Normalized defining polynomial
\( x^{27} - 8 x^{26} + 11 x^{25} + 79 x^{24} - 318 x^{23} + 272 x^{22} + 1817 x^{21} - 11487 x^{20} + 27792 x^{19} - 5214 x^{18} - 88324 x^{17} + 101476 x^{16} + 384139 x^{15} - 2894444 x^{14} + 8285833 x^{13} - 7065937 x^{12} - 6208737 x^{11} - 6594753 x^{10} + 51530498 x^{9} - 40765921 x^{8} + 72314518 x^{7} - 318855243 x^{6} + 340132420 x^{5} + 140246505 x^{4} - 148040525 x^{3} - 594053575 x^{2} + 805097125 x - 295863625 \)
Invariants
Degree: | $27$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
Signature: | $[1, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(-1089801024238603052304820653163822019730224609375\)\(\medspace = -\,5^{13}\cdot 7^{18}\cdot 67^{13}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
Root discriminant: | $60.14$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
Ramified primes: | $5, 7, 67$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
$|\Aut(K/\Q)|$: | $1$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} + \frac{1}{5} a^{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{6} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{7} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{8} + \frac{2}{5} a^{4}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{9} + \frac{2}{5} a^{5}$, $\frac{1}{25} a^{18} + \frac{1}{25} a^{17} + \frac{2}{25} a^{16} - \frac{2}{25} a^{15} + \frac{1}{25} a^{14} - \frac{1}{25} a^{13} - \frac{1}{25} a^{11} + \frac{9}{25} a^{9} - \frac{11}{25} a^{8} - \frac{12}{25} a^{7} - \frac{7}{25} a^{6} + \frac{11}{25} a^{5} - \frac{6}{25} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{19} + \frac{1}{25} a^{17} + \frac{1}{25} a^{16} - \frac{2}{25} a^{15} - \frac{2}{25} a^{14} + \frac{1}{25} a^{13} - \frac{1}{25} a^{12} + \frac{1}{25} a^{11} - \frac{1}{25} a^{10} + \frac{1}{5} a^{9} + \frac{9}{25} a^{8} - \frac{1}{5} a^{7} - \frac{12}{25} a^{6} + \frac{8}{25} a^{5} + \frac{11}{25} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{20} + \frac{1}{25} a^{16} + \frac{1}{25} a^{12} - \frac{9}{25} a^{8} + \frac{6}{25} a^{4}$, $\frac{1}{25} a^{21} + \frac{1}{25} a^{17} + \frac{1}{25} a^{13} - \frac{9}{25} a^{9} + \frac{6}{25} a^{5}$, $\frac{1}{2125} a^{22} - \frac{36}{2125} a^{21} - \frac{36}{2125} a^{20} + \frac{3}{425} a^{19} + \frac{43}{2125} a^{17} - \frac{108}{2125} a^{16} - \frac{138}{2125} a^{15} - \frac{2}{25} a^{14} - \frac{41}{425} a^{13} - \frac{46}{2125} a^{12} - \frac{94}{2125} a^{11} - \frac{149}{2125} a^{10} - \frac{137}{425} a^{9} + \frac{18}{425} a^{8} - \frac{863}{2125} a^{7} + \frac{1028}{2125} a^{6} - \frac{287}{2125} a^{5} + \frac{103}{425} a^{4} + \frac{22}{85} a^{3} - \frac{3}{17} a^{2} - \frac{2}{85} a + \frac{1}{17}$, $\frac{1}{1566125} a^{23} + \frac{1}{92125} a^{22} + \frac{27211}{1566125} a^{21} + \frac{28112}{1566125} a^{20} - \frac{52}{28475} a^{19} - \frac{10157}{1566125} a^{18} + \frac{52321}{1566125} a^{17} - \frac{75987}{1566125} a^{16} - \frac{54149}{1566125} a^{15} - \frac{31321}{313225} a^{14} + \frac{129764}{1566125} a^{13} + \frac{71503}{1566125} a^{12} - \frac{131611}{1566125} a^{11} - \frac{107352}{1566125} a^{10} - \frac{14587}{313225} a^{9} - \frac{494108}{1566125} a^{8} + \frac{610639}{1566125} a^{7} - \frac{669068}{1566125} a^{6} - \frac{269781}{1566125} a^{5} + \frac{154744}{313225} a^{4} + \frac{26787}{62645} a^{3} - \frac{17763}{62645} a^{2} + \frac{67}{935} a + \frac{49}{187}$, $\frac{1}{1566125} a^{24} - \frac{347}{1566125} a^{22} - \frac{16596}{1566125} a^{21} - \frac{48}{313225} a^{20} + \frac{5298}{1566125} a^{19} - \frac{5118}{313225} a^{18} - \frac{70726}{1566125} a^{17} + \frac{110757}{1566125} a^{16} - \frac{1976}{28475} a^{15} - \frac{152266}{1566125} a^{14} + \frac{14566}{313225} a^{13} - \frac{30143}{1566125} a^{12} - \frac{5054}{1566125} a^{11} + \frac{22887}{313225} a^{10} + \frac{5102}{1566125} a^{9} + \frac{120998}{313225} a^{8} - \frac{171074}{1566125} a^{7} - \frac{264587}{1566125} a^{6} + \frac{26288}{62645} a^{5} - \frac{18182}{313225} a^{4} + \frac{16963}{62645} a^{3} + \frac{1261}{5695} a^{2} - \frac{16}{187} a + \frac{69}{187}$, $\frac{1}{4019358014375} a^{25} - \frac{1092866}{4019358014375} a^{24} + \frac{476863}{4019358014375} a^{23} + \frac{733574131}{4019358014375} a^{22} + \frac{50923054101}{4019358014375} a^{21} - \frac{70866659987}{4019358014375} a^{20} - \frac{62376008723}{4019358014375} a^{19} + \frac{79492017619}{4019358014375} a^{18} - \frac{146356384442}{4019358014375} a^{17} + \frac{299501636793}{4019358014375} a^{16} - \frac{246968380411}{4019358014375} a^{15} + \frac{220948660616}{4019358014375} a^{14} - \frac{18324496293}{365396183125} a^{13} + \frac{51219224684}{4019358014375} a^{12} + \frac{281579707479}{4019358014375} a^{11} + \frac{16433562326}{236432824375} a^{10} - \frac{54779437001}{236432824375} a^{9} + \frac{114291735318}{236432824375} a^{8} - \frac{2289056378}{4019358014375} a^{7} + \frac{1574165690332}{4019358014375} a^{6} - \frac{71883675883}{160774320575} a^{5} + \frac{205226591736}{803871602875} a^{4} + \frac{2050424108}{14615847325} a^{3} + \frac{737604766}{3420730225} a^{2} + \frac{210878394}{479923345} a + \frac{112787458}{479923345}$, $\frac{1}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{26} - \frac{4513461660528888427991569493073674937760750255369714909153918022216}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{25} - \frac{410064483574045635555427618625359424317225932280089245690832798530276202}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{24} - \frac{13399847417108082311458417218431923098515592820899120678144928042639777364}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{23} - \frac{1841221176941680778636198050770345319651060199700292381177405602543606729749}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{22} - \frac{17950174076955799042632699913326695712812087451697353449869584155253777398224}{1972910292725710787163537901480013535502373171244563062237490501259674785410625} a^{21} - \frac{19888163621745446467741137189724647951559118230504029123502116433131122657451}{1972910292725710787163537901480013535502373171244563062237490501259674785410625} a^{20} + \frac{285621995123179985667257913057598047202089246293643884362254464342467803861949}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{19} + \frac{875561051395356442433580334904471846214900114514859876069603143486459204334948}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{18} - \frac{1933871007334067137722801421689650942924358227827162321600339491646442260535532}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{17} + \frac{2681929221981697407735213552767208180482240386252730548529618925636160544600169}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{16} + \frac{13614054681626008184042756617831329419018143550590812487394400738183663517943}{242657415682841433715301453123210220944142154752005082521188671277927914783125} a^{15} + \frac{657530197455033409434525410087951227499096546176220049161426305757021214399217}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{14} + \frac{1878086909965726154659734064671969072590048924463289061905855476077344701381054}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{13} + \frac{23135520372344323827708015652964702344909476369570164842156418663447652605562}{677267712428229076190468234836422556963501237889924633305407186999589851708125} a^{12} + \frac{573490749154096480385066661992177684256667115975829272681566937629485745927207}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{11} - \frac{42039817147728753323905301386643640373596870861296217645334134412358150564947}{743884208732645050897727405476070677320566933420081154614135762770041312531875} a^{10} - \frac{475859845084757160670604200505493372200828357686191699743619046310552105810424}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{9} + \frac{4486225755923723680824641095264153027638278496176094321053284068367888827340257}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{8} - \frac{12351244124047860835248035145842892509396271576389452552846027906886050504947943}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{7} - \frac{3687042986413462726163358486449831941687521586399218687824025864528562479317191}{9075387346538269620952274346808062263310916587724990086292456305794504012888875} a^{6} - \frac{3271419639977570543474932133456034215381419123251935882715551751305866733124269}{9075387346538269620952274346808062263310916587724990086292456305794504012888875} a^{5} + \frac{875979252537103876025278402267718262328618680962785412150265536987709578190937}{1815077469307653924190454869361612452662183317544998017258491261158900802577775} a^{4} + \frac{255528745868813682419974558160511843385927904247496537981509424044482191166587}{1815077469307653924190454869361612452662183317544998017258491261158900802577775} a^{3} + \frac{2751545162321599070427791141703515853346681554526408093301619196258586965047}{15783282341805686297308303211840108284018985369956504497899924010077398283285} a^{2} + \frac{560854573737487709118655268438947970635514241806287823972806669996870157088}{5418141699425832609523745878691380455708009903119397066443257495996718813665} a + \frac{121682527211440882607949895758835686999776433643535435147680465991331204285}{1083628339885166521904749175738276091141601980623879413288651499199343762733}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
Rank: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
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) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
Regulator: | \( 71364448730505.47 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
A solvable group of order 54 |
The 15 conjugacy class representatives for $D_{27}$ |
Character table for $D_{27}$ |
Intermediate fields
3.1.335.1, 9.1.12594450625.1 |
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 | $27$ | $27$ | R | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | $27$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | $27$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | $27$ | $27$ |
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 |
---|---|---|---|---|---|---|---|
$5$ | $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7 | Data not computed | ||||||
$67$ | $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |