008408740499703778337} a^{4} - \frac{542335464666831082628563032151014318931049466696190346195811441}{2466586928874577805181282359921918162895855008408740499703778337} a^{3} + \frac{420084309950644917296720734796005576692512657975868423324833923}{2466586928874577805181282359921918162895855008408740499703778337} a^{2} - \frac{154731064823687221733280278261826136640844021552180390729906274}{2466586928874577805181282359921918162895855008408740499703778337} a + \frac{1064263870253594120732238664167565471672091612434667519726}{2154952747238217463178630995735605102244911606288536387149}$
Class group and class number
$C_{5}\times C_{105}\times C_{86358510}$, which has order $45338217750$ (assuming GRH)
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{119965114913681054968989994797591538883345964163878}{85054721685330269144182150342135109065374310634784155162199253} a^{17} + \frac{389828738363091919327050596859804797246794940241519}{85054721685330269144182150342135109065374310634784155162199253} a^{16} - \frac{60878514532437295675657008753083634258258863567671197}{85054721685330269144182150342135109065374310634784155162199253} a^{15} - \frac{1680546995090014114194884820133591986050566538782268268}{85054721685330269144182150342135109065374310634784155162199253} a^{14} - \frac{991812280884255760295524142642961769657658816334055773}{2743700699526782875618779043294680937592719697896263069748363} a^{13} - \frac{9270789742386383597079646273752189393508469566083762885}{2298776261765682949842760820057705650415521909048220409789169} a^{12} - \frac{2896401574366933841299320750017196748893444374327921535849}{85054721685330269144182150342135109065374310634784155162199253} a^{11} - \frac{18581194934004445494589781082008484473296482791959867072895}{85054721685330269144182150342135109065374310634784155162199253} a^{10} - \frac{97611970985766161148201766440663161485558070235912187518123}{85054721685330269144182150342135109065374310634784155162199253} a^{9} - \frac{419053884273364621725786884060285146104590084204882901097503}{85054721685330269144182150342135109065374310634784155162199253} a^{8} - \frac{1545841773404692063960197237497003996966752102985043774258273}{85054721685330269144182150342135109065374310634784155162199253} a^{7} - \frac{4847490245113757087893123202965568139785323756031471364409122}{85054721685330269144182150342135109065374310634784155162199253} a^{6} - \frac{13532478093037159992315165155453198713752319441289936254172376}{85054721685330269144182150342135109065374310634784155162199253} a^{5} - \frac{32207364882748533828197890574840123899709731324681798682612148}{85054721685330269144182150342135109065374310634784155162199253} a^{4} - \frac{68471775601969907949458260484012404019386652779868494939547916}{85054721685330269144182150342135109065374310634784155162199253} a^{3} - \frac{3073979405819457258617818221972944975547302431529119560658251}{2298776261765682949842760820057705650415521909048220409789169} a^{2} - \frac{177293479937858998189777557807166291183249146023723266309040143}{85054721685330269144182150342135109065374310634784155162199253} a - \frac{83485160786253143652255671954343649002700452429683551377}{74308715422007498730297620542607072491203848492708151281} \) (order $6$) | 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: | \( 2974025056676.291 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_6\times S_3$ (as 18T6):
| A solvable group of order 36 |
| The 18 conjugacy class representatives for $S_3 \times C_6$ |
| Character table for $S_3 \times C_6$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.3.1274641.1, 3.3.130964.1, 6.0.43867161329787.1, 6.0.463092370992.1, 9.9.3649484816717273142496096064.1 |
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 12 sibling: | 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 | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$ |
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 | Data not computed | ||||||
| $3$ | 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.12.6.2 | $x^{12} + 108 x^{6} - 243 x^{2} + 2916$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| $29$ | 29.6.0.1 | $x^{6} - x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 29.12.6.1 | $x^{12} + 146334 x^{6} - 20511149 x^{2} + 5353409889$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| 1129 | Data not computed | ||||||