Normalized defining polynomial
\( x^{44} - x^{43} + 31 x^{42} + 2 x^{41} + 645 x^{40} - 1834 x^{39} + 15289 x^{38} - 49893 x^{37} + \cdots + 707281 \)
Invariants
| Degree: | $44$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 22]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(128\!\cdots\!125\)
\(\medspace = 5^{33}\cdot 67^{40}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(152.86\) | 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: | $5^{3/4}67^{10/11}\approx 152.8605809181423$ | ||
| Ramified primes: |
\(5\), \(67\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $44$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(335=5\cdot 67\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{335}(1,·)$, $\chi_{335}(131,·)$, $\chi_{335}(129,·)$, $\chi_{335}(9,·)$, $\chi_{335}(269,·)$, $\chi_{335}(14,·)$, $\chi_{335}(143,·)$, $\chi_{335}(148,·)$, $\chi_{335}(149,·)$, $\chi_{335}(22,·)$, $\chi_{335}(24,·)$, $\chi_{335}(68,·)$, $\chi_{335}(282,·)$, $\chi_{335}(283,·)$, $\chi_{335}(156,·)$, $\chi_{335}(158,·)$, $\chi_{335}(159,·)$, $\chi_{335}(292,·)$, $\chi_{335}(293,·)$, $\chi_{335}(263,·)$, $\chi_{335}(174,·)$, $\chi_{335}(308,·)$, $\chi_{335}(59,·)$, $\chi_{335}(62,·)$, $\chi_{335}(64,·)$, $\chi_{335}(193,·)$, $\chi_{335}(196,·)$, $\chi_{335}(198,·)$, $\chi_{335}(327,·)$, $\chi_{335}(76,·)$, $\chi_{335}(202,·)$, $\chi_{335}(332,·)$, $\chi_{335}(81,·)$, $\chi_{335}(82,·)$, $\chi_{335}(216,·)$, $\chi_{335}(89,·)$, $\chi_{335}(91,·)$, $\chi_{335}(92,·)$, $\chi_{335}(223,·)$, $\chi_{335}(226,·)$, $\chi_{335}(107,·)$, $\chi_{335}(241,·)$, $\chi_{335}(126,·)$, $\chi_{335}(277,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2097152}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{29}a^{37}-\frac{9}{29}a^{36}+\frac{10}{29}a^{35}-\frac{14}{29}a^{34}-\frac{9}{29}a^{33}-\frac{7}{29}a^{32}-\frac{7}{29}a^{31}+\frac{6}{29}a^{30}-\frac{7}{29}a^{29}-\frac{9}{29}a^{28}+\frac{2}{29}a^{27}-\frac{6}{29}a^{26}-\frac{10}{29}a^{25}+\frac{4}{29}a^{24}-\frac{9}{29}a^{23}-\frac{1}{29}a^{22}-\frac{3}{29}a^{21}-\frac{5}{29}a^{20}-\frac{12}{29}a^{19}+\frac{3}{29}a^{18}-\frac{2}{29}a^{17}-\frac{10}{29}a^{16}+\frac{5}{29}a^{15}+\frac{1}{29}a^{14}+\frac{7}{29}a^{12}-\frac{2}{29}a^{11}-\frac{10}{29}a^{10}-\frac{3}{29}a^{9}-\frac{1}{29}a^{8}+\frac{1}{29}a^{7}+\frac{5}{29}a^{6}-\frac{12}{29}a^{5}-\frac{14}{29}a^{4}+\frac{9}{29}a^{3}+\frac{12}{29}a^{2}-\frac{13}{29}a$, $\frac{1}{29}a^{38}-\frac{13}{29}a^{36}-\frac{11}{29}a^{35}+\frac{10}{29}a^{34}-\frac{1}{29}a^{33}-\frac{12}{29}a^{32}+\frac{1}{29}a^{31}-\frac{11}{29}a^{30}-\frac{14}{29}a^{29}+\frac{8}{29}a^{28}+\frac{12}{29}a^{27}-\frac{6}{29}a^{26}+\frac{1}{29}a^{25}-\frac{2}{29}a^{24}+\frac{5}{29}a^{23}-\frac{12}{29}a^{22}-\frac{3}{29}a^{21}+\frac{1}{29}a^{20}+\frac{11}{29}a^{19}-\frac{4}{29}a^{18}+\frac{1}{29}a^{17}+\frac{2}{29}a^{16}-\frac{12}{29}a^{15}+\frac{9}{29}a^{14}+\frac{7}{29}a^{13}+\frac{3}{29}a^{12}+\frac{1}{29}a^{11}-\frac{6}{29}a^{10}+\frac{1}{29}a^{9}-\frac{8}{29}a^{8}+\frac{14}{29}a^{7}+\frac{4}{29}a^{6}-\frac{6}{29}a^{5}-\frac{1}{29}a^{4}+\frac{6}{29}a^{3}+\frac{8}{29}a^{2}-\frac{1}{29}a$, $\frac{1}{29}a^{39}-\frac{12}{29}a^{36}-\frac{5}{29}a^{35}-\frac{9}{29}a^{34}-\frac{13}{29}a^{33}-\frac{3}{29}a^{32}+\frac{14}{29}a^{31}+\frac{6}{29}a^{30}+\frac{4}{29}a^{29}+\frac{11}{29}a^{28}-\frac{9}{29}a^{27}+\frac{10}{29}a^{26}+\frac{13}{29}a^{25}-\frac{1}{29}a^{24}-\frac{13}{29}a^{23}+\frac{13}{29}a^{22}-\frac{9}{29}a^{21}+\frac{4}{29}a^{20}+\frac{14}{29}a^{19}+\frac{11}{29}a^{18}+\frac{5}{29}a^{17}+\frac{3}{29}a^{16}-\frac{13}{29}a^{15}-\frac{9}{29}a^{14}+\frac{3}{29}a^{13}+\frac{5}{29}a^{12}-\frac{3}{29}a^{11}-\frac{13}{29}a^{10}+\frac{11}{29}a^{9}+\frac{1}{29}a^{8}-\frac{12}{29}a^{7}+\frac{1}{29}a^{6}-\frac{12}{29}a^{5}-\frac{2}{29}a^{4}+\frac{9}{29}a^{3}+\frac{10}{29}a^{2}+\frac{5}{29}a$, $\frac{1}{256447}a^{40}-\frac{150}{8843}a^{39}+\frac{4380}{256447}a^{38}-\frac{2172}{256447}a^{37}-\frac{61111}{256447}a^{36}-\frac{118538}{256447}a^{35}-\frac{95188}{256447}a^{34}-\frac{69855}{256447}a^{33}-\frac{41254}{256447}a^{32}+\frac{109928}{256447}a^{31}-\frac{98923}{256447}a^{30}+\frac{5257}{256447}a^{29}-\frac{101607}{256447}a^{28}-\frac{62240}{256447}a^{27}-\frac{50311}{256447}a^{26}-\frac{115947}{256447}a^{25}+\frac{39949}{256447}a^{24}+\frac{66467}{256447}a^{23}+\frac{103349}{256447}a^{22}-\frac{95396}{256447}a^{21}-\frac{21955}{256447}a^{20}+\frac{2061}{6931}a^{19}+\frac{45025}{256447}a^{18}-\frac{123073}{256447}a^{17}-\frac{116538}{256447}a^{16}+\frac{40973}{256447}a^{15}-\frac{96717}{256447}a^{14}+\frac{124886}{256447}a^{13}-\frac{5840}{256447}a^{12}-\frac{27505}{256447}a^{11}+\frac{3409}{8843}a^{10}+\frac{111752}{256447}a^{9}-\frac{52148}{256447}a^{8}+\frac{27000}{256447}a^{7}+\frac{104003}{256447}a^{6}+\frac{63554}{256447}a^{5}+\frac{997}{6931}a^{4}-\frac{105496}{256447}a^{3}-\frac{73467}{256447}a^{2}-\frac{103001}{256447}a-\frac{1128}{8843}$, $\frac{1}{23\!\cdots\!77}a^{41}-\frac{27\!\cdots\!95}{23\!\cdots\!77}a^{40}-\frac{62\!\cdots\!12}{23\!\cdots\!77}a^{39}+\frac{61\!\cdots\!25}{23\!\cdots\!77}a^{38}-\frac{32\!\cdots\!64}{23\!\cdots\!77}a^{37}+\frac{24\!\cdots\!04}{23\!\cdots\!77}a^{36}+\frac{65\!\cdots\!58}{23\!\cdots\!77}a^{35}+\frac{92\!\cdots\!42}{23\!\cdots\!77}a^{34}+\frac{53\!\cdots\!94}{23\!\cdots\!77}a^{33}+\frac{16\!\cdots\!42}{23\!\cdots\!77}a^{32}-\frac{76\!\cdots\!39}{23\!\cdots\!77}a^{31}+\frac{45\!\cdots\!72}{23\!\cdots\!77}a^{30}-\frac{11\!\cdots\!86}{23\!\cdots\!77}a^{29}+\frac{38\!\cdots\!16}{23\!\cdots\!77}a^{28}-\frac{32\!\cdots\!83}{23\!\cdots\!77}a^{27}+\frac{28\!\cdots\!66}{80\!\cdots\!13}a^{26}-\frac{12\!\cdots\!78}{62\!\cdots\!21}a^{25}+\frac{45\!\cdots\!74}{23\!\cdots\!77}a^{24}-\frac{10\!\cdots\!03}{23\!\cdots\!77}a^{23}+\frac{11\!\cdots\!16}{23\!\cdots\!77}a^{22}+\frac{60\!\cdots\!22}{23\!\cdots\!77}a^{21}-\frac{85\!\cdots\!02}{23\!\cdots\!77}a^{20}+\frac{94\!\cdots\!84}{23\!\cdots\!77}a^{19}+\frac{50\!\cdots\!68}{23\!\cdots\!77}a^{18}-\frac{20\!\cdots\!00}{23\!\cdots\!77}a^{17}-\frac{78\!\cdots\!71}{23\!\cdots\!77}a^{16}-\frac{33\!\cdots\!69}{23\!\cdots\!77}a^{15}+\frac{95\!\cdots\!92}{23\!\cdots\!77}a^{14}+\frac{10\!\cdots\!36}{23\!\cdots\!77}a^{13}-\frac{53\!\cdots\!52}{23\!\cdots\!77}a^{12}-\frac{76\!\cdots\!88}{23\!\cdots\!77}a^{11}+\frac{66\!\cdots\!26}{23\!\cdots\!77}a^{10}+\frac{70\!\cdots\!86}{23\!\cdots\!77}a^{9}+\frac{62\!\cdots\!82}{23\!\cdots\!77}a^{8}+\frac{58\!\cdots\!15}{23\!\cdots\!77}a^{7}-\frac{90\!\cdots\!72}{23\!\cdots\!77}a^{6}-\frac{76\!\cdots\!75}{23\!\cdots\!77}a^{5}+\frac{16\!\cdots\!06}{23\!\cdots\!77}a^{4}-\frac{20\!\cdots\!17}{23\!\cdots\!77}a^{3}+\frac{96\!\cdots\!63}{23\!\cdots\!77}a^{2}-\frac{37\!\cdots\!88}{23\!\cdots\!77}a+\frac{82\!\cdots\!35}{80\!\cdots\!13}$, $\frac{1}{67\!\cdots\!33}a^{42}-\frac{1}{67\!\cdots\!33}a^{41}+\frac{33\!\cdots\!35}{18\!\cdots\!09}a^{40}+\frac{31\!\cdots\!20}{67\!\cdots\!33}a^{39}-\frac{31\!\cdots\!53}{67\!\cdots\!33}a^{38}-\frac{27\!\cdots\!29}{18\!\cdots\!09}a^{37}-\frac{33\!\cdots\!21}{67\!\cdots\!33}a^{36}+\frac{25\!\cdots\!19}{67\!\cdots\!33}a^{35}+\frac{23\!\cdots\!99}{67\!\cdots\!33}a^{34}-\frac{32\!\cdots\!73}{67\!\cdots\!33}a^{33}+\frac{22\!\cdots\!89}{67\!\cdots\!33}a^{32}-\frac{22\!\cdots\!36}{67\!\cdots\!33}a^{31}+\frac{11\!\cdots\!33}{67\!\cdots\!33}a^{30}+\frac{16\!\cdots\!20}{67\!\cdots\!33}a^{29}-\frac{22\!\cdots\!54}{67\!\cdots\!33}a^{28}+\frac{26\!\cdots\!61}{67\!\cdots\!33}a^{27}-\frac{63\!\cdots\!06}{16\!\cdots\!33}a^{26}+\frac{15\!\cdots\!18}{67\!\cdots\!33}a^{25}-\frac{28\!\cdots\!13}{67\!\cdots\!33}a^{24}+\frac{21\!\cdots\!34}{67\!\cdots\!33}a^{23}+\frac{38\!\cdots\!97}{18\!\cdots\!09}a^{22}+\frac{16\!\cdots\!74}{67\!\cdots\!33}a^{21}-\frac{15\!\cdots\!66}{67\!\cdots\!33}a^{20}-\frac{20\!\cdots\!81}{67\!\cdots\!33}a^{19}-\frac{98\!\cdots\!95}{67\!\cdots\!33}a^{18}+\frac{16\!\cdots\!24}{67\!\cdots\!33}a^{17}-\frac{16\!\cdots\!78}{67\!\cdots\!33}a^{16}+\frac{56\!\cdots\!03}{67\!\cdots\!33}a^{15}+\frac{12\!\cdots\!22}{67\!\cdots\!33}a^{14}-\frac{43\!\cdots\!93}{67\!\cdots\!33}a^{13}-\frac{23\!\cdots\!52}{67\!\cdots\!33}a^{12}+\frac{44\!\cdots\!11}{67\!\cdots\!33}a^{11}-\frac{20\!\cdots\!62}{67\!\cdots\!33}a^{10}+\frac{30\!\cdots\!48}{67\!\cdots\!33}a^{9}-\frac{21\!\cdots\!38}{67\!\cdots\!33}a^{8}-\frac{16\!\cdots\!66}{67\!\cdots\!33}a^{7}-\frac{33\!\cdots\!80}{67\!\cdots\!33}a^{6}-\frac{32\!\cdots\!76}{67\!\cdots\!33}a^{5}-\frac{32\!\cdots\!48}{67\!\cdots\!33}a^{4}-\frac{13\!\cdots\!80}{67\!\cdots\!33}a^{3}+\frac{45\!\cdots\!30}{67\!\cdots\!33}a^{2}+\frac{54\!\cdots\!28}{23\!\cdots\!77}a-\frac{95\!\cdots\!35}{80\!\cdots\!13}$, $\frac{1}{19\!\cdots\!57}a^{43}-\frac{1}{19\!\cdots\!57}a^{42}+\frac{31}{19\!\cdots\!57}a^{41}-\frac{31\!\cdots\!84}{19\!\cdots\!57}a^{40}+\frac{33\!\cdots\!82}{19\!\cdots\!57}a^{39}-\frac{20\!\cdots\!90}{19\!\cdots\!57}a^{38}+\frac{12\!\cdots\!42}{19\!\cdots\!57}a^{37}-\frac{12\!\cdots\!67}{19\!\cdots\!57}a^{36}+\frac{59\!\cdots\!17}{19\!\cdots\!57}a^{35}+\frac{67\!\cdots\!63}{81\!\cdots\!63}a^{34}-\frac{54\!\cdots\!58}{19\!\cdots\!57}a^{33}+\frac{16\!\cdots\!44}{19\!\cdots\!57}a^{32}+\frac{68\!\cdots\!55}{19\!\cdots\!57}a^{31}+\frac{27\!\cdots\!93}{19\!\cdots\!57}a^{30}+\frac{70\!\cdots\!22}{19\!\cdots\!57}a^{29}+\frac{83\!\cdots\!44}{19\!\cdots\!57}a^{28}-\frac{28\!\cdots\!74}{19\!\cdots\!57}a^{27}+\frac{20\!\cdots\!70}{19\!\cdots\!57}a^{26}+\frac{56\!\cdots\!04}{19\!\cdots\!57}a^{25}+\frac{27\!\cdots\!82}{19\!\cdots\!57}a^{24}+\frac{67\!\cdots\!31}{19\!\cdots\!57}a^{23}+\frac{90\!\cdots\!24}{19\!\cdots\!57}a^{22}+\frac{18\!\cdots\!27}{19\!\cdots\!57}a^{21}+\frac{19\!\cdots\!50}{19\!\cdots\!57}a^{20}+\frac{39\!\cdots\!57}{19\!\cdots\!57}a^{19}+\frac{27\!\cdots\!52}{19\!\cdots\!57}a^{18}-\frac{90\!\cdots\!75}{19\!\cdots\!57}a^{17}-\frac{25\!\cdots\!53}{19\!\cdots\!57}a^{16}+\frac{79\!\cdots\!46}{19\!\cdots\!57}a^{15}-\frac{90\!\cdots\!72}{19\!\cdots\!57}a^{14}-\frac{55\!\cdots\!39}{19\!\cdots\!57}a^{13}-\frac{23\!\cdots\!90}{19\!\cdots\!57}a^{12}-\frac{75\!\cdots\!97}{19\!\cdots\!57}a^{11}-\frac{59\!\cdots\!00}{19\!\cdots\!57}a^{10}-\frac{63\!\cdots\!14}{19\!\cdots\!57}a^{9}+\frac{32\!\cdots\!15}{19\!\cdots\!57}a^{8}+\frac{29\!\cdots\!81}{19\!\cdots\!57}a^{7}-\frac{58\!\cdots\!45}{19\!\cdots\!57}a^{6}-\frac{75\!\cdots\!10}{19\!\cdots\!57}a^{5}-\frac{66\!\cdots\!76}{52\!\cdots\!61}a^{4}-\frac{36\!\cdots\!19}{19\!\cdots\!57}a^{3}+\frac{10\!\cdots\!89}{67\!\cdots\!33}a^{2}+\frac{47\!\cdots\!26}{23\!\cdots\!77}a+\frac{16\!\cdots\!35}{80\!\cdots\!13}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $21$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{28257716234339050816854957354039621869505384100799664930392194228113719235426754855930}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{43} - \frac{223970382531987922640090312966521487156305382037362988245926360609611614295736927598649}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{42} + \frac{1095054739333238771410295939162339751580638956975914550989942716583140507248767624656889}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{41} - \frac{6034349991556886539361572097751338692525309853481209115452305919469714509552215570983879}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{40} + \frac{18558783721532644676434341437210517234972262499889547998134936334455274627343924594403093}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{39} - \frac{178025262236781946958040401723461623154714987918493733686618799577333345684368942062341421}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{38} + \frac{806019396152117778706716840338218880926548629814713774399821467459832598467200254932015593}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{37} - \frac{4445214207425309587820417311973239133206221436690379216706741728483909800967837683436151083}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{36} + \frac{17261687886893707645735496403334530070698666437043623229488668257592961556652656820291797761}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{35} - \frac{72482252132029096102630325902423321881307343482295190776978737557703402956234691421731905199}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{34} + \frac{244925372614677460263952021681567076092594557232412262771186120386529405338667325471913354373}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{33} - \frac{879218729231240930193866718232257653038773281111919969282969173858540022194161907388139599762}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{32} + \frac{2726891673212468402007151809199963579588605618562869782875793392936751020707484702833500558006}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{31} - \frac{8955643884993043662800178753006333510771731210679123826144081440932072464913473146015675866793}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{30} + \frac{26517577957978245065131065366741600021992505356698407449748107362417446672424657972623381310083}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{29} - \frac{78790706336240969126470218192573704774067345730940915791738641920417186057396031718971764124337}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{28} + \frac{213265558870115821565547147205556493958787336313935185740209930026979280218034102797397414965689}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{27} - \frac{571192840529050207847019554630618077213452093813455524529043568878628355548975141500907759019767}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{26} + \frac{1398333202104702101760417798512504589629311326426824041967889309057003739233143971565739671964935}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{25} - \frac{3352539774770087478164901876605555679103609757381319081164375488619167834910435129647950261760423}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{24} + \frac{7489933585618106489734329456443498080902123223988671910795153711185428376051713665196260364814944}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{23} - \frac{16437958718529414728801682124401366301341074146175940861706950892285960803778715813901658750868317}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{22} + \frac{33933396715751765663879533084497287342474669604398007846419244182405909664948516447130774319206597}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{21} - \frac{68708782437673974589193352733431296680623374937675183207618843748069438745788018784998921730651690}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{20} + \frac{130556458077015508636468718340166849291614018074049164354280385519741098694303344936562440009168626}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{19} - \frac{241017868791292963675533063301350114082058124707681207176891471040939660661968758061670411336782908}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{18} + \frac{412124697330021859187684922407370396772789303029404598021751761840269981393197023751449322608858180}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{17} - \frac{667587188603175456737961401741946344844451032159431958804647385057566328000134440303897598400582708}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{16} + \frac{986001247746318766694303398018224241085289261684861092668741586251736292495328158519185318679170080}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{15} - \frac{1364922331009803381715933302146110203442176520606987667469572797139247880061261111678035091096353122}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{14} + \frac{1654030448840118790378034123884190778630738613014885017389917788213400918011703397106003367089500595}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{13} - \frac{1828652916450102040796234484916108190018381061528840308629240630558917110439319203951467357928409444}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{12} + \frac{1742769642511955111080973903550855803827402698888819046333380800106249038080767151160437507891230010}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{11} - \frac{1458868443899981180664018408771214577462436668858066766996446028233944532657317392827757294558707224}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{10} + \frac{849304349044390883743275166445741979360204116276947698510118255463322466233929307565389561304056052}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{9} - \frac{494734943547207906138868060994152545474933532128056046960544171126775613687383503902837280233395535}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{8} + \frac{218835201452428984533413654186967726541175855155895787930166823517425463034227856382274492730237643}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{7} - \frac{61871653592362029403071020556875348110306994184652203391793486059563200284597593362730078924495709}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{6} - \frac{14731249158214538699413587463345967968224196557164987155622156232042006681372689502136997135647736}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{5} - \frac{4169010794061413082033384187566839068490244417072667435506348597378831822399582065597588635961908}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{4} - \frac{20199643011390015614339040895453327111173606022028625546045051860376954879979443261931496520115}{262898413805518870214019707265866801974095238180518016759947350153347974958287060095120427539} a^{3} - \frac{439649209318937784203029697945105844959616412257166471573996859614996755941990172607827963216577}{7624054000360047236206571510710137257248761907235022486038473154447091273790324742758492398631} a^{2} - \frac{11313950540893919624549012108642578193463502458932904961191473841761681957464771566732997711}{9065462545017892076345507147098855240486042695879931612411977591494757757182312417073118191} a - \frac{1191667232004186609445664451790570088346009533210504483504669387970925834174359088450106343}{9065462545017892076345507147098855240486042695879931612411977591494757757182312417073118191} \)
(order $10$)
| 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 44 |
| The 44 conjugacy class representatives for $C_{44}$ |
| Character table for $C_{44}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 11.11.1822837804551761449.1, 22.22.162243049887845980095628744560672832080078125.1 |
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 | $44$ | $44$ | R | $44$ | ${\href{/padicField/11.11.0.1}{11} }^{4}$ | $44$ | $44$ | $22^{2}$ | $44$ | ${\href{/padicField/29.2.0.1}{2} }^{22}$ | ${\href{/padicField/31.11.0.1}{11} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{11}$ | ${\href{/padicField/41.11.0.1}{11} }^{4}$ | $44$ | $44$ | $44$ | $22^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $44$ | $4$ | $11$ | $33$ | |||
|
\(67\)
| Deg $44$ | $11$ | $4$ | $40$ |