Properties

Label 42.0.167...503.1
Degree $42$
Signature $[0, 21]$
Discriminant $-1.678\times 10^{82}$
Root discriminant $90.73$
Ramified primes $7, 29$
Class number not computed
Class group not computed
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - x^41 + 13*x^40 - 18*x^39 + 139*x^38 - 250*x^37 + 1451*x^36 + 321*x^35 + 11819*x^34 + 7069*x^33 + 101147*x^32 + 34723*x^31 + 906407*x^30 - 143662*x^29 + 2718926*x^28 - 434521*x^27 + 7093735*x^26 - 1731307*x^25 + 17818760*x^24 - 17036040*x^23 + 50814721*x^22 - 44435389*x^21 + 124550915*x^20 - 97201554*x^19 + 258557330*x^18 - 191477895*x^17 + 222793332*x^16 - 183429282*x^15 + 205234539*x^14 - 104619060*x^13 + 150123312*x^12 + 66144604*x^11 + 21062459*x^10 + 5929237*x^9 + 1586736*x^8 + 394344*x^7 + 106707*x^6 + 17635*x^5 + 2858*x^4 + 449*x^3 + 67*x^2 + 9*x + 1)
 
gp: K = bnfinit(x^42 - x^41 + 13*x^40 - 18*x^39 + 139*x^38 - 250*x^37 + 1451*x^36 + 321*x^35 + 11819*x^34 + 7069*x^33 + 101147*x^32 + 34723*x^31 + 906407*x^30 - 143662*x^29 + 2718926*x^28 - 434521*x^27 + 7093735*x^26 - 1731307*x^25 + 17818760*x^24 - 17036040*x^23 + 50814721*x^22 - 44435389*x^21 + 124550915*x^20 - 97201554*x^19 + 258557330*x^18 - 191477895*x^17 + 222793332*x^16 - 183429282*x^15 + 205234539*x^14 - 104619060*x^13 + 150123312*x^12 + 66144604*x^11 + 21062459*x^10 + 5929237*x^9 + 1586736*x^8 + 394344*x^7 + 106707*x^6 + 17635*x^5 + 2858*x^4 + 449*x^3 + 67*x^2 + 9*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 9, 67, 449, 2858, 17635, 106707, 394344, 1586736, 5929237, 21062459, 66144604, 150123312, -104619060, 205234539, -183429282, 222793332, -191477895, 258557330, -97201554, 124550915, -44435389, 50814721, -17036040, 17818760, -1731307, 7093735, -434521, 2718926, -143662, 906407, 34723, 101147, 7069, 11819, 321, 1451, -250, 139, -18, 13, -1, 1]);
 

\( x^{42} - x^{41} + 13 x^{40} - 18 x^{39} + 139 x^{38} - 250 x^{37} + 1451 x^{36} + 321 x^{35} + 11819 x^{34} + 7069 x^{33} + 101147 x^{32} + 34723 x^{31} + 906407 x^{30} - 143662 x^{29} + 2718926 x^{28} - 434521 x^{27} + 7093735 x^{26} - 1731307 x^{25} + 17818760 x^{24} - 17036040 x^{23} + 50814721 x^{22} - 44435389 x^{21} + 124550915 x^{20} - 97201554 x^{19} + 258557330 x^{18} - 191477895 x^{17} + 222793332 x^{16} - 183429282 x^{15} + 205234539 x^{14} - 104619060 x^{13} + 150123312 x^{12} + 66144604 x^{11} + 21062459 x^{10} + 5929237 x^{9} + 1586736 x^{8} + 394344 x^{7} + 106707 x^{6} + 17635 x^{5} + 2858 x^{4} + 449 x^{3} + 67 x^{2} + 9 x + 1 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-16\!\cdots\!503\)\(\medspace = -\,7^{35}\cdot 29^{36}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $90.73$
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:  $7, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $42$
This field is Galois and abelian over $\Q$.
Conductor:  \(203=7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{203}(1,·)$, $\chi_{203}(132,·)$, $\chi_{203}(136,·)$, $\chi_{203}(139,·)$, $\chi_{203}(141,·)$, $\chi_{203}(16,·)$, $\chi_{203}(152,·)$, $\chi_{203}(146,·)$, $\chi_{203}(20,·)$, $\chi_{203}(23,·)$, $\chi_{203}(24,·)$, $\chi_{203}(25,·)$, $\chi_{203}(30,·)$, $\chi_{203}(36,·)$, $\chi_{203}(165,·)$, $\chi_{203}(169,·)$, $\chi_{203}(170,·)$, $\chi_{203}(45,·)$, $\chi_{203}(52,·)$, $\chi_{203}(53,·)$, $\chi_{203}(54,·)$, $\chi_{203}(59,·)$, $\chi_{203}(190,·)$, $\chi_{203}(181,·)$, $\chi_{203}(65,·)$, $\chi_{203}(194,·)$, $\chi_{203}(197,·)$, $\chi_{203}(198,·)$, $\chi_{203}(199,·)$, $\chi_{203}(74,·)$, $\chi_{203}(78,·)$, $\chi_{203}(81,·)$, $\chi_{203}(82,·)$, $\chi_{203}(83,·)$, $\chi_{203}(88,·)$, $\chi_{203}(94,·)$, $\chi_{203}(103,·)$, $\chi_{203}(107,·)$, $\chi_{203}(110,·)$, $\chi_{203}(111,·)$, $\chi_{203}(117,·)$$\chi_{203}(123,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{17} a^{36} + \frac{2}{17} a^{35} - \frac{7}{17} a^{34} + \frac{4}{17} a^{33} - \frac{3}{17} a^{31} + \frac{4}{17} a^{30} + \frac{5}{17} a^{29} - \frac{7}{17} a^{28} + \frac{1}{17} a^{27} - \frac{5}{17} a^{26} - \frac{7}{17} a^{24} - \frac{2}{17} a^{23} + \frac{8}{17} a^{22} - \frac{1}{17} a^{21} - \frac{5}{17} a^{20} + \frac{4}{17} a^{19} - \frac{7}{17} a^{18} - \frac{7}{17} a^{17} - \frac{8}{17} a^{16} + \frac{2}{17} a^{15} - \frac{3}{17} a^{14} - \frac{5}{17} a^{12} + \frac{1}{17} a^{11} - \frac{2}{17} a^{10} - \frac{4}{17} a^{9} + \frac{6}{17} a^{8} + \frac{7}{17} a^{7} + \frac{8}{17} a^{5} - \frac{5}{17} a^{4} + \frac{8}{17} a^{3} - \frac{1}{17} a^{2} + \frac{1}{17} a + \frac{8}{17}$, $\frac{1}{1169817331145812311507795307473961104451} a^{37} - \frac{895343494208065975017290171578583050}{1169817331145812311507795307473961104451} a^{36} + \frac{160773916279889866607496026346993514828}{1169817331145812311507795307473961104451} a^{35} - \frac{171518038210386658307703508405936511421}{1169817331145812311507795307473961104451} a^{34} - \frac{245097033182016027243055821579113600923}{1169817331145812311507795307473961104451} a^{33} + \frac{507184937600243352611365337414991497504}{1169817331145812311507795307473961104451} a^{32} + \frac{220428198991352863020756029272060294606}{1169817331145812311507795307473961104451} a^{31} - \frac{298234449308169529603167248251484626365}{1169817331145812311507795307473961104451} a^{30} - \frac{575447777461496216427177510871090056307}{1169817331145812311507795307473961104451} a^{29} + \frac{378527888703165218136843044860168409650}{1169817331145812311507795307473961104451} a^{28} - \frac{112312367925292668841524885354113387328}{1169817331145812311507795307473961104451} a^{27} + \frac{172487876154503915266683479859616461115}{1169817331145812311507795307473961104451} a^{26} - \frac{502398780693905755582236233622162830865}{1169817331145812311507795307473961104451} a^{25} - \frac{151531529484216715885231650544226647491}{1169817331145812311507795307473961104451} a^{24} - \frac{578211099245688555749809817954266875908}{1169817331145812311507795307473961104451} a^{23} + \frac{221796158072143799643330633975719415774}{1169817331145812311507795307473961104451} a^{22} + \frac{566822837187294289377345176522154227266}{1169817331145812311507795307473961104451} a^{21} + \frac{460066393234877141703265628280247989828}{1169817331145812311507795307473961104451} a^{20} + \frac{454944088151697189068983821163902419062}{1169817331145812311507795307473961104451} a^{19} - \frac{471465510578853563459306734647781265837}{1169817331145812311507795307473961104451} a^{18} + \frac{330330607957686290375960454413275161275}{1169817331145812311507795307473961104451} a^{17} + \frac{31266270928603736161621278185553948083}{1169817331145812311507795307473961104451} a^{16} + \frac{10428536426635327641758251244366930869}{68812784185047783029870312204350653203} a^{15} - \frac{209320910948607168009734773491877121815}{1169817331145812311507795307473961104451} a^{14} + \frac{31844770464134543357371375911832330067}{1169817331145812311507795307473961104451} a^{13} + \frac{197255663610414606981409731029621410305}{1169817331145812311507795307473961104451} a^{12} + \frac{271817048875548584865614625341953361653}{1169817331145812311507795307473961104451} a^{11} - \frac{200862680609152994513333848103535414309}{1169817331145812311507795307473961104451} a^{10} - \frac{14189654967356037809633656159223254490}{68812784185047783029870312204350653203} a^{9} + \frac{419965285915048400012096509442848285570}{1169817331145812311507795307473961104451} a^{8} + \frac{358048364648146214739181429785597140571}{1169817331145812311507795307473961104451} a^{7} - \frac{463765286829607205257141048796346724559}{1169817331145812311507795307473961104451} a^{6} - \frac{371340896206970136039110352446164583128}{1169817331145812311507795307473961104451} a^{5} - \frac{280984648355901134418075199175633496396}{1169817331145812311507795307473961104451} a^{4} + \frac{439495428917787515093213682220157268240}{1169817331145812311507795307473961104451} a^{3} + \frac{293386143580406684986846186231003330636}{1169817331145812311507795307473961104451} a^{2} - \frac{461217233034247514498828151326151552360}{1169817331145812311507795307473961104451} a + \frac{476331801798878110671542047940029959226}{1169817331145812311507795307473961104451}$, $\frac{1}{1169817331145812311507795307473961104451} a^{38} - \frac{4975361606930414718316929104611310495}{1169817331145812311507795307473961104451} a^{36} - \frac{369467703901570116103492167974686422239}{1169817331145812311507795307473961104451} a^{35} + \frac{309763364618405139483689018719350696271}{1169817331145812311507795307473961104451} a^{34} - \frac{98934018102510189722632308252022518814}{1169817331145812311507795307473961104451} a^{33} + \frac{199313260060620351407534064516581507811}{1169817331145812311507795307473961104451} a^{32} - \frac{474654897574287254323264923515096067956}{1169817331145812311507795307473961104451} a^{31} - \frac{174965819321010102402442614039700077641}{1169817331145812311507795307473961104451} a^{30} + \frac{571212020614267547492869427180980921228}{1169817331145812311507795307473961104451} a^{29} + \frac{309809728638267636003209197738956257169}{1169817331145812311507795307473961104451} a^{28} + \frac{581947773426485833359143102670376428485}{1169817331145812311507795307473961104451} a^{27} - \frac{485493980415625195826321262843131559534}{1169817331145812311507795307473961104451} a^{26} - \frac{189365764912540707769995709286040594131}{1169817331145812311507795307473961104451} a^{25} - \frac{18817140558066635286493530222234128972}{1169817331145812311507795307473961104451} a^{24} - \frac{482423809866380811858464382102230321493}{1169817331145812311507795307473961104451} a^{23} - \frac{237159237854534616431942715407366908640}{1169817331145812311507795307473961104451} a^{22} + \frac{561247418216242737862292064006666165051}{1169817331145812311507795307473961104451} a^{21} - \frac{368796576280314536494563863324390123266}{1169817331145812311507795307473961104451} a^{20} + \frac{448899702980654855145812012685892196400}{1169817331145812311507795307473961104451} a^{19} - \frac{399313835488423938325830988622051292991}{1169817331145812311507795307473961104451} a^{18} + \frac{545511320162883605031050907203154140292}{1169817331145812311507795307473961104451} a^{17} - \frac{187466506566868487805319348852089127582}{1169817331145812311507795307473961104451} a^{16} - \frac{558497408586777062976512245984966222044}{1169817331145812311507795307473961104451} a^{15} + \frac{18080732254285868068547566800269454864}{68812784185047783029870312204350653203} a^{14} + \frac{441171711800886239071933076816556027686}{1169817331145812311507795307473961104451} a^{13} - \frac{74100725943657836841105169882600388143}{1169817331145812311507795307473961104451} a^{12} - \frac{493925285802328428626907835924871411987}{1169817331145812311507795307473961104451} a^{11} + \frac{343462464696216802171515305151876197485}{1169817331145812311507795307473961104451} a^{10} + \frac{462031213612229355663937309719231689430}{1169817331145812311507795307473961104451} a^{9} + \frac{509000114107179264597560220509445783958}{1169817331145812311507795307473961104451} a^{8} - \frac{270644055122219079400318669857221268519}{1169817331145812311507795307473961104451} a^{7} - \frac{46295991590365043987972500428906302965}{1169817331145812311507795307473961104451} a^{6} + \frac{471522087477669652222656550007636164442}{1169817331145812311507795307473961104451} a^{5} - \frac{170040460442725180104299945139690317480}{1169817331145812311507795307473961104451} a^{4} - \frac{197764359423215847877125399785564275864}{1169817331145812311507795307473961104451} a^{3} - \frac{476457867298832154726373927640680875173}{1169817331145812311507795307473961104451} a^{2} + \frac{262809540847662275708525794792639722231}{1169817331145812311507795307473961104451} a + \frac{328299396559587286351330845736395447906}{1169817331145812311507795307473961104451}$, $\frac{1}{1169817331145812311507795307473961104451} a^{39} + \frac{25978583512847027438143745054618464882}{1169817331145812311507795307473961104451} a^{36} + \frac{368662808095168412663385564373705127077}{1169817331145812311507795307473961104451} a^{35} - \frac{56919805941004083405660623718283548507}{1169817331145812311507795307473961104451} a^{34} - \frac{16545736910295018597887618310770090190}{1169817331145812311507795307473961104451} a^{33} + \frac{43875619883649282091607988278158943112}{1169817331145812311507795307473961104451} a^{32} + \frac{397996978324084542757030140449957779680}{1169817331145812311507795307473961104451} a^{31} - \frac{134922005280583014230397294183528016637}{1169817331145812311507795307473961104451} a^{30} - \frac{41516251582440733145781509292104722883}{1169817331145812311507795307473961104451} a^{29} + \frac{520359607073001296353322600588619149158}{1169817331145812311507795307473961104451} a^{28} + \frac{224633413026268313923378208737271015388}{1169817331145812311507795307473961104451} a^{27} - \frac{332916446628307778460097012960842574963}{1169817331145812311507795307473961104451} a^{26} - \frac{178601704041056458541154916839314852606}{1169817331145812311507795307473961104451} a^{25} + \frac{300873015117768611927257756485941691487}{1169817331145812311507795307473961104451} a^{24} - \frac{36557395334128187465690383986193251136}{1169817331145812311507795307473961104451} a^{23} - \frac{443371789909147822216177785459972216319}{1169817331145812311507795307473961104451} a^{22} + \frac{202617466686826722781077260927031004533}{1169817331145812311507795307473961104451} a^{21} + \frac{283742592556293113514995951570630384343}{1169817331145812311507795307473961104451} a^{20} + \frac{430037102016282695531088140920977930574}{1169817331145812311507795307473961104451} a^{19} + \frac{339838775672848166803546776372131351315}{1169817331145812311507795307473961104451} a^{18} - \frac{287208334636550796973889985933739406017}{1169817331145812311507795307473961104451} a^{17} + \frac{578653186408672851727219797674862167563}{1169817331145812311507795307473961104451} a^{16} + \frac{298884314363770512994344717817284762212}{1169817331145812311507795307473961104451} a^{15} + \frac{124298174589592783861589274485593898703}{1169817331145812311507795307473961104451} a^{14} + \frac{345759764880414722351604287011360608259}{1169817331145812311507795307473961104451} a^{13} - \frac{465765469021624052540005116489019742349}{1169817331145812311507795307473961104451} a^{12} - \frac{370940067289640936844422681756746267357}{1169817331145812311507795307473961104451} a^{11} + \frac{382418460190998997701048322945768058808}{1169817331145812311507795307473961104451} a^{10} + \frac{16656357643028596142544281599146386974}{68812784185047783029870312204350653203} a^{9} - \frac{416872841341266189268763577492929992288}{1169817331145812311507795307473961104451} a^{8} - \frac{468980393591489491346892299219465003750}{1169817331145812311507795307473961104451} a^{7} + \frac{264398570252603572651161428516701102135}{1169817331145812311507795307473961104451} a^{6} - \frac{399089046018024362757066186403737586480}{1169817331145812311507795307473961104451} a^{5} - \frac{498541213060397958503631236611875938569}{1169817331145812311507795307473961104451} a^{4} + \frac{536368060922559588621804478630166033445}{1169817331145812311507795307473961104451} a^{3} + \frac{157686134719178913729386028996176838123}{1169817331145812311507795307473961104451} a^{2} - \frac{492847699247909294237992817498148464367}{1169817331145812311507795307473961104451} a - \frac{327855519300206673465522497597813420178}{1169817331145812311507795307473961104451}$, $\frac{1}{1169817331145812311507795307473961104451} a^{40} - \frac{6087611092784999202840150859442583390}{1169817331145812311507795307473961104451} a^{36} - \frac{27689539438786735622638423470694699062}{1169817331145812311507795307473961104451} a^{35} - \frac{45361793674633254811443386842616301609}{1169817331145812311507795307473961104451} a^{34} - \frac{302580390135549111586329677847684504251}{1169817331145812311507795307473961104451} a^{33} - \frac{238189232328823662336166631520048006956}{1169817331145812311507795307473961104451} a^{32} + \frac{19784167220306799675994223846673347050}{68812784185047783029870312204350653203} a^{31} - \frac{388836062738100136951415314516867884621}{1169817331145812311507795307473961104451} a^{30} + \frac{17878677132905867571724992502728347687}{1169817331145812311507795307473961104451} a^{29} + \frac{299878268459777205296327734028094871746}{1169817331145812311507795307473961104451} a^{28} + \frac{130510667618701239449976625603795817595}{1169817331145812311507795307473961104451} a^{27} + \frac{20048502776631888728040288203883176975}{68812784185047783029870312204350653203} a^{26} + \frac{433258081614332054603560560857419664772}{1169817331145812311507795307473961104451} a^{25} + \frac{110552291107997296199218768452653788039}{1169817331145812311507795307473961104451} a^{24} - \frac{510060980597425888013017140537852866714}{1169817331145812311507795307473961104451} a^{23} - \frac{79197921432743037549005196795280090627}{1169817331145812311507795307473961104451} a^{22} - \frac{348331525838719930187146725391282921268}{1169817331145812311507795307473961104451} a^{21} + \frac{131718548357537744990336248260750594485}{1169817331145812311507795307473961104451} a^{20} - \frac{250917413917564060665764275174278353897}{1169817331145812311507795307473961104451} a^{19} - \frac{24346631541381784218251645098187932752}{1169817331145812311507795307473961104451} a^{18} + \frac{336492367256933132921318529827346831239}{1169817331145812311507795307473961104451} a^{17} - \frac{375507278744986810754096951998954448036}{1169817331145812311507795307473961104451} a^{16} - \frac{492344910200757170362937256721876690157}{1169817331145812311507795307473961104451} a^{15} - \frac{228093667819576286323681298168147600283}{1169817331145812311507795307473961104451} a^{14} + \frac{107389708443994828598665251949948962092}{1169817331145812311507795307473961104451} a^{13} + \frac{12641112901468084839122732398270930282}{68812784185047783029870312204350653203} a^{12} + \frac{16848560101781735605468943301713499153}{1169817331145812311507795307473961104451} a^{11} + \frac{8774543276568671866014729608220050982}{68812784185047783029870312204350653203} a^{10} + \frac{11984169739363061107193086985339543285}{1169817331145812311507795307473961104451} a^{9} + \frac{250517261549517197129506276901050671728}{1169817331145812311507795307473961104451} a^{8} + \frac{226186010007589824499721866959257906980}{1169817331145812311507795307473961104451} a^{7} + \frac{278048441171148013892241369999497072231}{1169817331145812311507795307473961104451} a^{6} - \frac{4557790314593438867514265439417142337}{68812784185047783029870312204350653203} a^{5} - \frac{299731690957928577093741089029414534259}{1169817331145812311507795307473961104451} a^{4} - \frac{71944567415700592822887334118627739702}{1169817331145812311507795307473961104451} a^{3} - \frac{315612980946720789647198327600536970743}{1169817331145812311507795307473961104451} a^{2} - \frac{106261080706962696318394646026217701905}{1169817331145812311507795307473961104451} a + \frac{581060035490262322872377266510728018978}{1169817331145812311507795307473961104451}$, $\frac{1}{1169817331145812311507795307473961104451} a^{41} + \frac{17698925526954109832814708263047448343}{1169817331145812311507795307473961104451} a^{36} - \frac{529034224095175291265563947106050946047}{1169817331145812311507795307473961104451} a^{35} - \frac{401450433622434246754685788237207111674}{1169817331145812311507795307473961104451} a^{34} + \frac{52964446002824533430366991149867086381}{1169817331145812311507795307473961104451} a^{33} - \frac{33987288983520582527222327001952766174}{1169817331145812311507795307473961104451} a^{32} + \frac{537788641756528570650873720868123522727}{1169817331145812311507795307473961104451} a^{31} + \frac{268895201564295918080266121343817288473}{1169817331145812311507795307473961104451} a^{30} + \frac{116892471268323355072113316152138320758}{1169817331145812311507795307473961104451} a^{29} - \frac{492970475980634862478277779020011633611}{1169817331145812311507795307473961104451} a^{28} + \frac{31067029535968596261887380749976366240}{1169817331145812311507795307473961104451} a^{27} - \frac{59816174208169306889571316415860276537}{1169817331145812311507795307473961104451} a^{26} + \frac{501845118943880820399562321430768242800}{1169817331145812311507795307473961104451} a^{25} - \frac{502740947725029257415567930106352855163}{1169817331145812311507795307473961104451} a^{24} + \frac{439499545088617049483076954021887819980}{1169817331145812311507795307473961104451} a^{23} + \frac{492971624314649305456978479768581621183}{1169817331145812311507795307473961104451} a^{22} - \frac{350394995317697192074196174949874054299}{1169817331145812311507795307473961104451} a^{21} - \frac{532524422200840669535112574740032758784}{1169817331145812311507795307473961104451} a^{20} + \frac{278076067558788846989376464479380799391}{1169817331145812311507795307473961104451} a^{19} - \frac{429519907880901061228722812858064246532}{1169817331145812311507795307473961104451} a^{18} + \frac{241499296170525606727921580921670605635}{1169817331145812311507795307473961104451} a^{17} - \frac{108249000047524415940393169261009654843}{1169817331145812311507795307473961104451} a^{16} + \frac{520920847108358011452819872123024079112}{1169817331145812311507795307473961104451} a^{15} - \frac{309743274799464126362948834550667200860}{1169817331145812311507795307473961104451} a^{14} + \frac{571555189235714389290017202255750226707}{1169817331145812311507795307473961104451} a^{13} + \frac{214186003670371231018475314209186117686}{1169817331145812311507795307473961104451} a^{12} - \frac{1826264059027358167602489300843234977}{1169817331145812311507795307473961104451} a^{11} - \frac{297540829178715134559668476712055569584}{1169817331145812311507795307473961104451} a^{10} - \frac{353361763330275318058669483040538400135}{1169817331145812311507795307473961104451} a^{9} + \frac{260540470263817890863326995188264939609}{1169817331145812311507795307473961104451} a^{8} - \frac{341505156323359942659994657011136863527}{1169817331145812311507795307473961104451} a^{7} - \frac{431377449541424897389646858916835692462}{1169817331145812311507795307473961104451} a^{6} - \frac{322305527436583930935880785865504352422}{1169817331145812311507795307473961104451} a^{5} - \frac{265884119640500866684250193161945252968}{1169817331145812311507795307473961104451} a^{4} - \frac{487211153726662897101788030194450809652}{1169817331145812311507795307473961104451} a^{3} - \frac{150706665295296961685895221889798651206}{1169817331145812311507795307473961104451} a^{2} + \frac{91916369458217261216961183473240202199}{1169817331145812311507795307473961104451} a + \frac{319972639222328185904946439189087033536}{1169817331145812311507795307473961104451}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $20$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{1186100698624582900239519268600628928}{68812784185047783029870312204350653203} a^{41} - \frac{593050349312291450119759634300314464}{68812784185047783029870312204350653203} a^{40} + \frac{14445012079677956035059859664029088016}{68812784185047783029870312204350653203} a^{39} - \frac{13216551487513116603976761521165240859}{68812784185047783029870312204350653203} a^{38} + \frac{149194523591277891951556673714693395872}{68812784185047783029870312204350653203} a^{37} - \frac{206678046735333570366736232553659590704}{68812784185047783029870312204350653203} a^{36} + \frac{1519013748284961364988890051883926883184}{68812784185047783029870312204350653203} a^{35} + \frac{1342454187146844881838950183615090405616}{68812784185047783029870312204350653203} a^{34} + \frac{13645114240673384645587615294422456728592}{68812784185047783029870312204350653203} a^{33} + \frac{15332893174076929520382014087952987384960}{68812784185047783029870312204350653203} a^{32} + \frac{119670440938100233700292971106932990040763}{68812784185047783029870312204350653203} a^{31} + \frac{98975867226297783800344170395906053224000}{68812784185047783029870312204350653203} a^{30} + \frac{1057419956058974115916590829139136066670464}{68812784185047783029870312204350653203} a^{29} + \frac{358194023558255190607540268231133403237152}{68812784185047783029870312204350653203} a^{28} + \frac{2795627485651155650035544520898996377006720}{68812784185047783029870312204350653203} a^{27} + \frac{1190241079638306662998530370487496573430416}{68812784185047783029870312204350653203} a^{26} + \frac{7113524142397604779198672202817346720523040}{68812784185047783029870312204350653203} a^{25} + \frac{2434277806836715207885795474461170260281703}{68812784185047783029870312204350653203} a^{24} + \frac{17385222105796744832808365236469576900524896}{68812784185047783029870312204350653203} a^{23} - \frac{8678486245646583861159940991647593367748832}{68812784185047783029870312204350653203} a^{22} + \frac{43301464548334823254625474578828974696664656}{68812784185047783029870312204350653203} a^{21} - \frac{15319408818841659003311655364645148477759328}{68812784185047783029870312204350653203} a^{20} + \frac{101282902768538641735555424445091306775909520}{68812784185047783029870312204350653203} a^{19} - \frac{22332474976468556109191641243489096330596480}{68812784185047783029870312204350653203} a^{18} + \frac{199662557300016500661691621818281765635048966}{68812784185047783029870312204350653203} a^{17} - \frac{31440714183447289528059198000241584795594864}{68812784185047783029870312204350653203} a^{16} + \frac{48007647831825071103750365809467813164258592}{68812784185047783029870312204350653203} a^{15} - \frac{1484949910517465785715679112380544599459888}{68812784185047783029870312204350653203} a^{14} + \frac{41595672981393094624801081200177216917697936}{68812784185047783029870312204350653203} a^{13} + \frac{76995164978369564809491651286082015477473536}{68812784185047783029870312204350653203} a^{12} + \frac{30001814081864313097746114540830513478258288}{68812784185047783029870312204350653203} a^{11} + \frac{216061521390126675204192078224130140696762755}{68812784185047783029870312204350653203} a^{10} + \frac{2543526865447208945659207155440512415341440}{68812784185047783029870312204350653203} a^{9} + \frac{665696390737214484040151087709436798733088}{68812784185047783029870312204350653203} a^{8} + \frac{170307073666056230983573679570096426482224}{68812784185047783029870312204350653203} a^{7} + \frac{40465604484625582516021559127213356822112}{68812784185047783029870312204350653203} a^{6} + \frac{6660167226473215946077086321633352971600}{68812784185047783029870312204350653203} a^{5} + \frac{1072489195992042495295148172943954399968}{68812784185047783029870312204350653203} a^{4} - \frac{14199422276057202244392332908220744896433}{68812784185047783029870312204350653203} a^{3} + \frac{24399785800277133947784396382641509376}{68812784185047783029870312204350653203} a^{2} + \frac{3177055442744418482784426612323113200}{68812784185047783029870312204350653203} a + \frac{296525174656145725059879817150157232}{68812784185047783029870312204350653203} \) (order $14$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$C_{42}$ (as 42T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 7.7.594823321.1, 14.0.291381688005381590432263.1, 21.21.142736986105602839685204351151303673689.1

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 $21^{2}$ $42$ $42$ R $21^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{7}$ $42$ $21^{2}$ R $42$ $21^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{21}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{6}$ $42$ $21^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{7}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$