Normalized defining polynomial
\( x^{42} - 4 x^{41} + 62 x^{40} - 200 x^{39} + 2132 x^{38} - 6208 x^{37} + 46584 x^{36} - 115568 x^{35} + \cdots + 8082649 \)
Invariants
| Degree: | $42$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 21]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-213\!\cdots\!363\)
\(\medspace = -\,3^{21}\cdot 7^{28}\cdot 29^{36}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(113.62\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $3^{1/2}7^{2/3}29^{6/7}\approx 113.61700029758396$ | ||
| Ramified primes: |
\(3\), \(7\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $42$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(609=3\cdot 7\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{609}(256,·)$, $\chi_{609}(1,·)$, $\chi_{609}(23,·)$, $\chi_{609}(268,·)$, $\chi_{609}(16,·)$, $\chi_{609}(529,·)$, $\chi_{609}(277,·)$, $\chi_{609}(407,·)$, $\chi_{609}(25,·)$, $\chi_{609}(284,·)$, $\chi_{609}(547,·)$, $\chi_{609}(422,·)$, $\chi_{609}(169,·)$, $\chi_{609}(170,·)$, $\chi_{609}(431,·)$, $\chi_{609}(436,·)$, $\chi_{609}(53,·)$, $\chi_{609}(310,·)$, $\chi_{609}(442,·)$, $\chi_{609}(571,·)$, $\chi_{609}(190,·)$, $\chi_{609}(575,·)$, $\chi_{609}(65,·)$, $\chi_{609}(197,·)$, $\chi_{609}(326,·)$, $\chi_{609}(74,·)$, $\chi_{609}(401,·)$, $\chi_{609}(596,·)$, $\chi_{609}(88,·)$, $\chi_{609}(604,·)$, $\chi_{609}(344,·)$, $\chi_{609}(400,·)$, $\chi_{609}(226,·)$, $\chi_{609}(484,·)$, $\chi_{609}(487,·)$, $\chi_{609}(233,·)$, $\chi_{609}(107,·)$, $\chi_{609}(494,·)$, $\chi_{609}(239,·)$, $\chi_{609}(368,·)$, $\chi_{609}(373,·)$, $\chi_{609}(281,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{1048576}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{17}a^{36}+\frac{5}{17}a^{35}-\frac{6}{17}a^{34}+\frac{7}{17}a^{33}-\frac{3}{17}a^{32}+\frac{1}{17}a^{31}-\frac{1}{17}a^{30}-\frac{7}{17}a^{29}-\frac{3}{17}a^{28}-\frac{8}{17}a^{27}+\frac{5}{17}a^{26}-\frac{6}{17}a^{25}-\frac{1}{17}a^{24}+\frac{8}{17}a^{23}+\frac{2}{17}a^{22}+\frac{4}{17}a^{21}+\frac{3}{17}a^{20}-\frac{3}{17}a^{19}+\frac{8}{17}a^{18}+\frac{6}{17}a^{16}+\frac{8}{17}a^{15}+\frac{7}{17}a^{14}+\frac{2}{17}a^{13}+\frac{5}{17}a^{12}+\frac{1}{17}a^{11}+\frac{6}{17}a^{10}+\frac{5}{17}a^{9}+\frac{1}{17}a^{7}-\frac{7}{17}a^{6}-\frac{7}{17}a^{5}-\frac{7}{17}a^{3}-\frac{7}{17}a^{2}+\frac{8}{17}a+\frac{1}{17}$, $\frac{1}{17}a^{37}+\frac{3}{17}a^{35}+\frac{3}{17}a^{34}-\frac{4}{17}a^{33}-\frac{1}{17}a^{32}-\frac{6}{17}a^{31}-\frac{2}{17}a^{30}-\frac{2}{17}a^{29}+\frac{7}{17}a^{28}-\frac{6}{17}a^{27}+\frac{3}{17}a^{26}-\frac{5}{17}a^{25}-\frac{4}{17}a^{24}-\frac{4}{17}a^{23}-\frac{6}{17}a^{22}-\frac{1}{17}a^{20}+\frac{6}{17}a^{19}-\frac{6}{17}a^{18}+\frac{6}{17}a^{17}-\frac{5}{17}a^{16}+\frac{1}{17}a^{15}+\frac{1}{17}a^{14}-\frac{5}{17}a^{13}-\frac{7}{17}a^{12}+\frac{1}{17}a^{11}-\frac{8}{17}a^{10}-\frac{8}{17}a^{9}+\frac{1}{17}a^{8}+\frac{5}{17}a^{7}-\frac{6}{17}a^{6}+\frac{1}{17}a^{5}-\frac{7}{17}a^{4}-\frac{6}{17}a^{3}-\frac{8}{17}a^{2}-\frac{5}{17}a-\frac{5}{17}$, $\frac{1}{697}a^{38}-\frac{20}{697}a^{37}-\frac{6}{697}a^{36}+\frac{3}{41}a^{35}+\frac{109}{697}a^{34}+\frac{322}{697}a^{33}+\frac{1}{17}a^{32}-\frac{282}{697}a^{31}+\frac{319}{697}a^{30}+\frac{195}{697}a^{29}+\frac{18}{41}a^{28}+\frac{42}{697}a^{27}+\frac{264}{697}a^{26}-\frac{258}{697}a^{25}+\frac{18}{41}a^{24}+\frac{19}{697}a^{23}-\frac{4}{41}a^{22}+\frac{48}{697}a^{21}+\frac{101}{697}a^{20}+\frac{20}{697}a^{19}+\frac{156}{697}a^{18}+\frac{300}{697}a^{17}+\frac{200}{697}a^{16}+\frac{4}{17}a^{15}-\frac{20}{697}a^{14}+\frac{58}{697}a^{13}+\frac{147}{697}a^{12}-\frac{173}{697}a^{11}-\frac{174}{697}a^{10}+\frac{269}{697}a^{9}+\frac{2}{697}a^{8}+\frac{242}{697}a^{7}+\frac{65}{697}a^{6}+\frac{274}{697}a^{5}-\frac{87}{697}a^{4}-\frac{335}{697}a^{3}-\frac{139}{697}a^{2}+\frac{40}{697}a+\frac{91}{697}$, $\frac{1}{697}a^{39}+\frac{4}{697}a^{37}+\frac{13}{697}a^{36}-\frac{19}{697}a^{35}-\frac{245}{697}a^{34}-\frac{161}{697}a^{33}-\frac{118}{697}a^{32}-\frac{32}{697}a^{31}+\frac{97}{697}a^{30}+\frac{24}{697}a^{29}-\frac{275}{697}a^{28}+\frac{79}{697}a^{27}-\frac{308}{697}a^{26}+\frac{271}{697}a^{25}+\frac{235}{697}a^{24}+\frac{25}{697}a^{23}-\frac{3}{17}a^{22}-\frac{5}{697}a^{21}-\frac{215}{697}a^{20}-\frac{18}{697}a^{19}+\frac{222}{697}a^{18}+\frac{296}{697}a^{17}-\frac{182}{697}a^{16}+\frac{144}{697}a^{15}-\frac{55}{697}a^{14}+\frac{118}{697}a^{13}+\frac{307}{697}a^{12}+\frac{343}{697}a^{11}+\frac{274}{697}a^{10}-\frac{276}{697}a^{9}-\frac{5}{697}a^{8}+\frac{67}{697}a^{7}-\frac{66}{697}a^{6}-\frac{347}{697}a^{5}-\frac{66}{697}a^{4}-\frac{115}{697}a^{3}-\frac{321}{697}a^{2}+\frac{194}{697}a-\frac{148}{697}$, $\frac{1}{10\!\cdots\!63}a^{40}+\frac{44\!\cdots\!35}{10\!\cdots\!63}a^{39}-\frac{39\!\cdots\!53}{10\!\cdots\!63}a^{38}-\frac{23\!\cdots\!67}{10\!\cdots\!63}a^{37}-\frac{15\!\cdots\!82}{10\!\cdots\!63}a^{36}-\frac{42\!\cdots\!32}{10\!\cdots\!63}a^{35}+\frac{20\!\cdots\!63}{10\!\cdots\!63}a^{34}+\frac{69\!\cdots\!62}{10\!\cdots\!63}a^{33}-\frac{12\!\cdots\!50}{10\!\cdots\!63}a^{32}-\frac{23\!\cdots\!70}{59\!\cdots\!39}a^{31}-\frac{23\!\cdots\!79}{10\!\cdots\!63}a^{30}+\frac{12\!\cdots\!42}{59\!\cdots\!39}a^{29}+\frac{36\!\cdots\!94}{10\!\cdots\!63}a^{28}+\frac{32\!\cdots\!82}{10\!\cdots\!63}a^{27}-\frac{10\!\cdots\!13}{10\!\cdots\!63}a^{26}+\frac{50\!\cdots\!07}{10\!\cdots\!63}a^{25}+\frac{48\!\cdots\!48}{10\!\cdots\!63}a^{24}+\frac{14\!\cdots\!66}{10\!\cdots\!63}a^{23}-\frac{48\!\cdots\!18}{10\!\cdots\!63}a^{22}-\frac{40\!\cdots\!10}{10\!\cdots\!63}a^{21}-\frac{42\!\cdots\!41}{10\!\cdots\!63}a^{20}-\frac{50\!\cdots\!45}{10\!\cdots\!63}a^{19}+\frac{90\!\cdots\!30}{24\!\cdots\!43}a^{18}+\frac{39\!\cdots\!66}{10\!\cdots\!63}a^{17}-\frac{90\!\cdots\!77}{10\!\cdots\!63}a^{16}-\frac{50\!\cdots\!56}{10\!\cdots\!63}a^{15}+\frac{51\!\cdots\!57}{10\!\cdots\!63}a^{14}+\frac{85\!\cdots\!32}{10\!\cdots\!63}a^{13}-\frac{21\!\cdots\!19}{10\!\cdots\!63}a^{12}+\frac{47\!\cdots\!64}{10\!\cdots\!63}a^{11}+\frac{13\!\cdots\!32}{10\!\cdots\!63}a^{10}+\frac{16\!\cdots\!54}{10\!\cdots\!63}a^{9}-\frac{30\!\cdots\!54}{10\!\cdots\!63}a^{8}-\frac{51\!\cdots\!84}{10\!\cdots\!63}a^{7}-\frac{16\!\cdots\!44}{10\!\cdots\!63}a^{6}-\frac{52\!\cdots\!46}{10\!\cdots\!63}a^{5}-\frac{41\!\cdots\!00}{10\!\cdots\!63}a^{4}+\frac{24\!\cdots\!07}{10\!\cdots\!63}a^{3}+\frac{24\!\cdots\!66}{10\!\cdots\!63}a^{2}+\frac{10\!\cdots\!05}{10\!\cdots\!63}a-\frac{35\!\cdots\!63}{10\!\cdots\!63}$, $\frac{1}{13\!\cdots\!77}a^{41}+\frac{24\!\cdots\!92}{13\!\cdots\!77}a^{40}-\frac{78\!\cdots\!11}{13\!\cdots\!77}a^{39}+\frac{45\!\cdots\!22}{13\!\cdots\!77}a^{38}+\frac{56\!\cdots\!76}{13\!\cdots\!77}a^{37}-\frac{23\!\cdots\!38}{13\!\cdots\!77}a^{36}+\frac{44\!\cdots\!40}{13\!\cdots\!77}a^{35}+\frac{35\!\cdots\!00}{13\!\cdots\!77}a^{34}-\frac{27\!\cdots\!92}{13\!\cdots\!77}a^{33}+\frac{50\!\cdots\!07}{13\!\cdots\!77}a^{32}+\frac{14\!\cdots\!73}{13\!\cdots\!77}a^{31}+\frac{38\!\cdots\!17}{13\!\cdots\!77}a^{30}+\frac{37\!\cdots\!06}{80\!\cdots\!81}a^{29}-\frac{56\!\cdots\!78}{13\!\cdots\!77}a^{28}+\frac{35\!\cdots\!89}{13\!\cdots\!77}a^{27}+\frac{16\!\cdots\!11}{13\!\cdots\!77}a^{26}-\frac{30\!\cdots\!56}{48\!\cdots\!39}a^{25}+\frac{41\!\cdots\!65}{13\!\cdots\!77}a^{24}-\frac{36\!\cdots\!90}{13\!\cdots\!77}a^{23}-\frac{52\!\cdots\!72}{13\!\cdots\!77}a^{22}-\frac{51\!\cdots\!14}{13\!\cdots\!77}a^{21}-\frac{61\!\cdots\!25}{13\!\cdots\!77}a^{20}+\frac{52\!\cdots\!23}{13\!\cdots\!77}a^{19}-\frac{18\!\cdots\!49}{13\!\cdots\!77}a^{18}-\frac{57\!\cdots\!07}{13\!\cdots\!77}a^{17}+\frac{43\!\cdots\!93}{13\!\cdots\!77}a^{16}-\frac{29\!\cdots\!00}{13\!\cdots\!77}a^{15}-\frac{23\!\cdots\!18}{13\!\cdots\!77}a^{14}-\frac{13\!\cdots\!97}{13\!\cdots\!77}a^{13}-\frac{67\!\cdots\!91}{13\!\cdots\!77}a^{12}+\frac{26\!\cdots\!56}{13\!\cdots\!77}a^{11}+\frac{15\!\cdots\!33}{13\!\cdots\!77}a^{10}+\frac{39\!\cdots\!19}{13\!\cdots\!77}a^{9}-\frac{65\!\cdots\!88}{13\!\cdots\!77}a^{8}-\frac{73\!\cdots\!01}{13\!\cdots\!77}a^{7}+\frac{33\!\cdots\!66}{13\!\cdots\!77}a^{6}-\frac{42\!\cdots\!18}{13\!\cdots\!77}a^{5}-\frac{12\!\cdots\!57}{13\!\cdots\!77}a^{4}+\frac{61\!\cdots\!40}{13\!\cdots\!77}a^{3}+\frac{27\!\cdots\!84}{80\!\cdots\!81}a^{2}+\frac{46\!\cdots\!36}{13\!\cdots\!77}a+\frac{23\!\cdots\!49}{48\!\cdots\!39}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $20$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{23244626942559013839518193205665855954922164033896175847293245683639459464666228708284477530067308823365968592800989502788514915808587179676905868958708833}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{41} - \frac{94352440396107911065946397662558284508733960876432027467182318607341956784392045490685629384023750505473911164027303198421941188249104306453437619193239608}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{40} + \frac{1453594829709800581572362139132869314550102564563439922880470963409419851414513191006206665741405093981524684106647881874739031684295466339368453619752270777}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{39} - \frac{4762023070726910992469064067551889259842369492885629713434011572797805061326028475377724667177169917230038394220225500076557323126439028184848600679499645127}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{38} + \frac{50261028051126330034518869467298910662818183526847469864642527562042258886572224813981728839304949485851949909632984474304520534128587069502961780487007102144}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{37} - \frac{148621418179674418312048878727187492518972884890186213707223904886058770148232316100824597484070814439033326055878304445785286222766591352538255889138396310724}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{36} + \frac{1106056080834321615847744271055463105229052896391418855575805916619854524233661379139374911859045646480023441089435174823981397475435276207317116778455631874014}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{35} - \frac{2793340984210869276603431567662313231034250240674064125846146762434163575278462430478496947129302088084174698201629744207644377007180279597423781690059499016918}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{34} + \frac{16781444709796823661064804808718929340877960603170980952402338469686615795034382760760458765602268097436541507079867972001735411390476430707063088149747612982781}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{33} - \frac{37500776463679319881982240317723207053449088156846048993664626283845171632639219240338753114135541393880908717868030338818984994808432657324517593517253016492621}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{32} + \frac{187511968030403224923540971187859649584688304698463765036779598146542289897982321653192106174282463644522687534891085212035073731862902862626969021354619903853722}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{31} - \frac{366976362762856421325325184104048129548023766932028160473276521583830122536789246499101395779836751666856978667549395887239366983804360717666611359826574341976096}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{30} + \frac{1576120375471169258745300677660117273880781671197486593907512287306083564185245306779238473099482446740114660582088294771545239828027808659326607978154899348683586}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{29} - \frac{2745181342922386465200045499381689421882518596810222271033500433412352314676316918337064744244346775425690040634952889635277164231025012885286398755380489112308024}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{28} + \frac{10213920653842296523860782238590356756297863975885564725309912350607833700307515668858375048955885248056849390972337173548629846408089881556938555659216573925057403}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{27} - \frac{15677666072465580694385953381919000655649055830103526897225279758481589504463608833225005821130064377574731397478932462293813069736144712734607821978252957217372645}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{26} + \frac{17960929359823005604739911517128436067443456798278388480518542208472387753379632121446139489422818340665748476678154292690279599751981838598464448113679477361410}{507231012026095645298915164876224678552305506825643496076942329505720924613472399192518824659480731896495954723102395916157914835702126104585537364977291545079} a^{25} - \frac{69278862961158478476706381110464339082640968604378208601265052459724176085275033162374357965641453757340823713837199148967566613246893696375931772119110847722222522}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{24} + \frac{198822193230577184359740676610298273459880570287275576647429209900752081488493137602532696622807371013764386551844199714157700149182139493751919893832456534467009852}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{23} - \frac{235205899684931804980514569678875705855005567360914654768825703006084047588533481273574505511660650168155003264799419219309153287182055502762441402366030471097358711}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{22} + \frac{596933422424965198688892976033895699150689721828596823003276581999882900833464950048125331059587597352711706346428454728464240709363580657420851444973239359445500183}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{21} - \frac{617406373829303564128988204563650539068748366309364296792930415448075719599882387983708069741672679914174677718511465790582274195059976779886312127504827179953419154}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{20} + \frac{1397896975101670867524163961255973219675686068232944453894975175515299724916921311213703646592675565076820285878129731328903637132477809134780375255826603948479433248}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{19} - \frac{1268824936992983435682643809882101849797010528539027664700480659287715660014723913237035632069922483519005158780979937331834490289582841405146460563920797771441006141}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{18} + \frac{2529489898937790207777285592459978108594748389685862097940710035921139700047293881004822711952026512963469506556703267992405191692977479178775696718181061774188859546}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{17} - \frac{2054011397941005008958961784400084076631957475412381773956725184082268874292396496537784601011545060778668488969063208490054556580867333069629323692142773401559437780}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{16} + \frac{3570089415207978889051503562789172137754167103106394185185746282950047878765446906648534383630500159607964865232070418086059248058078942362915628887445233378534064296}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{15} - \frac{2660149896499471261958721684877166958470043396603707697096883903406128324182625184496605136763584341872703320480760367593898472043496052236015735946032285766629527661}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{14} + \frac{3883084645457170033560115288541603103413489901307947204099972936189339212983473216416083284919253529381862019116163880444159964081910952640555249218931058184917807701}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{13} - \frac{2727722152546382693589829887976949759405544243493215031343112304706142240152454041221923912972166143807304962204576784634839673136722078858614173301392970717737181883}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{12} + \frac{3294181821811200744939537260934680510996212443091451429703383394114050264607400741090878453502737438916812446108249390970178271424195964974728483379239978749671387012}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{11} - \frac{2155767150839941447774712700437570204734966993972466463338280103434567413244840420535925474011143719212774620221587333585707021348470115929376088397127625994242099752}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{10} + \frac{2142468409592192104260950939129937838139766474939750089115539479017390563607195669109766643108039727946624777625504411515862774434703184546860534893355437921919757890}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{9} - \frac{1282074029010904832545011196153685692207694216825331193720731696890500155669433011950755730103329782976065005759872865578025602717941068711212795814278059242003220895}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{8} + \frac{1046223473585309315717939724793986950497038564144931851079847802960687288943965135508868005599509518696471738414322160653327761988820514498105614697960326076040823974}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{7} - \frac{547217370045546818082611988800516810418946358564238586620848482031673442359442513663305427284319261926577845847087491038374222043397274832182655054909937636518330459}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{6} + \frac{362148791375792996392488477934550881546678551816900526488911205228061431808255974748787037829430355024340755274558414687185809150577001138304487191859550478556680684}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{5} - \frac{155642864613956454797237088850514555782519383131476192460041066862154533214602002683855171066423943265218244584802529232945068627445760276359481006148688021494054868}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{4} + \frac{80912004914011140624450296416553973699450064333082917097357249883055563268055533066007467028238933697234975239580258840569658092632719002845215896997532504930431180}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{3} - \frac{24196630737225863278434433168045725064234202178169427789105499083617850921226267168302820505783526888410797895103123781568098546326180343745686371419845697024917276}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a^{2} + \frac{10164381964980591613888335314711966733949500773985232096878540191722038670524583016837281585774519679171372302950285644946784966891616403788746391422575782894822985}{1442057767190189919584815813743106761124204555905304459346747042784764588676102030904331018506903720781737999277780111589636951877901144515336682728630439862659597} a - \frac{187850124031816049415602078850936406750633485495841316881957874143079979868254263831695876420096437071627534865490724172823134865080973801150206213884987750849}{507231012026095645298915164876224678552305506825643496076942329505720924613472399192518824659480731896495954723102395916157914835702126104585537364977291545079} \)
(order $6$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
Galois group
| A cyclic group of order 42 |
| The 42 conjugacy class representatives for $C_{42}$ |
| Character table for $C_{42}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{7})^+\), 6.0.64827.1, 7.7.594823321.1, 14.0.773792930870360792667.1, 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 | $42$ | R | $42$ | R | $42$ | ${\href{/padicField/13.7.0.1}{7} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{7}$ | $21^{2}$ | $42$ | R | $21^{2}$ | $21^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{21}$ | ${\href{/padicField/43.7.0.1}{7} }^{6}$ | $42$ | $42$ | ${\href{/padicField/59.6.0.1}{6} }^{7}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $42$ | $2$ | $21$ | $21$ | |||
|
\(7\)
| Deg $21$ | $3$ | $7$ | $14$ | |||
| Deg $21$ | $3$ | $7$ | $14$ | ||||
|
\(29\)
| 29.14.12.1 | $x^{14} + 168 x^{13} + 12110 x^{12} + 485856 x^{11} + 11733204 x^{10} + 171095904 x^{9} + 1407877912 x^{8} + 5266970938 x^{7} + 2815760696 x^{6} + 684731964 x^{5} + 107750832 x^{4} + 339857336 x^{3} + 4765729696 x^{2} + 37989914704 x + 129797258121$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ |
| 29.14.12.1 | $x^{14} + 168 x^{13} + 12110 x^{12} + 485856 x^{11} + 11733204 x^{10} + 171095904 x^{9} + 1407877912 x^{8} + 5266970938 x^{7} + 2815760696 x^{6} + 684731964 x^{5} + 107750832 x^{4} + 339857336 x^{3} + 4765729696 x^{2} + 37989914704 x + 129797258121$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ | |
| 29.14.12.1 | $x^{14} + 168 x^{13} + 12110 x^{12} + 485856 x^{11} + 11733204 x^{10} + 171095904 x^{9} + 1407877912 x^{8} + 5266970938 x^{7} + 2815760696 x^{6} + 684731964 x^{5} + 107750832 x^{4} + 339857336 x^{3} + 4765729696 x^{2} + 37989914704 x + 129797258121$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ |