Normalized defining polynomial
\( x^{28} - x^{27} - x^{26} - 50 x^{25} + 45 x^{24} + 41 x^{23} + 923 x^{22} - 740 x^{21} - 566 x^{20} + \cdots + 6241 \)
Invariants
| Degree: | $28$ |
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| Signature: | $(0, 14)$ |
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| Discriminant: |
\(14121388821225670988853483488774192350843817726481\)
\(\medspace = 3^{14}\cdot 43^{26}\)
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| Root discriminant: | \(56.93\) |
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| Galois root discriminant: | $3^{1/2}43^{13/14}\approx 56.93151811241331$ | ||
| Ramified primes: |
\(3\), \(43\)
|
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2\times C_{14}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(129=3\cdot 43\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{129}(128,·)$, $\chi_{129}(1,·)$, $\chi_{129}(2,·)$, $\chi_{129}(4,·)$, $\chi_{129}(70,·)$, $\chi_{129}(65,·)$, $\chi_{129}(8,·)$, $\chi_{129}(11,·)$, $\chi_{129}(64,·)$, $\chi_{129}(16,·)$, $\chi_{129}(82,·)$, $\chi_{129}(85,·)$, $\chi_{129}(22,·)$, $\chi_{129}(88,·)$, $\chi_{129}(94,·)$, $\chi_{129}(32,·)$, $\chi_{129}(97,·)$, $\chi_{129}(35,·)$, $\chi_{129}(41,·)$, $\chi_{129}(107,·)$, $\chi_{129}(44,·)$, $\chi_{129}(47,·)$, $\chi_{129}(113,·)$, $\chi_{129}(118,·)$, $\chi_{129}(121,·)$, $\chi_{129}(59,·)$, $\chi_{129}(125,·)$, $\chi_{129}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{8192}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{79}a^{23}+\frac{4}{79}a^{22}+\frac{30}{79}a^{21}-\frac{33}{79}a^{20}+\frac{15}{79}a^{19}-\frac{29}{79}a^{18}-\frac{25}{79}a^{17}+\frac{9}{79}a^{16}+\frac{24}{79}a^{15}+\frac{30}{79}a^{14}+\frac{10}{79}a^{13}+\frac{1}{79}a^{12}+\frac{14}{79}a^{11}+\frac{18}{79}a^{10}-\frac{36}{79}a^{9}-\frac{21}{79}a^{8}+\frac{5}{79}a^{7}+\frac{8}{79}a^{6}+\frac{21}{79}a^{5}+\frac{13}{79}a^{4}+\frac{1}{79}a^{3}+\frac{23}{79}a^{2}-\frac{37}{79}a$, $\frac{1}{553}a^{24}+\frac{3}{553}a^{23}+\frac{26}{553}a^{22}-\frac{142}{553}a^{21}-\frac{27}{79}a^{20}+\frac{114}{553}a^{19}-\frac{75}{553}a^{18}+\frac{113}{553}a^{17}+\frac{15}{553}a^{16}-\frac{73}{553}a^{15}+\frac{138}{553}a^{14}-\frac{9}{553}a^{13}-\frac{66}{553}a^{12}-\frac{22}{79}a^{11}-\frac{19}{79}a^{10}-\frac{64}{553}a^{9}+\frac{184}{553}a^{8}+\frac{23}{79}a^{7}+\frac{92}{553}a^{6}-\frac{166}{553}a^{5}+\frac{67}{553}a^{4}+\frac{37}{79}a^{3}+\frac{256}{553}a^{2}+\frac{116}{553}a-\frac{1}{7}$, $\frac{1}{169771}a^{25}-\frac{150}{169771}a^{24}+\frac{463}{169771}a^{23}-\frac{48094}{169771}a^{22}-\frac{72137}{169771}a^{21}+\frac{38726}{169771}a^{20}+\frac{66707}{169771}a^{19}+\frac{19337}{169771}a^{18}-\frac{12024}{169771}a^{17}+\frac{22839}{169771}a^{16}-\frac{6452}{169771}a^{15}+\frac{26771}{169771}a^{14}+\frac{83267}{169771}a^{13}+\frac{30748}{169771}a^{12}+\frac{1979}{24253}a^{11}-\frac{33265}{169771}a^{10}+\frac{31361}{169771}a^{9}+\frac{72641}{169771}a^{8}+\frac{75608}{169771}a^{7}-\frac{65139}{169771}a^{6}+\frac{6677}{169771}a^{5}-\frac{67469}{169771}a^{4}+\frac{79314}{169771}a^{3}+\frac{10312}{169771}a^{2}-\frac{37154}{169771}a+\frac{132}{2149}$, $\frac{1}{12\cdots 43}a^{26}+\frac{25500050657}{12\cdots 43}a^{25}+\frac{8172513473303}{12\cdots 43}a^{24}-\frac{35401708760840}{12\cdots 43}a^{23}-\frac{54\cdots 08}{12\cdots 43}a^{22}-\frac{45333011933414}{154539746166617}a^{21}-\frac{545176098296287}{12\cdots 43}a^{20}-\frac{21\cdots 61}{12\cdots 43}a^{19}-\frac{50\cdots 87}{12\cdots 43}a^{18}+\frac{56\cdots 82}{12\cdots 43}a^{17}-\frac{585010836262925}{12\cdots 43}a^{16}+\frac{426512719512321}{12\cdots 43}a^{15}-\frac{49\cdots 76}{12\cdots 43}a^{14}+\frac{51\cdots 38}{12\cdots 43}a^{13}-\frac{36109037429781}{154539746166617}a^{12}-\frac{35\cdots 48}{12\cdots 43}a^{11}+\frac{49\cdots 56}{12\cdots 43}a^{10}-\frac{43\cdots 68}{12\cdots 43}a^{9}+\frac{440703115311966}{17\cdots 49}a^{8}+\frac{26\cdots 75}{12\cdots 43}a^{7}-\frac{22\cdots 43}{12\cdots 43}a^{6}+\frac{39\cdots 45}{12\cdots 43}a^{5}-\frac{419010392445237}{12\cdots 43}a^{4}-\frac{58\cdots 33}{12\cdots 43}a^{3}-\frac{477653372780291}{17\cdots 49}a^{2}-\frac{25\cdots 99}{12\cdots 43}a+\frac{4722382228694}{154539746166617}$, $\frac{1}{97\cdots 71}a^{27}-\frac{30\cdots 27}{97\cdots 71}a^{26}+\frac{21\cdots 95}{97\cdots 71}a^{25}-\frac{71\cdots 09}{97\cdots 71}a^{24}-\frac{43\cdots 44}{13\cdots 53}a^{23}+\frac{22\cdots 20}{97\cdots 71}a^{22}-\frac{48\cdots 76}{97\cdots 71}a^{21}+\frac{60\cdots 73}{97\cdots 71}a^{20}-\frac{44\cdots 08}{97\cdots 71}a^{19}-\frac{23\cdots 20}{97\cdots 71}a^{18}+\frac{28\cdots 02}{97\cdots 71}a^{17}-\frac{36\cdots 31}{97\cdots 71}a^{16}-\frac{60\cdots 99}{13\cdots 53}a^{15}+\frac{35\cdots 30}{13\cdots 53}a^{14}+\frac{43\cdots 46}{97\cdots 71}a^{13}-\frac{14\cdots 98}{12\cdots 49}a^{12}-\frac{76\cdots 02}{97\cdots 71}a^{11}-\frac{24\cdots 71}{97\cdots 71}a^{10}-\frac{20\cdots 43}{97\cdots 71}a^{9}+\frac{63\cdots 66}{13\cdots 53}a^{8}+\frac{51\cdots 34}{97\cdots 71}a^{7}+\frac{35\cdots 39}{97\cdots 71}a^{6}+\frac{21\cdots 34}{97\cdots 71}a^{5}-\frac{23\cdots 46}{97\cdots 71}a^{4}-\frac{70\cdots 42}{97\cdots 71}a^{3}+\frac{74\cdots 46}{97\cdots 71}a^{2}+\frac{23\cdots 71}{97\cdots 71}a+\frac{25\cdots 18}{12\cdots 49}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{203}$, which has order $203$ (assuming GRH) |
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| Narrow class group: | $C_{203}$, which has order $203$ (assuming GRH) |
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| Relative class number: | $203$ (assuming GRH) |
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( \frac{1469565974509958327034868919014095}{13589686546556770354122015967062022987} a^{27} - \frac{2185707542180747637832708906726247}{13589686546556770354122015967062022987} a^{26} - \frac{7423076916778607575505668606219}{172021348690592029799012860342557253} a^{25} - \frac{73439535746083352207161767503807445}{13589686546556770354122015967062022987} a^{24} + \frac{102056386983756908318367238772596210}{13589686546556770354122015967062022987} a^{23} + \frac{20112019365816358240629209106930190}{13589686546556770354122015967062022987} a^{22} + \frac{1360145432505719268638469568936869224}{13589686546556770354122015967062022987} a^{21} - \frac{1753206778039274686307086151665928270}{13589686546556770354122015967062022987} a^{20} - \frac{163205604854518903203388760690261033}{13589686546556770354122015967062022987} a^{19} - \frac{11924696640706567412378496798402125083}{13589686546556770354122015967062022987} a^{18} + \frac{13848196158548853725689842851966575139}{13589686546556770354122015967062022987} a^{17} + \frac{417485373565694113786508502403038620}{13589686546556770354122015967062022987} a^{16} + \frac{52384474480375976058742825612964545713}{13589686546556770354122015967062022987} a^{15} - \frac{47527746331131048620434744262972068572}{13589686546556770354122015967062022987} a^{14} - \frac{6234873556739011773170419315120310751}{13589686546556770354122015967062022987} a^{13} - \frac{113173553420117357450659172523095744476}{13589686546556770354122015967062022987} a^{12} + \frac{44128646302373508697263147156693373974}{13589686546556770354122015967062022987} a^{11} + \frac{57441928696070197496832368274180043024}{13589686546556770354122015967062022987} a^{10} + \frac{17870280564843121166570103024525552756}{1941383792365252907731716566723146141} a^{9} + \frac{29156360210991818865782673380619600489}{13589686546556770354122015967062022987} a^{8} - \frac{40840584820498036407055156919440751331}{13589686546556770354122015967062022987} a^{7} - \frac{164990259650080922193773411795086730354}{13589686546556770354122015967062022987} a^{6} - \frac{30186355712350454417212570511258960916}{13589686546556770354122015967062022987} a^{5} - \frac{91841198497284455494610331339211512711}{13589686546556770354122015967062022987} a^{4} + \frac{19657519881538952482738490037871420732}{1941383792365252907731716566723146141} a^{3} + \frac{40280007276170701259769319385155426104}{13589686546556770354122015967062022987} a^{2} + \frac{22718577442160810286994597934338804912}{13589686546556770354122015967062022987} a + \frac{50996908933488818958189033790849093}{24574478384370289971287551477508179} \)
(order $6$)
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| Fundamental units: |
$\frac{14\cdots 63}{97\cdots 71}a^{27}-\frac{16\cdots 29}{97\cdots 71}a^{26}-\frac{64\cdots 81}{97\cdots 71}a^{25}-\frac{72\cdots 52}{97\cdots 71}a^{24}+\frac{74\cdots 70}{97\cdots 71}a^{23}+\frac{19\cdots 76}{97\cdots 71}a^{22}+\frac{19\cdots 62}{13\cdots 53}a^{21}-\frac{17\cdots 53}{13\cdots 53}a^{20}-\frac{15\cdots 71}{13\cdots 53}a^{19}-\frac{11\cdots 53}{97\cdots 71}a^{18}+\frac{94\cdots 60}{97\cdots 71}a^{17}-\frac{78\cdots 04}{13\cdots 53}a^{16}+\frac{49\cdots 74}{97\cdots 71}a^{15}-\frac{29\cdots 27}{97\cdots 71}a^{14}+\frac{14\cdots 47}{97\cdots 71}a^{13}-\frac{98\cdots 64}{97\cdots 71}a^{12}+\frac{13\cdots 72}{97\cdots 71}a^{11}+\frac{27\cdots 06}{97\cdots 71}a^{10}+\frac{95\cdots 57}{97\cdots 71}a^{9}+\frac{44\cdots 01}{97\cdots 71}a^{8}+\frac{67\cdots 20}{97\cdots 71}a^{7}-\frac{10\cdots 78}{97\cdots 71}a^{6}-\frac{14\cdots 31}{97\cdots 71}a^{5}-\frac{94\cdots 63}{97\cdots 71}a^{4}+\frac{81\cdots 62}{97\cdots 71}a^{3}+\frac{93\cdots 18}{97\cdots 71}a^{2}+\frac{28\cdots 93}{13\cdots 53}a+\frac{15\cdots 74}{12\cdots 49}$, $\frac{42\cdots 31}{97\cdots 71}a^{27}-\frac{37\cdots 25}{97\cdots 71}a^{26}-\frac{59\cdots 51}{97\cdots 71}a^{25}-\frac{21\cdots 42}{97\cdots 71}a^{24}+\frac{21\cdots 60}{12\cdots 49}a^{23}+\frac{25\cdots 50}{97\cdots 71}a^{22}+\frac{55\cdots 02}{13\cdots 53}a^{21}-\frac{40\cdots 47}{13\cdots 53}a^{20}-\frac{53\cdots 65}{13\cdots 53}a^{19}-\frac{32\cdots 19}{97\cdots 71}a^{18}+\frac{21\cdots 30}{97\cdots 71}a^{17}+\frac{37\cdots 40}{13\cdots 53}a^{16}+\frac{13\cdots 92}{97\cdots 71}a^{15}-\frac{64\cdots 73}{97\cdots 71}a^{14}-\frac{99\cdots 59}{97\cdots 71}a^{13}-\frac{27\cdots 00}{97\cdots 71}a^{12}+\frac{52\cdots 16}{97\cdots 71}a^{11}+\frac{22\cdots 08}{97\cdots 71}a^{10}+\frac{33\cdots 71}{97\cdots 71}a^{9}+\frac{13\cdots 45}{97\cdots 71}a^{8}-\frac{85\cdots 14}{97\cdots 71}a^{7}-\frac{39\cdots 04}{97\cdots 71}a^{6}-\frac{21\cdots 67}{97\cdots 71}a^{5}-\frac{17\cdots 19}{97\cdots 71}a^{4}+\frac{26\cdots 46}{97\cdots 71}a^{3}+\frac{27\cdots 20}{12\cdots 49}a^{2}+\frac{67\cdots 22}{13\cdots 53}a-\frac{83\cdots 56}{12\cdots 49}$, $\frac{81\cdots 63}{97\cdots 71}a^{27}-\frac{49\cdots 95}{97\cdots 71}a^{26}-\frac{11\cdots 71}{97\cdots 71}a^{25}-\frac{58\cdots 80}{13\cdots 53}a^{24}+\frac{29\cdots 56}{13\cdots 53}a^{23}+\frac{47\cdots 72}{97\cdots 71}a^{22}+\frac{75\cdots 76}{97\cdots 71}a^{21}-\frac{31\cdots 55}{97\cdots 71}a^{20}-\frac{68\cdots 50}{97\cdots 71}a^{19}-\frac{65\cdots 59}{97\cdots 71}a^{18}+\frac{20\cdots 73}{97\cdots 71}a^{17}+\frac{45\cdots 72}{97\cdots 71}a^{16}+\frac{40\cdots 08}{13\cdots 53}a^{15}-\frac{29\cdots 01}{97\cdots 71}a^{14}-\frac{15\cdots 77}{97\cdots 71}a^{13}-\frac{59\cdots 44}{97\cdots 71}a^{12}-\frac{20\cdots 21}{97\cdots 71}a^{11}+\frac{43\cdots 56}{13\cdots 53}a^{10}+\frac{75\cdots 57}{97\cdots 71}a^{9}+\frac{53\cdots 39}{97\cdots 71}a^{8}+\frac{87\cdots 72}{97\cdots 71}a^{7}-\frac{70\cdots 23}{97\cdots 71}a^{6}-\frac{55\cdots 88}{97\cdots 71}a^{5}-\frac{57\cdots 73}{97\cdots 71}a^{4}+\frac{15\cdots 96}{97\cdots 71}a^{3}+\frac{59\cdots 03}{13\cdots 53}a^{2}+\frac{27\cdots 67}{97\cdots 71}a+\frac{98\cdots 53}{12\cdots 49}$, $\frac{93\cdots 39}{97\cdots 71}a^{27}-\frac{13\cdots 48}{97\cdots 71}a^{26}-\frac{14\cdots 40}{13\cdots 53}a^{25}-\frac{46\cdots 95}{97\cdots 71}a^{24}+\frac{60\cdots 65}{97\cdots 71}a^{23}-\frac{15\cdots 56}{97\cdots 71}a^{22}+\frac{86\cdots 41}{97\cdots 71}a^{21}-\frac{18\cdots 08}{17\cdots 07}a^{20}+\frac{17\cdots 67}{97\cdots 71}a^{19}-\frac{75\cdots 68}{97\cdots 71}a^{18}+\frac{81\cdots 07}{97\cdots 71}a^{17}-\frac{21\cdots 65}{97\cdots 71}a^{16}+\frac{32\cdots 93}{97\cdots 71}a^{15}-\frac{40\cdots 20}{13\cdots 53}a^{14}+\frac{67\cdots 93}{97\cdots 71}a^{13}-\frac{66\cdots 78}{97\cdots 71}a^{12}+\frac{27\cdots 28}{97\cdots 71}a^{11}+\frac{14\cdots 01}{97\cdots 71}a^{10}+\frac{57\cdots 13}{97\cdots 71}a^{9}+\frac{13\cdots 70}{97\cdots 71}a^{8}-\frac{86\cdots 29}{97\cdots 71}a^{7}-\frac{70\cdots 01}{97\cdots 71}a^{6}+\frac{68\cdots 06}{97\cdots 71}a^{5}-\frac{56\cdots 19}{97\cdots 71}a^{4}+\frac{84\cdots 31}{97\cdots 71}a^{3}-\frac{14\cdots 18}{97\cdots 71}a^{2}+\frac{12\cdots 14}{97\cdots 71}a+\frac{44\cdots 79}{12\cdots 49}$, $\frac{22\cdots 04}{97\cdots 71}a^{27}-\frac{11\cdots 97}{45\cdots 79}a^{26}-\frac{15\cdots 55}{13\cdots 53}a^{25}-\frac{15\cdots 58}{13\cdots 53}a^{24}+\frac{10\cdots 30}{97\cdots 71}a^{23}+\frac{34\cdots 91}{97\cdots 71}a^{22}+\frac{29\cdots 88}{13\cdots 53}a^{21}-\frac{17\cdots 65}{97\cdots 71}a^{20}-\frac{22\cdots 03}{97\cdots 71}a^{19}-\frac{17\cdots 82}{97\cdots 71}a^{18}+\frac{13\cdots 85}{97\cdots 71}a^{17}-\frac{33\cdots 41}{97\cdots 71}a^{16}+\frac{75\cdots 57}{97\cdots 71}a^{15}-\frac{38\cdots 75}{97\cdots 71}a^{14}+\frac{51\cdots 35}{97\cdots 71}a^{13}-\frac{15\cdots 64}{97\cdots 71}a^{12}+\frac{27\cdots 79}{97\cdots 71}a^{11}+\frac{42\cdots 04}{97\cdots 71}a^{10}+\frac{14\cdots 40}{97\cdots 71}a^{9}+\frac{94\cdots 75}{97\cdots 71}a^{8}+\frac{11\cdots 51}{97\cdots 71}a^{7}-\frac{15\cdots 88}{97\cdots 71}a^{6}-\frac{51\cdots 32}{97\cdots 71}a^{5}-\frac{14\cdots 25}{97\cdots 71}a^{4}+\frac{98\cdots 63}{97\cdots 71}a^{3}+\frac{38\cdots 65}{13\cdots 53}a^{2}-\frac{30\cdots 25}{97\cdots 71}a+\frac{34\cdots 54}{12\cdots 49}$, $\frac{12\cdots 42}{97\cdots 71}a^{27}-\frac{15\cdots 85}{97\cdots 71}a^{26}-\frac{66\cdots 44}{97\cdots 71}a^{25}-\frac{62\cdots 77}{97\cdots 71}a^{24}+\frac{68\cdots 50}{97\cdots 71}a^{23}+\frac{33\cdots 39}{13\cdots 53}a^{22}+\frac{11\cdots 37}{97\cdots 71}a^{21}-\frac{11\cdots 82}{97\cdots 71}a^{20}-\frac{21\cdots 69}{97\cdots 71}a^{19}-\frac{10\cdots 77}{97\cdots 71}a^{18}+\frac{85\cdots 43}{97\cdots 71}a^{17}+\frac{65\cdots 13}{97\cdots 71}a^{16}+\frac{43\cdots 92}{97\cdots 71}a^{15}-\frac{25\cdots 23}{97\cdots 71}a^{14}-\frac{40\cdots 41}{97\cdots 71}a^{13}-\frac{92\cdots 89}{97\cdots 71}a^{12}+\frac{44\cdots 21}{97\cdots 71}a^{11}+\frac{34\cdots 14}{97\cdots 71}a^{10}+\frac{10\cdots 81}{97\cdots 71}a^{9}+\frac{59\cdots 02}{97\cdots 71}a^{8}+\frac{74\cdots 96}{97\cdots 71}a^{7}-\frac{13\cdots 23}{97\cdots 71}a^{6}-\frac{87\cdots 10}{13\cdots 53}a^{5}-\frac{87\cdots 51}{97\cdots 71}a^{4}+\frac{85\cdots 84}{97\cdots 71}a^{3}+\frac{47\cdots 38}{97\cdots 71}a^{2}+\frac{29\cdots 27}{97\cdots 71}a+\frac{21\cdots 35}{12\cdots 49}$, $\frac{55\cdots 68}{97\cdots 71}a^{27}-\frac{74\cdots 17}{97\cdots 71}a^{26}-\frac{54\cdots 51}{13\cdots 53}a^{25}-\frac{27\cdots 61}{97\cdots 71}a^{24}+\frac{34\cdots 64}{97\cdots 71}a^{23}+\frac{14\cdots 65}{97\cdots 71}a^{22}+\frac{49\cdots 00}{97\cdots 71}a^{21}-\frac{57\cdots 79}{97\cdots 71}a^{20}-\frac{32\cdots 58}{17\cdots 07}a^{19}-\frac{41\cdots 70}{97\cdots 71}a^{18}+\frac{43\cdots 46}{97\cdots 71}a^{17}+\frac{10\cdots 84}{97\cdots 71}a^{16}+\frac{16\cdots 09}{97\cdots 71}a^{15}-\frac{13\cdots 64}{97\cdots 71}a^{14}-\frac{54\cdots 31}{97\cdots 71}a^{13}-\frac{32\cdots 43}{97\cdots 71}a^{12}+\frac{58\cdots 15}{97\cdots 71}a^{11}+\frac{24\cdots 71}{97\cdots 71}a^{10}+\frac{61\cdots 62}{17\cdots 07}a^{9}+\frac{15\cdots 85}{97\cdots 71}a^{8}-\frac{10\cdots 71}{97\cdots 71}a^{7}-\frac{52\cdots 88}{97\cdots 71}a^{6}-\frac{13\cdots 65}{13\cdots 53}a^{5}-\frac{25\cdots 39}{97\cdots 71}a^{4}+\frac{29\cdots 74}{97\cdots 71}a^{3}+\frac{30\cdots 16}{97\cdots 71}a^{2}-\frac{81\cdots 59}{97\cdots 71}a+\frac{13\cdots 29}{12\cdots 49}$, $\frac{24\cdots 56}{97\cdots 71}a^{27}-\frac{31\cdots 83}{97\cdots 71}a^{26}-\frac{11\cdots 84}{97\cdots 71}a^{25}-\frac{12\cdots 42}{97\cdots 71}a^{24}+\frac{14\cdots 92}{97\cdots 71}a^{23}+\frac{53\cdots 20}{13\cdots 53}a^{22}+\frac{22\cdots 10}{97\cdots 71}a^{21}-\frac{24\cdots 78}{97\cdots 71}a^{20}-\frac{29\cdots 42}{97\cdots 71}a^{19}-\frac{28\cdots 05}{13\cdots 53}a^{18}+\frac{19\cdots 75}{97\cdots 71}a^{17}+\frac{47\cdots 13}{97\cdots 71}a^{16}+\frac{88\cdots 10}{97\cdots 71}a^{15}-\frac{64\cdots 96}{97\cdots 71}a^{14}-\frac{59\cdots 14}{97\cdots 71}a^{13}-\frac{19\cdots 38}{97\cdots 71}a^{12}+\frac{49\cdots 98}{97\cdots 71}a^{11}+\frac{73\cdots 53}{97\cdots 71}a^{10}+\frac{31\cdots 35}{13\cdots 53}a^{9}+\frac{65\cdots 97}{97\cdots 71}a^{8}-\frac{41\cdots 90}{97\cdots 71}a^{7}-\frac{26\cdots 37}{97\cdots 71}a^{6}-\frac{60\cdots 72}{97\cdots 71}a^{5}-\frac{14\cdots 07}{97\cdots 71}a^{4}+\frac{21\cdots 11}{97\cdots 71}a^{3}+\frac{58\cdots 18}{97\cdots 71}a^{2}+\frac{35\cdots 69}{97\cdots 71}a+\frac{26\cdots 31}{12\cdots 49}$, $\frac{23\cdots 06}{97\cdots 71}a^{27}-\frac{31\cdots 27}{97\cdots 71}a^{26}-\frac{12\cdots 46}{97\cdots 71}a^{25}-\frac{11\cdots 05}{97\cdots 71}a^{24}+\frac{14\cdots 99}{97\cdots 71}a^{23}+\frac{43\cdots 36}{97\cdots 71}a^{22}+\frac{22\cdots 22}{97\cdots 71}a^{21}-\frac{24\cdots 37}{97\cdots 71}a^{20}-\frac{41\cdots 36}{97\cdots 71}a^{19}-\frac{19\cdots 94}{97\cdots 71}a^{18}+\frac{19\cdots 79}{97\cdots 71}a^{17}+\frac{21\cdots 20}{13\cdots 53}a^{16}+\frac{86\cdots 02}{97\cdots 71}a^{15}-\frac{67\cdots 67}{97\cdots 71}a^{14}-\frac{10\cdots 30}{97\cdots 71}a^{13}-\frac{18\cdots 86}{97\cdots 71}a^{12}+\frac{58\cdots 90}{97\cdots 71}a^{11}+\frac{79\cdots 24}{97\cdots 71}a^{10}+\frac{21\cdots 42}{97\cdots 71}a^{9}+\frac{53\cdots 64}{97\cdots 71}a^{8}-\frac{48\cdots 70}{97\cdots 71}a^{7}-\frac{26\cdots 18}{97\cdots 71}a^{6}-\frac{63\cdots 97}{97\cdots 71}a^{5}-\frac{13\cdots 35}{97\cdots 71}a^{4}+\frac{21\cdots 44}{97\cdots 71}a^{3}+\frac{91\cdots 17}{13\cdots 53}a^{2}+\frac{36\cdots 28}{97\cdots 71}a+\frac{24\cdots 30}{12\cdots 49}$, $\frac{75\cdots 45}{13\cdots 53}a^{27}-\frac{40\cdots 10}{97\cdots 71}a^{26}-\frac{47\cdots 03}{97\cdots 71}a^{25}-\frac{26\cdots 82}{97\cdots 71}a^{24}+\frac{25\cdots 12}{13\cdots 53}a^{23}+\frac{18\cdots 86}{97\cdots 71}a^{22}+\frac{48\cdots 65}{97\cdots 71}a^{21}-\frac{26\cdots 71}{97\cdots 71}a^{20}-\frac{22\cdots 90}{97\cdots 71}a^{19}-\frac{57\cdots 05}{13\cdots 53}a^{18}+\frac{17\cdots 53}{97\cdots 71}a^{17}+\frac{12\cdots 07}{97\cdots 71}a^{16}+\frac{16\cdots 43}{97\cdots 71}a^{15}-\frac{46\cdots 69}{13\cdots 53}a^{14}-\frac{41\cdots 62}{97\cdots 71}a^{13}-\frac{28\cdots 32}{97\cdots 71}a^{12}-\frac{11\cdots 82}{97\cdots 71}a^{11}+\frac{12\cdots 91}{97\cdots 71}a^{10}+\frac{20\cdots 64}{97\cdots 71}a^{9}+\frac{32\cdots 83}{12\cdots 49}a^{8}+\frac{16\cdots 52}{13\cdots 53}a^{7}-\frac{20\cdots 22}{97\cdots 71}a^{6}+\frac{13\cdots 59}{13\cdots 53}a^{5}-\frac{39\cdots 39}{97\cdots 71}a^{4}+\frac{14\cdots 54}{97\cdots 71}a^{3}-\frac{33\cdots 48}{97\cdots 71}a^{2}-\frac{60\cdots 28}{97\cdots 71}a+\frac{22\cdots 86}{12\cdots 49}$, $\frac{51\cdots 33}{13\cdots 53}a^{27}-\frac{34\cdots 94}{97\cdots 71}a^{26}-\frac{63\cdots 60}{97\cdots 71}a^{25}-\frac{17\cdots 65}{97\cdots 71}a^{24}+\frac{15\cdots 57}{97\cdots 71}a^{23}+\frac{28\cdots 76}{97\cdots 71}a^{22}+\frac{31\cdots 22}{97\cdots 71}a^{21}-\frac{26\cdots 91}{97\cdots 71}a^{20}-\frac{44\cdots 40}{97\cdots 71}a^{19}-\frac{25\cdots 39}{97\cdots 71}a^{18}+\frac{21\cdots 73}{97\cdots 71}a^{17}+\frac{32\cdots 26}{97\cdots 71}a^{16}+\frac{97\cdots 24}{97\cdots 71}a^{15}-\frac{74\cdots 02}{97\cdots 71}a^{14}-\frac{12\cdots 80}{97\cdots 71}a^{13}-\frac{15\cdots 25}{97\cdots 71}a^{12}+\frac{29\cdots 08}{38\cdots 21}a^{11}+\frac{26\cdots 38}{97\cdots 71}a^{10}+\frac{14\cdots 02}{97\cdots 71}a^{9}-\frac{51\cdots 79}{13\cdots 53}a^{8}-\frac{12\cdots 97}{97\cdots 71}a^{7}-\frac{21\cdots 72}{97\cdots 71}a^{6}+\frac{20\cdots 82}{97\cdots 71}a^{5}-\frac{10\cdots 06}{97\cdots 71}a^{4}+\frac{12\cdots 60}{97\cdots 71}a^{3}+\frac{13\cdots 72}{13\cdots 53}a^{2}+\frac{36\cdots 91}{97\cdots 71}a+\frac{13\cdots 50}{12\cdots 49}$, $\frac{73\cdots 46}{97\cdots 71}a^{27}-\frac{13\cdots 64}{97\cdots 71}a^{26}-\frac{46\cdots 12}{97\cdots 71}a^{25}-\frac{35\cdots 47}{97\cdots 71}a^{24}+\frac{64\cdots 23}{97\cdots 71}a^{23}+\frac{20\cdots 96}{97\cdots 71}a^{22}+\frac{64\cdots 62}{97\cdots 71}a^{21}-\frac{11\cdots 18}{97\cdots 71}a^{20}-\frac{29\cdots 14}{97\cdots 71}a^{19}-\frac{53\cdots 55}{97\cdots 71}a^{18}+\frac{91\cdots 95}{97\cdots 71}a^{17}+\frac{34\cdots 16}{13\cdots 53}a^{16}+\frac{21\cdots 88}{97\cdots 71}a^{15}-\frac{32\cdots 65}{97\cdots 71}a^{14}-\frac{15\cdots 51}{97\cdots 71}a^{13}-\frac{40\cdots 65}{97\cdots 71}a^{12}+\frac{42\cdots 77}{97\cdots 71}a^{11}+\frac{64\cdots 56}{97\cdots 71}a^{10}+\frac{58\cdots 72}{13\cdots 53}a^{9}-\frac{32\cdots 41}{13\cdots 53}a^{8}-\frac{71\cdots 73}{97\cdots 71}a^{7}-\frac{88\cdots 66}{97\cdots 71}a^{6}+\frac{19\cdots 70}{97\cdots 71}a^{5}+\frac{33\cdots 00}{13\cdots 53}a^{4}+\frac{88\cdots 58}{97\cdots 71}a^{3}+\frac{29\cdots 60}{97\cdots 71}a^{2}-\frac{41\cdots 52}{97\cdots 71}a-\frac{23\cdots 30}{12\cdots 49}$, $\frac{38\cdots 34}{97\cdots 71}a^{27}-\frac{58\cdots 52}{97\cdots 71}a^{26}-\frac{16\cdots 59}{97\cdots 71}a^{25}-\frac{19\cdots 37}{97\cdots 71}a^{24}+\frac{27\cdots 11}{97\cdots 71}a^{23}+\frac{59\cdots 17}{97\cdots 71}a^{22}+\frac{35\cdots 29}{97\cdots 71}a^{21}-\frac{46\cdots 13}{97\cdots 71}a^{20}-\frac{55\cdots 93}{97\cdots 71}a^{19}-\frac{30\cdots 78}{97\cdots 71}a^{18}+\frac{36\cdots 19}{97\cdots 71}a^{17}+\frac{22\cdots 50}{97\cdots 71}a^{16}+\frac{13\cdots 28}{97\cdots 71}a^{15}-\frac{12\cdots 23}{97\cdots 71}a^{14}-\frac{22\cdots 61}{97\cdots 71}a^{13}-\frac{27\cdots 50}{97\cdots 71}a^{12}+\frac{16\cdots 78}{13\cdots 53}a^{11}+\frac{16\cdots 31}{97\cdots 71}a^{10}+\frac{29\cdots 15}{97\cdots 71}a^{9}+\frac{56\cdots 33}{97\cdots 71}a^{8}-\frac{12\cdots 95}{97\cdots 71}a^{7}-\frac{52\cdots 08}{12\cdots 49}a^{6}-\frac{46\cdots 12}{97\cdots 71}a^{5}-\frac{18\cdots 84}{97\cdots 71}a^{4}+\frac{34\cdots 20}{97\cdots 71}a^{3}+\frac{10\cdots 67}{97\cdots 71}a^{2}-\frac{30\cdots 03}{97\cdots 71}a+\frac{79\cdots 24}{12\cdots 49}$
|
| |
| Regulator: | \( 158620092291.7724 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 158620092291.7724 \cdot 203}{6\cdot\sqrt{14121388821225670988853483488774192350843817726481}}\cr\approx \mathstrut & 0.213443255955932 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_{14}$ (as 28T2):
| An abelian group of order 28 |
| The 28 conjugacy class representatives for $C_2\times C_{14}$ |
| Character table for $C_2\times C_{14}$ |
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 | ${\href{/padicField/2.14.0.1}{14} }^{2}$ | R | ${\href{/padicField/5.14.0.1}{14} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{14}$ | ${\href{/padicField/11.14.0.1}{14} }^{2}$ | ${\href{/padicField/13.7.0.1}{7} }^{4}$ | ${\href{/padicField/17.14.0.1}{14} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }^{2}$ | ${\href{/padicField/23.14.0.1}{14} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{14}$ | ${\href{/padicField/41.14.0.1}{14} }^{2}$ | R | ${\href{/padicField/47.14.0.1}{14} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }^{2}$ | ${\href{/padicField/59.14.0.1}{14} }^{2}$ |
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 $28$ | $2$ | $14$ | $14$ | |||
|
\(43\)
| 43.1.14.13a1.1 | $x^{14} + 43$ | $14$ | $1$ | $13$ | $C_{14}$ | $$[\ ]_{14}$$ |
| 43.1.14.13a1.1 | $x^{14} + 43$ | $14$ | $1$ | $13$ | $C_{14}$ | $$[\ ]_{14}$$ |