Normalized defining polynomial
\( x^{32} + 12 x^{30} - 20 x^{29} + 134 x^{28} + 160 x^{27} + 806 x^{26} + 1660 x^{25} - 4368 x^{24} + 14920 x^{23} - 18238 x^{22} + 72540 x^{21} + 51278 x^{20} - 604520 x^{19} + 778586 x^{18} - 4167900 x^{17} + 6106776 x^{16} + 2072120 x^{15} - 7073306 x^{14} + 52233500 x^{13} - 159058772 x^{12} + 187842120 x^{11} - 109316122 x^{10} - 115761180 x^{9} + 821797637 x^{8} - 1733801640 x^{7} + 1884896234 x^{6} - 1217828880 x^{5} + 571687194 x^{4} - 291053480 x^{3} + 149005988 x^{2} - 41560600 x + 4354996 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $61.29$ | 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: | $2, 3, 5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(840=2^{3}\cdot 3\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(643,·)$, $\chi_{840}(517,·)$, $\chi_{840}(391,·)$, $\chi_{840}(13,·)$, $\chi_{840}(533,·)$, $\chi_{840}(281,·)$, $\chi_{840}(797,·)$, $\chi_{840}(799,·)$, $\chi_{840}(547,·)$, $\chi_{840}(293,·)$, $\chi_{840}(41,·)$, $\chi_{840}(43,·)$, $\chi_{840}(559,·)$, $\chi_{840}(307,·)$, $\chi_{840}(671,·)$, $\chi_{840}(769,·)$, $\chi_{840}(169,·)$, $\chi_{840}(449,·)$, $\chi_{840}(323,·)$, $\chi_{840}(197,·)$, $\chi_{840}(71,·)$, $\chi_{840}(587,·)$, $\chi_{840}(209,·)$, $\chi_{840}(83,·)$, $\chi_{840}(601,·)$, $\chi_{840}(827,·)$, $\chi_{840}(239,·)$, $\chi_{840}(839,·)$, $\chi_{840}(757,·)$, $\chi_{840}(631,·)$, $\chi_{840}(253,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{11}$, $\frac{1}{40} a^{24} - \frac{1}{8} a^{20} - \frac{1}{20} a^{18} - \frac{1}{4} a^{16} - \frac{11}{40} a^{12} - \frac{1}{2} a^{10} - \frac{1}{8} a^{8} + \frac{1}{20} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{10}$, $\frac{1}{40} a^{25} - \frac{1}{8} a^{21} - \frac{1}{20} a^{19} - \frac{1}{4} a^{17} - \frac{11}{40} a^{13} - \frac{1}{2} a^{11} - \frac{1}{8} a^{9} + \frac{1}{20} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{10} a$, $\frac{1}{1160} a^{26} + \frac{9}{1160} a^{25} - \frac{1}{1160} a^{24} + \frac{5}{58} a^{23} - \frac{57}{232} a^{22} - \frac{33}{232} a^{21} + \frac{83}{1160} a^{20} - \frac{119}{580} a^{19} + \frac{14}{145} a^{18} + \frac{23}{116} a^{17} + \frac{23}{116} a^{16} + \frac{3}{29} a^{15} - \frac{571}{1160} a^{14} + \frac{21}{1160} a^{13} - \frac{209}{1160} a^{12} - \frac{10}{29} a^{11} + \frac{19}{232} a^{10} + \frac{7}{232} a^{9} + \frac{167}{1160} a^{8} - \frac{41}{580} a^{7} - \frac{13}{290} a^{6} + \frac{35}{116} a^{5} + \frac{13}{116} a^{4} + \frac{5}{58} a^{3} - \frac{8}{145} a^{2} - \frac{129}{290} a + \frac{31}{290}$, $\frac{1}{1160} a^{27} + \frac{1}{232} a^{25} - \frac{7}{1160} a^{24} - \frac{5}{232} a^{23} + \frac{2}{29} a^{22} - \frac{27}{1160} a^{21} + \frac{35}{232} a^{20} - \frac{6}{29} a^{19} + \frac{17}{580} a^{18} + \frac{19}{116} a^{17} - \frac{21}{116} a^{16} - \frac{491}{1160} a^{15} + \frac{13}{29} a^{14} - \frac{39}{232} a^{13} + \frac{437}{1160} a^{12} - \frac{73}{232} a^{11} + \frac{17}{58} a^{10} + \frac{577}{1160} a^{9} - \frac{85}{232} a^{8} - \frac{15}{58} a^{7} + \frac{3}{580} a^{6} + \frac{17}{116} a^{5} - \frac{49}{116} a^{4} - \frac{48}{145} a^{3} + \frac{3}{58} a^{2} - \frac{11}{58} a + \frac{127}{290}$, $\frac{1}{25920200} a^{28} - \frac{105}{1036808} a^{27} - \frac{5537}{12960100} a^{26} + \frac{17301}{2592020} a^{25} - \frac{31181}{25920200} a^{24} - \frac{80839}{5184040} a^{23} + \frac{128316}{3240025} a^{22} - \frac{98991}{648005} a^{21} + \frac{2985149}{12960100} a^{20} - \frac{93857}{2592020} a^{19} - \frac{638141}{3240025} a^{18} + \frac{59793}{1296010} a^{17} + \frac{5999049}{25920200} a^{16} + \frac{2049127}{5184040} a^{15} - \frac{63927}{446900} a^{14} + \frac{977927}{2592020} a^{13} - \frac{13109}{632200} a^{12} - \frac{978319}{5184040} a^{11} - \frac{2610067}{6480050} a^{10} + \frac{64381}{1296010} a^{9} + \frac{4623901}{12960100} a^{8} + \frac{209079}{2592020} a^{7} - \frac{1710793}{6480050} a^{6} - \frac{89151}{648005} a^{5} + \frac{1310277}{3240025} a^{4} - \frac{69}{290} a^{3} - \frac{76663}{3240025} a^{2} - \frac{44288}{648005} a - \frac{512847}{3240025}$, $\frac{1}{25920200} a^{29} + \frac{1453}{12960100} a^{27} - \frac{831}{5184040} a^{26} + \frac{1949}{223450} a^{25} - \frac{6941}{2592020} a^{24} - \frac{571193}{6480050} a^{23} - \frac{312193}{5184040} a^{22} - \frac{184687}{25920200} a^{21} - \frac{35307}{1296010} a^{20} - \frac{2813979}{12960100} a^{19} + \frac{8599}{63220} a^{18} + \frac{1860399}{25920200} a^{17} - \frac{175997}{1296010} a^{16} + \frac{5482627}{12960100} a^{15} - \frac{351563}{5184040} a^{14} + \frac{1418727}{3240025} a^{13} + \frac{15673}{89380} a^{12} - \frac{2441067}{6480050} a^{11} - \frac{1554001}{5184040} a^{10} + \frac{8965737}{25920200} a^{9} - \frac{120746}{648005} a^{8} - \frac{1169971}{12960100} a^{7} + \frac{711263}{2592020} a^{6} + \frac{5532483}{12960100} a^{5} + \frac{129732}{648005} a^{4} + \frac{539369}{6480050} a^{3} + \frac{603597}{1296010} a^{2} + \frac{1102741}{6480050} a + \frac{285184}{648005}$, $\frac{1}{50670210999337163731191756841500560274200} a^{30} + \frac{75739585666066182421596408556197}{5067021099933716373119175684150056027420} a^{29} - \frac{68803095163353308753553563589417}{10134042199867432746238351368300112054840} a^{28} - \frac{11249114563458746908017139285605818}{1266755274983429093279793921037514006855} a^{27} + \frac{1090324501179521513132571718970603581}{6333776374917145466398969605187570034275} a^{26} - \frac{41542208215405223057401410335984738719}{5067021099933716373119175684150056027420} a^{25} + \frac{113723991469529726098854807659711020957}{25335105499668581865595878420750280137100} a^{24} - \frac{157031497365971845573516587855775222653}{2533510549966858186559587842075028013710} a^{23} - \frac{1838927245151310533230766549844457849057}{10134042199867432746238351368300112054840} a^{22} + \frac{70357262587369690832229664169264587542}{1266755274983429093279793921037514006855} a^{21} - \frac{11647692298212763830403363177933291182111}{50670210999337163731191756841500560274200} a^{20} - \frac{43439954224744409978750471537314099629}{253351054996685818655958784207502801371} a^{19} + \frac{11801087135163301839510307239842654469617}{50670210999337163731191756841500560274200} a^{18} + \frac{315612363543470281461798604587796784073}{5067021099933716373119175684150056027420} a^{17} - \frac{1588253321265724331931232778680297341391}{10134042199867432746238351368300112054840} a^{16} - \frac{169890042638800544631368123374250808464}{1266755274983429093279793921037514006855} a^{15} + \frac{2692136115390042551429682334015000080974}{6333776374917145466398969605187570034275} a^{14} - \frac{1965380689079233703585829581793546896823}{5067021099933716373119175684150056027420} a^{13} - \frac{6426127017789595121424550203642224060577}{25335105499668581865595878420750280137100} a^{12} + \frac{806485930860876188836824425824662523589}{2533510549966858186559587842075028013710} a^{11} + \frac{2709193505777894044812801165766994799}{8850691877613478380994193334759923192} a^{10} + \frac{484057738113255140950152105396951939294}{1266755274983429093279793921037514006855} a^{9} - \frac{342378541815437645339985037850413236439}{50670210999337163731191756841500560274200} a^{8} + \frac{25143600805963127222322647069189889854}{253351054996685818655958784207502801371} a^{7} + \frac{146293336920273173515160907399149735106}{6333776374917145466398969605187570034275} a^{6} + \frac{9163572261410937942928046629321222516}{1266755274983429093279793921037514006855} a^{5} + \frac{59182623971845820738046600774429452513}{174724865514955737004109506350001931980} a^{4} - \frac{75501654467319060211639652504022732122}{1266755274983429093279793921037514006855} a^{3} + \frac{2475667578988636995149600580565923085301}{6333776374917145466398969605187570034275} a^{2} - \frac{418092069011204659703297587466807703696}{1266755274983429093279793921037514006855} a + \frac{4595022260514342960265113879548176192289}{12667552749834290932797939210375140068550}$, $\frac{1}{200157682528817938405730073779144553160931929058848556416297626491539510095148030518200} a^{31} + \frac{98958942563266569497959248302807925510477479}{25019710316102242300716259222393069145116491132356069552037203311442438761893503814775} a^{30} + \frac{404156885748328494820873636355191070403323000927730967131187363277727102056881}{50039420632204484601432518444786138290232982264712139104074406622884877523787007629550} a^{29} - \frac{228266568956072792744197946671538656989733647704677471671260908195639743809581}{20015768252881793840573007377914455316093192905884855641629762649153951009514803051820} a^{28} - \frac{165137951923014351687592494950790578020637962617500586582294574874230484839543389}{862748631589732493128146869737692039486775556288140329380593217635946164203224269475} a^{27} - \frac{956452529718601997955757785557673268182937602972270099474652495502872096058838421}{50039420632204484601432518444786138290232982264712139104074406622884877523787007629550} a^{26} + \frac{119980019779061064420520413872450152340400263298504680930765556312996423143581967943}{40031536505763587681146014755828910632186385811769711283259525298307902019029606103640} a^{25} - \frac{250274277170130617162739463686352139460827954085999239055442569285955965371263555984}{25019710316102242300716259222393069145116491132356069552037203311442438761893503814775} a^{24} - \frac{9843279434952654241970705067188607302326272403660815504845359515228832611037256934523}{200157682528817938405730073779144553160931929058848556416297626491539510095148030518200} a^{23} + \frac{802315774204747954471235867204180640524338839142415963126169753364027165037052683}{5594121926462211805638068020658036700976297626015890341428105827041350198299274190} a^{22} + \frac{46273099853325465732301029157157924051292018863914283523450837185314890431509270546861}{200157682528817938405730073779144553160931929058848556416297626491539510095148030518200} a^{21} + \frac{24217943219250413383247966068628256791921986533484291103217882902163779858454071049079}{100078841264408969202865036889572276580465964529424278208148813245769755047574015259100} a^{20} - \frac{4217191575739284311282615840309317574251551894190053569869280501101897628879846054211}{40031536505763587681146014755828910632186385811769711283259525298307902019029606103640} a^{19} + \frac{4396883518037009954778187124788711780611330774185250125466539358199075761385895921453}{25019710316102242300716259222393069145116491132356069552037203311442438761893503814775} a^{18} - \frac{10369737113017511818291969431743322466424435159744272100802526790212291874497677536807}{100078841264408969202865036889572276580465964529424278208148813245769755047574015259100} a^{17} + \frac{157755723623987856362422989780740927337627436193832213386377977215513761043366416329}{20015768252881793840573007377914455316093192905884855641629762649153951009514803051820} a^{16} - \frac{15278261076041179507886481918415238986917975115897401512008097985037830696545161634183}{50039420632204484601432518444786138290232982264712139104074406622884877523787007629550} a^{15} - \frac{19106417572016084713251798652206630277660194259102207230361396036022168477211530096109}{50039420632204484601432518444786138290232982264712139104074406622884877523787007629550} a^{14} - \frac{12820213016695067183000094841147182709691123055738262564897474788563937539146358857509}{40031536505763587681146014755828910632186385811769711283259525298307902019029606103640} a^{13} + \frac{10567892320562446594923324597502450286050186811911015373963564868675294816221663385969}{25019710316102242300716259222393069145116491132356069552037203311442438761893503814775} a^{12} - \frac{23541711535100224459704876271283432387532063397589725085515305274180133262490937921607}{200157682528817938405730073779144553160931929058848556416297626491539510095148030518200} a^{11} - \frac{961342962904838613462679543248998038447116216583218344586479082983701569919724598897}{2001576825288179384057300737791445531609319290588485564162976264915395100951480305182} a^{10} + \frac{11013000639648161411510896697890062938984954903560218634736837615835590466142547718469}{200157682528817938405730073779144553160931929058848556416297626491539510095148030518200} a^{9} + \frac{26203970361239728994359597389441449039619524971217058618832020568807125220370307613411}{100078841264408969202865036889572276580465964529424278208148813245769755047574015259100} a^{8} + \frac{2562418970806786113464292963350598524687169229854069833107028014787018419607693194021}{10007884126440896920286503688957227658046596452942427820814881324576975504757401525910} a^{7} - \frac{11045902363365226817483912534162697610929367698660936274560900759328425275484245416218}{25019710316102242300716259222393069145116491132356069552037203311442438761893503814775} a^{6} + \frac{34842106085993201583865822974480635900556968408107326887438239048225232722692949825527}{100078841264408969202865036889572276580465964529424278208148813245769755047574015259100} a^{5} - \frac{2441878669170431076222227396464720368927232864333420275921709471520814443269419637237}{10007884126440896920286503688957227658046596452942427820814881324576975504757401525910} a^{4} - \frac{1693981408175051010561457746464731825946802600334213610554142374069659898406346920591}{25019710316102242300716259222393069145116491132356069552037203311442438761893503814775} a^{3} + \frac{3933299092660164726164016831345080639757484855173130879911165652193483408355902730227}{25019710316102242300716259222393069145116491132356069552037203311442438761893503814775} a^{2} - \frac{14486096573282350448281135114051519390735351350540258857549859940142698697141568141887}{50039420632204484601432518444786138290232982264712139104074406622884877523787007629550} a + \frac{3049401502612632284876722182153427262216676575499346317487261409817762374476235875709}{25019710316102242300716259222393069145116491132356069552037203311442438761893503814775}$
Class group and class number
Not computed
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{8620533481647496013026052917087836055803739086882867446496726677}{104654553424854690647346669369168194441658049056063111754900401134511100} a^{31} + \frac{24281969803498107446184722328852176169706569103937406518156608173}{209309106849709381294693338738336388883316098112126223509800802269022200} a^{30} - \frac{171393675551043716443505762110958941873269339550997153398987214611}{209309106849709381294693338738336388883316098112126223509800802269022200} a^{29} + \frac{135190026162135727433479682991015392533965440926145305285206992589}{41861821369941876258938667747667277776663219622425244701960160453804440} a^{28} - \frac{2324971716496830544235448028924776267303730906918660225572603267213}{209309106849709381294693338738336388883316098112126223509800802269022200} a^{27} + \frac{39493214385406464724986426603810257817053631685137732344913045868}{26163638356213672661836667342292048610414512264015777938725100283627775} a^{26} - \frac{1088324731860635527688952865774840607764154633431542167480434325061}{41861821369941876258938667747667277776663219622425244701960160453804440} a^{25} + \frac{895809394776166118651244413994031506045243508904572815513750099101}{104654553424854690647346669369168194441658049056063111754900401134511100} a^{24} + \frac{156471666930885209768795832811088477341216922044140919917422439794177}{209309106849709381294693338738336388883316098112126223509800802269022200} a^{23} - \frac{10322672195387847093378684671234267131751362550988687943389908144589}{8372364273988375251787733549533455555332643924485048940392032090760888} a^{22} + \frac{160354797827775414121305114375590086488873787001547832818187784851689}{52327276712427345323673334684584097220829024528031555877450200567255550} a^{21} - \frac{1196909926228053699603921976326471927326403532456452466820914579558513}{209309106849709381294693338738336388883316098112126223509800802269022200} a^{20} + \frac{40000318740257547384253966673832940764765297163330587766805808533469}{10465455342485469064734666936916819444165804905606311175490040113451110} a^{19} + \frac{14177607982564873385408296101734286252698694065738937819743178083065121}{209309106849709381294693338738336388883316098112126223509800802269022200} a^{18} - \frac{23419675390393675392796288664241892552814609626580773990049301290656479}{209309106849709381294693338738336388883316098112126223509800802269022200} a^{17} + \frac{14926745908594399928229421834150559126721681175975476189294428682837359}{41861821369941876258938667747667277776663219622425244701960160453804440} a^{16} - \frac{196586648579701424608292620144028814215565050529838485832827590199398397}{209309106849709381294693338738336388883316098112126223509800802269022200} a^{15} - \frac{102818024953898634864192471515919233476110969303914860147244316950082}{902194426076333540063333356630760296910845250483302687542244837366475} a^{14} + \frac{47198694076996413276047947012819100430727011655016080859319651681772831}{41861821369941876258938667747667277776663219622425244701960160453804440} a^{13} - \frac{16023709938599423307774624243803539991759920182242008519725282870185389}{3608777704305334160253333426523041187643381001933210750168979349465900} a^{12} + \frac{3908909668949449952410227920536111234863814140759557050797556130360037213}{209309106849709381294693338738336388883316098112126223509800802269022200} a^{11} - \frac{26183110164668345395156640862987679060616152833077646882164987402553569}{1021020033413216494120455310918714092113737063961591334194150254970840} a^{10} + \frac{342479146943788367559097233012048495335805854790062581756109648668735683}{26163638356213672661836667342292048610414512264015777938725100283627775} a^{9} + \frac{1889531750983304897140259820711775420593862819983736984304102029736834943}{209309106849709381294693338738336388883316098112126223509800802269022200} a^{8} - \frac{1791109594704404187053495946769225056742321111913691186721728264030197159}{20930910684970938129469333873833638888331609811212622350980080226902220} a^{7} + \frac{11153935038414814335564266867537223623867320522005900454991010482017206731}{52327276712427345323673334684584097220829024528031555877450200567255550} a^{6} - \frac{6408327528908318906191374461264854133393327802082223510621987229095897042}{26163638356213672661836667342292048610414512264015777938725100283627775} a^{5} + \frac{3114005994991580671115476483341581216098849362559738554989170730087220517}{20930910684970938129469333873833638888331609811212622350980080226902220} a^{4} - \frac{3127722942077513746617842365845176917654917390863841491089826079881516107}{52327276712427345323673334684584097220829024528031555877450200567255550} a^{3} + \frac{20046340959028980686468052433584973409693514279149387207403249397563493}{638137520883260308825284569324196307571085664975994583871343909356775} a^{2} - \frac{479115958625461793413063025581728158677991487621671983479141692211841191}{26163638356213672661836667342292048610414512264015777938725100283627775} a + \frac{219341043778539117120910390354438475099459788081422549210362252885779317}{52327276712427345323673334684584097220829024528031555877450200567255550} \) (order $12$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3\times C_4$ (as 32T34):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2^3\times C_4$ |
| Character table for $C_2^3\times C_4$ is not computed |
Intermediate fields
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 | R | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.2 | $x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 20$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ |
| 2.8.16.2 | $x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 20$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ | |
| 2.8.16.2 | $x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 20$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ | |
| 2.8.16.2 | $x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 20$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ | |
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $7$ | 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |