Normalized defining polynomial
\( x^{38} - x^{37} + 91 x^{36} - 24 x^{35} + 5113 x^{34} - 166 x^{33} + 173717 x^{32} + 27194 x^{31} + \cdots + 13841287201 \)
Invariants
Degree: | $38$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 19]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-152\!\cdots\!747\) \(\medspace = -\,3^{19}\cdot 191^{36}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(250.92\) | 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: | $3^{1/2}191^{18/19}\approx 250.92250057347005$ | ||
Ramified primes: | \(3\), \(191\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Gal(K/\Q) }$: | $38$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(573=3\cdot 191\) | ||
Dirichlet character group: | $\lbrace$$\chi_{573}(1,·)$, $\chi_{573}(260,·)$, $\chi_{573}(5,·)$, $\chi_{573}(518,·)$, $\chi_{573}(136,·)$, $\chi_{573}(535,·)$, $\chi_{573}(532,·)$, $\chi_{573}(407,·)$, $\chi_{573}(536,·)$, $\chi_{573}(388,·)$, $\chi_{573}(154,·)$, $\chi_{573}(412,·)$, $\chi_{573}(542,·)$, $\chi_{573}(32,·)$, $\chi_{573}(418,·)$, $\chi_{573}(298,·)$, $\chi_{573}(562,·)$, $\chi_{573}(559,·)$, $\chi_{573}(434,·)$, $\chi_{573}(52,·)$, $\chi_{573}(316,·)$, $\chi_{573}(160,·)$, $\chi_{573}(451,·)$, $\chi_{573}(196,·)$, $\chi_{573}(197,·)$, $\chi_{573}(341,·)$, $\chi_{573}(344,·)$, $\chi_{573}(221,·)$, $\chi_{573}(223,·)$, $\chi_{573}(227,·)$, $\chi_{573}(107,·)$, $\chi_{573}(368,·)$, $\chi_{573}(371,·)$, $\chi_{573}(25,·)$, $\chi_{573}(503,·)$, $\chi_{573}(121,·)$, $\chi_{573}(125,·)$, $\chi_{573}(383,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{262144}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{3}$, $\frac{1}{7}a^{10}-\frac{1}{7}a^{4}$, $\frac{1}{7}a^{11}-\frac{1}{7}a^{5}$, $\frac{1}{7}a^{12}-\frac{1}{7}a^{6}$, $\frac{1}{7}a^{13}-\frac{1}{7}a$, $\frac{1}{49}a^{14}-\frac{2}{49}a^{8}+\frac{1}{49}a^{2}$, $\frac{1}{49}a^{15}-\frac{2}{49}a^{9}+\frac{1}{49}a^{3}$, $\frac{1}{49}a^{16}-\frac{2}{49}a^{10}+\frac{1}{49}a^{4}$, $\frac{1}{49}a^{17}-\frac{2}{49}a^{11}+\frac{1}{49}a^{5}$, $\frac{1}{49}a^{18}-\frac{2}{49}a^{12}+\frac{1}{49}a^{6}$, $\frac{1}{49}a^{19}-\frac{2}{49}a^{13}+\frac{1}{49}a^{7}$, $\frac{1}{49}a^{20}-\frac{3}{49}a^{8}+\frac{2}{49}a^{2}$, $\frac{1}{343}a^{21}-\frac{3}{343}a^{15}+\frac{3}{343}a^{9}-\frac{1}{343}a^{3}$, $\frac{1}{343}a^{22}-\frac{3}{343}a^{16}+\frac{3}{343}a^{10}-\frac{1}{343}a^{4}$, $\frac{1}{343}a^{23}-\frac{3}{343}a^{17}+\frac{3}{343}a^{11}-\frac{1}{343}a^{5}$, $\frac{1}{343}a^{24}-\frac{3}{343}a^{18}+\frac{3}{343}a^{12}-\frac{1}{343}a^{6}$, $\frac{1}{343}a^{25}-\frac{3}{343}a^{19}+\frac{3}{343}a^{13}-\frac{1}{343}a^{7}$, $\frac{1}{2401}a^{26}-\frac{2}{2401}a^{25}+\frac{1}{2401}a^{23}-\frac{2}{2401}a^{22}-\frac{17}{2401}a^{20}-\frac{15}{2401}a^{19}-\frac{17}{2401}a^{17}-\frac{15}{2401}a^{16}-\frac{18}{2401}a^{14}+\frac{134}{2401}a^{13}-\frac{2}{49}a^{12}-\frac{165}{2401}a^{11}+\frac{134}{2401}a^{10}-\frac{2}{49}a^{9}-\frac{113}{2401}a^{8}-\frac{117}{2401}a^{7}-\frac{5}{49}a^{6}-\frac{1191}{2401}a^{5}-\frac{803}{2401}a^{4}-\frac{5}{49}a^{3}+\frac{24}{49}a^{2}-\frac{2}{7}a$, $\frac{1}{2401}a^{27}+\frac{3}{2401}a^{25}+\frac{1}{2401}a^{24}+\frac{3}{2401}a^{22}-\frac{3}{2401}a^{21}-\frac{2}{2401}a^{19}-\frac{17}{2401}a^{18}-\frac{2}{2401}a^{16}-\frac{11}{2401}a^{15}+\frac{93}{2401}a^{13}-\frac{18}{2401}a^{12}+\frac{1}{49}a^{11}+\frac{93}{2401}a^{10}-\frac{22}{2401}a^{9}+\frac{1}{49}a^{8}-\frac{94}{2401}a^{7}+\frac{377}{2401}a^{6}-\frac{22}{49}a^{5}+\frac{592}{2401}a^{4}+\frac{54}{343}a^{3}-\frac{22}{49}a^{2}+\frac{2}{7}a$, $\frac{1}{2401}a^{28}+\frac{3}{2401}a^{22}-\frac{15}{2401}a^{16}+\frac{17}{2401}a^{10}-\frac{6}{2401}a^{4}$, $\frac{1}{16807}a^{29}+\frac{3}{16807}a^{28}+\frac{1}{16807}a^{27}-\frac{4}{16807}a^{25}+\frac{15}{16807}a^{24}+\frac{3}{16807}a^{23}-\frac{2}{16807}a^{22}-\frac{17}{16807}a^{21}-\frac{1}{343}a^{20}-\frac{79}{16807}a^{19}-\frac{10}{16807}a^{18}+\frac{132}{16807}a^{17}-\frac{54}{16807}a^{16}-\frac{116}{16807}a^{15}+\frac{3}{343}a^{14}-\frac{761}{16807}a^{13}+\frac{612}{16807}a^{12}-\frac{571}{16807}a^{11}+\frac{200}{16807}a^{10}+\frac{916}{16807}a^{9}+\frac{12}{343}a^{8}-\frac{871}{16807}a^{7}-\frac{7477}{16807}a^{6}+\frac{6609}{16807}a^{5}+\frac{158}{343}a^{4}-\frac{9}{343}a^{3}+\frac{23}{49}a^{2}+\frac{2}{7}a$, $\frac{1}{16807}a^{30}-\frac{1}{16807}a^{28}-\frac{3}{16807}a^{27}+\frac{3}{16807}a^{26}+\frac{13}{16807}a^{25}+\frac{1}{2401}a^{24}-\frac{4}{16807}a^{23}-\frac{4}{16807}a^{22}+\frac{2}{16807}a^{21}-\frac{51}{16807}a^{20}+\frac{122}{16807}a^{19}+\frac{15}{16807}a^{18}+\frac{117}{16807}a^{17}-\frac{164}{16807}a^{16}+\frac{152}{16807}a^{15}+\frac{44}{16807}a^{14}-\frac{969}{16807}a^{13}-\frac{545}{16807}a^{12}-\frac{614}{16807}a^{11}-\frac{1028}{16807}a^{10}+\frac{241}{16807}a^{9}+\frac{1033}{16807}a^{8}-\frac{881}{16807}a^{7}+\frac{8068}{16807}a^{6}-\frac{2929}{16807}a^{5}+\frac{955}{2401}a^{4}+\frac{97}{343}a^{3}+\frac{1}{49}a^{2}-\frac{1}{7}a$, $\frac{1}{117649}a^{31}-\frac{1}{117649}a^{30}+\frac{3}{117649}a^{29}-\frac{11}{117649}a^{28}-\frac{11}{117649}a^{27}+\frac{24}{117649}a^{26}+\frac{34}{117649}a^{25}-\frac{24}{16807}a^{24}-\frac{72}{117649}a^{23}+\frac{89}{117649}a^{22}-\frac{107}{117649}a^{21}+\frac{768}{117649}a^{20}-\frac{3}{117649}a^{19}+\frac{664}{117649}a^{18}-\frac{726}{117649}a^{17}-\frac{1174}{117649}a^{16}-\frac{194}{117649}a^{15}-\frac{1020}{117649}a^{14}+\frac{488}{117649}a^{13}-\frac{5720}{117649}a^{12}+\frac{529}{117649}a^{11}-\frac{1060}{117649}a^{10}+\frac{998}{117649}a^{9}+\frac{228}{117649}a^{8}-\frac{520}{117649}a^{7}+\frac{38839}{117649}a^{6}-\frac{4421}{16807}a^{5}+\frac{191}{2401}a^{4}+\frac{103}{343}a^{3}+\frac{13}{49}a^{2}-\frac{1}{7}a$, $\frac{1}{117649}a^{32}+\frac{2}{117649}a^{30}-\frac{1}{117649}a^{29}-\frac{1}{117649}a^{28}+\frac{20}{117649}a^{27}+\frac{9}{117649}a^{26}-\frac{64}{117649}a^{25}-\frac{135}{117649}a^{24}-\frac{11}{117649}a^{23}+\frac{66}{117649}a^{22}-\frac{144}{117649}a^{21}-\frac{1146}{117649}a^{20}+\frac{843}{117649}a^{19}-\frac{132}{117649}a^{18}-\frac{143}{117649}a^{17}-\frac{1011}{117649}a^{16}+\frac{32}{117649}a^{15}-\frac{146}{16807}a^{14}-\frac{318}{117649}a^{13}+\frac{3895}{117649}a^{12}+\frac{3557}{117649}a^{11}-\frac{5228}{117649}a^{10}-\frac{6425}{117649}a^{9}+\frac{4559}{117649}a^{8}+\frac{4341}{117649}a^{7}-\frac{32442}{117649}a^{6}-\frac{4945}{16807}a^{5}+\frac{60}{343}a^{4}-\frac{72}{343}a^{3}+\frac{9}{49}a^{2}-\frac{3}{7}a$, $\frac{1}{89766187}a^{33}+\frac{117}{89766187}a^{32}-\frac{75}{89766187}a^{31}-\frac{1762}{89766187}a^{30}-\frac{692}{89766187}a^{29}+\frac{14337}{89766187}a^{28}-\frac{1221}{89766187}a^{27}-\frac{14131}{89766187}a^{26}+\frac{14492}{12823741}a^{25}+\frac{16285}{12823741}a^{24}+\frac{113943}{89766187}a^{23}+\frac{108434}{89766187}a^{22}-\frac{123659}{89766187}a^{21}-\frac{161232}{89766187}a^{20}-\frac{875530}{89766187}a^{19}-\frac{691185}{89766187}a^{18}-\frac{97689}{89766187}a^{17}-\frac{553095}{89766187}a^{16}-\frac{574085}{89766187}a^{15}+\frac{229786}{89766187}a^{14}+\frac{944729}{89766187}a^{13}-\frac{6066877}{89766187}a^{12}-\frac{5574549}{89766187}a^{11}+\frac{2371851}{89766187}a^{10}+\frac{567753}{12823741}a^{9}-\frac{2407520}{89766187}a^{8}-\frac{2924515}{89766187}a^{7}-\frac{33501292}{89766187}a^{6}-\frac{395726}{12823741}a^{5}+\frac{824345}{1831963}a^{4}-\frac{31312}{261709}a^{3}-\frac{6900}{37387}a^{2}+\frac{507}{5341}a-\frac{54}{109}$, $\frac{1}{89766187}a^{34}-\frac{30}{89766187}a^{32}+\frac{146}{89766187}a^{31}-\frac{548}{89766187}a^{30}+\frac{2215}{89766187}a^{29}+\frac{17499}{89766187}a^{28}-\frac{17770}{89766187}a^{27}-\frac{16915}{89766187}a^{26}+\frac{8259}{12823741}a^{25}-\frac{112080}{89766187}a^{24}-\frac{18419}{89766187}a^{23}-\frac{70626}{89766187}a^{22}+\frac{47927}{89766187}a^{21}-\frac{5541}{12823741}a^{20}-\frac{654116}{89766187}a^{19}+\frac{1276}{12823741}a^{18}-\frac{106104}{89766187}a^{17}+\frac{856336}{89766187}a^{16}+\frac{644387}{89766187}a^{15}+\frac{793761}{89766187}a^{14}-\frac{4974033}{89766187}a^{13}+\frac{76597}{89766187}a^{12}+\frac{5889195}{89766187}a^{11}-\frac{1635720}{89766187}a^{10}+\frac{1876004}{89766187}a^{9}+\frac{4340086}{89766187}a^{8}-\frac{50268}{823543}a^{7}-\frac{2746056}{89766187}a^{6}-\frac{3427211}{12823741}a^{5}+\frac{734742}{1831963}a^{4}+\frac{101589}{261709}a^{3}+\frac{14277}{37387}a^{2}-\frac{925}{5341}a-\frac{4}{109}$, $\frac{1}{28\!\cdots\!29}a^{35}-\frac{13722960588}{28\!\cdots\!29}a^{34}-\frac{13395831}{58\!\cdots\!21}a^{33}-\frac{4660720226802}{28\!\cdots\!29}a^{32}+\frac{10608979324751}{28\!\cdots\!29}a^{31}-\frac{47028408171421}{28\!\cdots\!29}a^{30}+\frac{31483027476815}{28\!\cdots\!29}a^{29}-\frac{57860024469234}{28\!\cdots\!29}a^{28}+\frac{300888128345415}{28\!\cdots\!29}a^{27}-\frac{4880325001574}{58\!\cdots\!21}a^{26}-\frac{35\!\cdots\!73}{28\!\cdots\!29}a^{25}+\frac{22\!\cdots\!10}{28\!\cdots\!29}a^{24}+\frac{582194426089153}{40\!\cdots\!47}a^{23}-\frac{33\!\cdots\!10}{28\!\cdots\!29}a^{22}+\frac{556061758226451}{28\!\cdots\!29}a^{21}+\frac{10\!\cdots\!39}{28\!\cdots\!29}a^{20}+\frac{39\!\cdots\!55}{28\!\cdots\!29}a^{19}-\frac{22\!\cdots\!20}{28\!\cdots\!29}a^{18}-\frac{20\!\cdots\!56}{28\!\cdots\!29}a^{17}+\frac{10\!\cdots\!88}{28\!\cdots\!29}a^{16}-\frac{19\!\cdots\!56}{28\!\cdots\!29}a^{15}+\frac{76\!\cdots\!31}{28\!\cdots\!29}a^{14}-\frac{74\!\cdots\!01}{28\!\cdots\!29}a^{13}+\frac{17\!\cdots\!64}{28\!\cdots\!29}a^{12}+\frac{16\!\cdots\!28}{28\!\cdots\!29}a^{11}-\frac{95\!\cdots\!64}{28\!\cdots\!29}a^{10}+\frac{16\!\cdots\!31}{28\!\cdots\!29}a^{9}-\frac{18\!\cdots\!34}{28\!\cdots\!29}a^{8}-\frac{52\!\cdots\!24}{28\!\cdots\!29}a^{7}+\frac{587509322012215}{40\!\cdots\!47}a^{6}+\frac{51\!\cdots\!88}{58\!\cdots\!21}a^{5}+\frac{35\!\cdots\!01}{83\!\cdots\!03}a^{4}+\frac{23419799942713}{11\!\cdots\!29}a^{3}-\frac{77010725974068}{169765516786747}a^{2}-\frac{9954329976680}{24252216683821}a+\frac{205248013810}{494943197629}$, $\frac{1}{10\!\cdots\!41}a^{36}-\frac{27\!\cdots\!84}{10\!\cdots\!41}a^{35}+\frac{43\!\cdots\!32}{14\!\cdots\!63}a^{34}-\frac{31\!\cdots\!29}{10\!\cdots\!41}a^{33}-\frac{43\!\cdots\!04}{10\!\cdots\!41}a^{32}-\frac{15\!\cdots\!11}{10\!\cdots\!41}a^{31}+\frac{84\!\cdots\!06}{10\!\cdots\!41}a^{30}-\frac{10\!\cdots\!08}{10\!\cdots\!41}a^{29}-\frac{42\!\cdots\!05}{10\!\cdots\!41}a^{28}-\frac{10\!\cdots\!26}{14\!\cdots\!63}a^{27}-\frac{11\!\cdots\!17}{10\!\cdots\!41}a^{26}-\frac{13\!\cdots\!85}{10\!\cdots\!41}a^{25}+\frac{10\!\cdots\!39}{14\!\cdots\!63}a^{24}+\frac{12\!\cdots\!10}{10\!\cdots\!41}a^{23}+\frac{13\!\cdots\!30}{10\!\cdots\!41}a^{22}+\frac{57\!\cdots\!26}{10\!\cdots\!41}a^{21}+\frac{80\!\cdots\!74}{10\!\cdots\!41}a^{20}+\frac{10\!\cdots\!25}{10\!\cdots\!41}a^{19}+\frac{10\!\cdots\!22}{10\!\cdots\!41}a^{18}-\frac{15\!\cdots\!25}{10\!\cdots\!41}a^{17}-\frac{66\!\cdots\!43}{10\!\cdots\!41}a^{16}-\frac{36\!\cdots\!49}{10\!\cdots\!41}a^{15}+\frac{34\!\cdots\!51}{10\!\cdots\!41}a^{14}+\frac{55\!\cdots\!53}{10\!\cdots\!41}a^{13}-\frac{15\!\cdots\!91}{10\!\cdots\!41}a^{12}-\frac{71\!\cdots\!10}{10\!\cdots\!41}a^{11}-\frac{51\!\cdots\!52}{10\!\cdots\!41}a^{10}+\frac{54\!\cdots\!82}{10\!\cdots\!41}a^{9}-\frac{22\!\cdots\!72}{10\!\cdots\!41}a^{8}+\frac{74\!\cdots\!85}{14\!\cdots\!63}a^{7}+\frac{64\!\cdots\!22}{21\!\cdots\!09}a^{6}+\frac{47\!\cdots\!78}{30\!\cdots\!87}a^{5}+\frac{64\!\cdots\!09}{43\!\cdots\!41}a^{4}+\frac{20\!\cdots\!17}{61\!\cdots\!63}a^{3}+\frac{10\!\cdots\!70}{25\!\cdots\!63}a^{2}+\frac{55\!\cdots\!94}{12\!\cdots\!87}a+\frac{54\!\cdots\!69}{25\!\cdots\!63}$, $\frac{1}{30\!\cdots\!01}a^{37}+\frac{10\!\cdots\!04}{30\!\cdots\!01}a^{36}+\frac{22\!\cdots\!11}{43\!\cdots\!43}a^{35}-\frac{11\!\cdots\!41}{30\!\cdots\!01}a^{34}-\frac{12\!\cdots\!79}{30\!\cdots\!01}a^{33}-\frac{14\!\cdots\!55}{30\!\cdots\!01}a^{32}+\frac{34\!\cdots\!89}{30\!\cdots\!01}a^{31}+\frac{27\!\cdots\!92}{30\!\cdots\!01}a^{30}+\frac{47\!\cdots\!96}{30\!\cdots\!01}a^{29}-\frac{82\!\cdots\!72}{43\!\cdots\!43}a^{28}+\frac{27\!\cdots\!17}{30\!\cdots\!01}a^{27}-\frac{20\!\cdots\!00}{30\!\cdots\!01}a^{26}-\frac{54\!\cdots\!41}{43\!\cdots\!43}a^{25}+\frac{24\!\cdots\!40}{30\!\cdots\!01}a^{24}+\frac{76\!\cdots\!60}{30\!\cdots\!01}a^{23}+\frac{22\!\cdots\!90}{30\!\cdots\!01}a^{22}+\frac{32\!\cdots\!62}{30\!\cdots\!01}a^{21}-\frac{18\!\cdots\!29}{30\!\cdots\!01}a^{20}+\frac{18\!\cdots\!38}{30\!\cdots\!01}a^{19}+\frac{38\!\cdots\!13}{30\!\cdots\!01}a^{18}-\frac{22\!\cdots\!36}{30\!\cdots\!01}a^{17}+\frac{13\!\cdots\!76}{30\!\cdots\!01}a^{16}-\frac{26\!\cdots\!36}{30\!\cdots\!01}a^{15}-\frac{58\!\cdots\!71}{30\!\cdots\!01}a^{14}+\frac{77\!\cdots\!92}{30\!\cdots\!01}a^{13}-\frac{58\!\cdots\!32}{30\!\cdots\!01}a^{12}+\frac{10\!\cdots\!74}{30\!\cdots\!01}a^{11}-\frac{17\!\cdots\!29}{30\!\cdots\!01}a^{10}-\frac{12\!\cdots\!11}{30\!\cdots\!01}a^{9}+\frac{13\!\cdots\!55}{43\!\cdots\!43}a^{8}-\frac{42\!\cdots\!16}{62\!\cdots\!49}a^{7}-\frac{74\!\cdots\!05}{81\!\cdots\!23}a^{6}-\frac{47\!\cdots\!35}{12\!\cdots\!01}a^{5}+\frac{90\!\cdots\!74}{18\!\cdots\!43}a^{4}-\frac{58\!\cdots\!40}{25\!\cdots\!49}a^{3}+\frac{96\!\cdots\!36}{37\!\cdots\!07}a^{2}-\frac{18\!\cdots\!05}{75\!\cdots\!43}a+\frac{41\!\cdots\!25}{15\!\cdots\!07}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $7$ |
Class group and class number
not computed
Unit group
Rank: | $18$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{4816252170315163138117851302261472656312674300013663836892486465989579986185475594712658494600501569654350537598801945091168658119}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{37} + \frac{5322726934380524893507794006270716809335601686347935155502693999574734824891508410899939111857665950269546714043375237115588293953}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{36} - \frac{62699942078416586227456131556328821444612307045940474841454923130814403655199239979582031090553059628994332347953378104448228635600}{83757877717200665823588768258884108516069938406400008166704773414845491540297010507299068356256451283016271438742194601760510504088259226069} a^{35} + \frac{161810822297524181622244179389560041884974294813201144439592946232036056377188745592350621504857814930439481200555680878723601799807}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{34} - \frac{24648104360251243649571866306262349559364502139989739299749020622901118282349373460402517614155196058657598485939586595811497950882279}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{33} + \frac{3393469072260192012227163856798827458065317167181502196009082504853183371106704178776202306822149252603730078447686620253645985674620}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{32} - \frac{837337159972834517409998212545347779319415773814819495854493488558366667621485183746983940732548258208666075721626609986161259703718419}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{31} - \frac{42881083074839588382148043877569979505745219856359788127578987490511314252218759670987125608899623894791040786338969084826473293887781}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{30} - \frac{20426220454507742514281548342578706918226457857762096136565120014670759957126122841732185203759689245841512376997652355178362575701863913}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{29} - \frac{605113323629505739423050330371045799545316931463014943764958132881508815531173582765034717533977693484076516429760615258073761410278541}{83757877717200665823588768258884108516069938406400008166704773414845491540297010507299068356256451283016271438742194601760510504088259226069} a^{28} - \frac{356353975508648648653018110275245937535449379596165698608645849619831101358123626112434155374083569175835049593988385131335471413962010428}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{27} - \frac{160049162708613813105546888375621411946182664802341591003260804211181644555177055841358933810007425776318033529137752066224655324929217312}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{26} - \frac{674828161901453630664970064357378382435987295081754204888334741295276882683036095968017278816214197853866289835757344247480500498630778466}{83757877717200665823588768258884108516069938406400008166704773414845491540297010507299068356256451283016271438742194601760510504088259226069} a^{25} - \frac{3084479648192895625537243717703731586707048574393750142339792328873410019083279731841138471879754376070024957729714462730467878321472393809}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{24} - \frac{48188632676554974733388031346103905843971270072755195741933769785989624491585660702218625349153506767425997447872859727279166178425189027715}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{23} - \frac{41034663886899958848618149982160284988841847256497448439174619782315873725828201627158361494297584153690506333932293360352887596177237384031}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{22} - \frac{386788604520632762885559604114299737526043346116537560171853774595349340858931094957132605027155557670605869844596774401705756385738565185504}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{21} - \frac{381539754942230958208249674963275597691820354222836865843610463879796470970047718147508448276875190510877412045722609257777034881734076299464}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{20} - \frac{2430527595750453517842471163667942180358856600649144416430133364566092790929553682801470279783910461199281554402294858509652219643385293909655}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{19} - \frac{2645921216988351373616159265619382240576611069101197772384446731203831212766702806583995858296586858131754314000624476784249531577071886825617}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{18} - \frac{11988456980390686780596927166393750251294965721267237719285429597408936785117972946469673512447349764503627563195075103563778651355889226811198}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{17} - \frac{13431919259598909144192443421468359178308486500666190637490650590619533668543536142873166141876842197998943064299206425467483923579758839882372}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{16} - \frac{45356749873547266105218594988002779801138338422966310197110038725718689769958859191784930022416777122505240161670894859652888563757230527192116}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{15} - \frac{51021107358094205687977605697227186707156520942090890438054837737852601172751156755910170410834963216787774987088290810560818817701868146497548}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{14} - \frac{131455693959226034773872729049835164094003639165367031140349743027788904599926489378072824636755259505243376149774652393641318573775015410820632}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{13} - \frac{139989938874582848256063980520816376903731445279317771421483300807601377302140069194305296036469607896693179404120923899090746841396748195594895}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{12} - \frac{279168014469598699481761519458683396456467392845648092174175139990587384670676847317306555192090121600441917380224108779012886901176265842233475}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{11} - \frac{276642036853305343783171609511449739831586555685510770275742654519359387284221350262451148101162915771442261269622098089903305598275130990224033}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{10} - \frac{433058219993507668231164979948760971594327033523264519105135622685414186625657980684748164617001819606064922025157950656797766749957137809029255}{586305144020404660765121377812188759612489568844800057166933413903918440782079073551093478493795158981113900071195362212323573528617814582483} a^{9} - \frac{53159053139607724247715022550899839192639978637908365535973223962229297245865396567781155630093898966900800457146651894986723219688071083366698}{83757877717200665823588768258884108516069938406400008166704773414845491540297010507299068356256451283016271438742194601760510504088259226069} a^{8} - \frac{9028772860596264994217584592312218823059504020379664836846030969253134056753962836551358074259431085822793513687680564782205144524856150155281}{11965411102457237974798395465554872645152848343771429738100681916406498791471001501042724050893778754716610205534599228822930072012608460867} a^{7} - \frac{8546357135105725499824157633704846995921311482238421734848381144357878458896931904002098358046128270523185596655582248806569984795677294647}{15682059112001622509565393794960514607015528628796107127261706312459369320407603540029782504447940700808139194671820745508427355193458009} a^{6} - \frac{124561947418983864851106122337044286057495131787218535089069173638586650755805197831110893572945122191646198567952787301330091307129233393843}{244192063315453836220375417664385156023527517219825096695932284008295893703489826551892327569260790912583881745604065894345511673726703283} a^{5} - \frac{9982556606653137019831078956978523405134387784520640552374997499733070614896677705147478519620938928861980085071856913664307954037816347488}{34884580473636262317196488237769308003361073888546442385133183429756556243355689507413189652751541558940554535086295127763644524818100469} a^{4} - \frac{853369750001272626668738746509746344739200133143330905094032592604568944929462354641301800963747856286015992994896256997269987408460300873}{4983511496233751759599498319681329714765867698363777483590454775679508034765098501059027093250220222705793505012327875394806360688300067} a^{3} - \frac{33772555497226671779131143537965752008003084787420153947975886377433743707268186917161392588338940583260471077352760384817741708510218371}{711930213747678822799928331383047102109409671194825354798636396525644004966442643008432441892888603243684786430332553627829480098328581} a^{2} - \frac{199117059709795191421615459452698422692815704720546612869550275744663217948475373500198233935107218711359345520400968830657419359528131}{14529188035666914751018945538429532696110401452955619485686457071951918468702911081804743712099767413136424212863929665874071022414869} a - \frac{115242852397386778884348276934071391588388119795602314623487518800913183838389282603316061088609731946913687484389672824969796806040}{296514041544222750020794806906725157063477580672563662973193001468406499361283899628668239022444232921151514548243462568858592294181} \) (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 38 |
The 38 conjugacy class representatives for $C_{38}$ |
Character table for $C_{38}$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 19.19.114445997944945591651333831028437092270721.1 |
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 | $38$ | R | $38$ | ${\href{/padicField/7.1.0.1}{1} }^{38}$ | $38$ | $19^{2}$ | $38$ | $19^{2}$ | $38$ | $38$ | $19^{2}$ | $19^{2}$ | $38$ | $19^{2}$ | $38$ | $38$ | $38$ |
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 $38$ | $2$ | $19$ | $19$ | |||
\(191\) | Deg $38$ | $19$ | $2$ | $36$ |