Normalized defining polynomial
\( x^{45} - 3 x^{44} - 126 x^{43} + 350 x^{42} + 7095 x^{41} - 18207 x^{40} - 237254 x^{39} + 560949 x^{38} + \cdots + 756289 \)
Invariants
| Degree: | $45$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[45, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(408\!\cdots\!321\)
\(\medspace = 3^{60}\cdot 61^{42}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(200.66\) | 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: | $3^{4/3}61^{14/15}\approx 200.66402486963622$ | ||
| Ramified primes: |
\(3\), \(61\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $45$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(549=3^{2}\cdot 61\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{549}(256,·)$, $\chi_{549}(1,·)$, $\chi_{549}(259,·)$, $\chi_{549}(388,·)$, $\chi_{549}(391,·)$, $\chi_{549}(13,·)$, $\chi_{549}(142,·)$, $\chi_{549}(16,·)$, $\chi_{549}(22,·)$, $\chi_{549}(535,·)$, $\chi_{549}(25,·)$, $\chi_{549}(286,·)$, $\chi_{549}(544,·)$, $\chi_{549}(34,·)$, $\chi_{549}(424,·)$, $\chi_{549}(169,·)$, $\chi_{549}(301,·)$, $\chi_{549}(178,·)$, $\chi_{549}(436,·)$, $\chi_{549}(439,·)$, $\chi_{549}(184,·)$, $\chi_{549}(58,·)$, $\chi_{549}(196,·)$, $\chi_{549}(325,·)$, $\chi_{549}(70,·)$, $\chi_{549}(199,·)$, $\chi_{549}(73,·)$, $\chi_{549}(76,·)$, $\chi_{549}(205,·)$, $\chi_{549}(208,·)$, $\chi_{549}(469,·)$, $\chi_{549}(217,·)$, $\chi_{549}(442,·)$, $\chi_{549}(352,·)$, $\chi_{549}(400,·)$, $\chi_{549}(484,·)$, $\chi_{549}(103,·)$, $\chi_{549}(361,·)$, $\chi_{549}(367,·)$, $\chi_{549}(241,·)$, $\chi_{549}(118,·)$, $\chi_{549}(379,·)$, $\chi_{549}(508,·)$, $\chi_{549}(253,·)$, $\chi_{549}(382,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{11}a^{27}-\frac{1}{11}a^{26}+\frac{3}{11}a^{25}+\frac{3}{11}a^{24}-\frac{1}{11}a^{23}+\frac{5}{11}a^{22}+\frac{3}{11}a^{21}+\frac{4}{11}a^{20}+\frac{4}{11}a^{19}+\frac{3}{11}a^{18}-\frac{5}{11}a^{17}-\frac{5}{11}a^{16}+\frac{1}{11}a^{15}+\frac{3}{11}a^{13}-\frac{5}{11}a^{12}+\frac{2}{11}a^{11}-\frac{1}{11}a^{10}-\frac{5}{11}a^{9}-\frac{1}{11}a^{8}+\frac{4}{11}a^{7}+\frac{3}{11}a^{6}+\frac{4}{11}a^{5}+\frac{1}{11}a^{4}+\frac{1}{11}a^{3}+\frac{5}{11}a^{2}+\frac{5}{11}a+\frac{5}{11}$, $\frac{1}{11}a^{28}+\frac{2}{11}a^{26}-\frac{5}{11}a^{25}+\frac{2}{11}a^{24}+\frac{4}{11}a^{23}-\frac{3}{11}a^{22}-\frac{4}{11}a^{21}-\frac{3}{11}a^{20}-\frac{4}{11}a^{19}-\frac{2}{11}a^{18}+\frac{1}{11}a^{17}-\frac{4}{11}a^{16}+\frac{1}{11}a^{15}+\frac{3}{11}a^{14}-\frac{2}{11}a^{13}-\frac{3}{11}a^{12}+\frac{1}{11}a^{11}+\frac{5}{11}a^{10}+\frac{5}{11}a^{9}+\frac{3}{11}a^{8}-\frac{4}{11}a^{7}-\frac{4}{11}a^{6}+\frac{5}{11}a^{5}+\frac{2}{11}a^{4}-\frac{5}{11}a^{3}-\frac{1}{11}a^{2}-\frac{1}{11}a+\frac{5}{11}$, $\frac{1}{11}a^{29}-\frac{3}{11}a^{26}-\frac{4}{11}a^{25}-\frac{2}{11}a^{24}-\frac{1}{11}a^{23}-\frac{3}{11}a^{22}+\frac{2}{11}a^{21}-\frac{1}{11}a^{20}+\frac{1}{11}a^{19}-\frac{5}{11}a^{18}-\frac{5}{11}a^{17}+\frac{1}{11}a^{15}-\frac{2}{11}a^{14}+\frac{2}{11}a^{13}+\frac{1}{11}a^{11}-\frac{4}{11}a^{10}+\frac{2}{11}a^{9}-\frac{2}{11}a^{8}-\frac{1}{11}a^{7}-\frac{1}{11}a^{6}+\frac{5}{11}a^{5}+\frac{4}{11}a^{4}-\frac{3}{11}a^{3}-\frac{5}{11}a+\frac{1}{11}$, $\frac{1}{11}a^{30}+\frac{4}{11}a^{26}-\frac{4}{11}a^{25}-\frac{3}{11}a^{24}+\frac{5}{11}a^{23}-\frac{5}{11}a^{22}-\frac{3}{11}a^{21}+\frac{2}{11}a^{20}-\frac{4}{11}a^{19}+\frac{4}{11}a^{18}-\frac{4}{11}a^{17}-\frac{3}{11}a^{16}+\frac{1}{11}a^{15}+\frac{2}{11}a^{14}-\frac{2}{11}a^{13}-\frac{3}{11}a^{12}+\frac{2}{11}a^{11}-\frac{1}{11}a^{10}+\frac{5}{11}a^{9}-\frac{4}{11}a^{8}+\frac{3}{11}a^{6}+\frac{5}{11}a^{5}+\frac{3}{11}a^{3}-\frac{1}{11}a^{2}+\frac{5}{11}a+\frac{4}{11}$, $\frac{1}{11}a^{31}-\frac{4}{11}a^{25}+\frac{4}{11}a^{24}-\frac{1}{11}a^{23}-\frac{1}{11}a^{22}+\frac{1}{11}a^{21}+\frac{2}{11}a^{20}-\frac{1}{11}a^{19}-\frac{5}{11}a^{18}-\frac{5}{11}a^{17}-\frac{1}{11}a^{16}-\frac{2}{11}a^{15}-\frac{2}{11}a^{14}-\frac{4}{11}a^{13}+\frac{2}{11}a^{11}-\frac{2}{11}a^{10}+\frac{5}{11}a^{9}+\frac{4}{11}a^{8}-\frac{2}{11}a^{7}+\frac{4}{11}a^{6}-\frac{5}{11}a^{5}-\frac{1}{11}a^{4}-\frac{5}{11}a^{3}-\frac{4}{11}a^{2}-\frac{5}{11}a+\frac{2}{11}$, $\frac{1}{11}a^{32}-\frac{4}{11}a^{26}+\frac{4}{11}a^{25}-\frac{1}{11}a^{24}-\frac{1}{11}a^{23}+\frac{1}{11}a^{22}+\frac{2}{11}a^{21}-\frac{1}{11}a^{20}-\frac{5}{11}a^{19}-\frac{5}{11}a^{18}-\frac{1}{11}a^{17}-\frac{2}{11}a^{16}-\frac{2}{11}a^{15}-\frac{4}{11}a^{14}+\frac{2}{11}a^{12}-\frac{2}{11}a^{11}+\frac{5}{11}a^{10}+\frac{4}{11}a^{9}-\frac{2}{11}a^{8}+\frac{4}{11}a^{7}-\frac{5}{11}a^{6}-\frac{1}{11}a^{5}-\frac{5}{11}a^{4}-\frac{4}{11}a^{3}-\frac{5}{11}a^{2}+\frac{2}{11}a$, $\frac{1}{11}a^{33}-\frac{3}{11}a^{23}+\frac{3}{11}a^{13}+\frac{2}{11}a^{11}-\frac{1}{11}a^{3}-\frac{2}{11}a-\frac{2}{11}$, $\frac{1}{11}a^{34}-\frac{3}{11}a^{24}+\frac{3}{11}a^{14}+\frac{2}{11}a^{12}-\frac{1}{11}a^{4}-\frac{2}{11}a^{2}-\frac{2}{11}a$, $\frac{1}{11}a^{35}-\frac{3}{11}a^{25}+\frac{3}{11}a^{15}+\frac{2}{11}a^{13}-\frac{1}{11}a^{5}-\frac{2}{11}a^{3}-\frac{2}{11}a^{2}$, $\frac{1}{11}a^{36}-\frac{3}{11}a^{26}+\frac{3}{11}a^{16}+\frac{2}{11}a^{14}-\frac{1}{11}a^{6}-\frac{2}{11}a^{4}-\frac{2}{11}a^{3}$, $\frac{1}{11}a^{37}-\frac{3}{11}a^{26}-\frac{2}{11}a^{25}-\frac{2}{11}a^{24}-\frac{3}{11}a^{23}+\frac{4}{11}a^{22}-\frac{2}{11}a^{21}+\frac{1}{11}a^{20}+\frac{1}{11}a^{19}-\frac{2}{11}a^{18}-\frac{1}{11}a^{17}-\frac{4}{11}a^{16}+\frac{5}{11}a^{15}-\frac{2}{11}a^{13}-\frac{4}{11}a^{12}-\frac{5}{11}a^{11}-\frac{3}{11}a^{10}-\frac{4}{11}a^{9}-\frac{3}{11}a^{8}-\frac{2}{11}a^{6}-\frac{1}{11}a^{5}+\frac{1}{11}a^{4}+\frac{3}{11}a^{3}+\frac{4}{11}a^{2}+\frac{4}{11}a+\frac{4}{11}$, $\frac{1}{11}a^{38}-\frac{5}{11}a^{26}-\frac{4}{11}a^{25}-\frac{5}{11}a^{24}+\frac{1}{11}a^{23}+\frac{2}{11}a^{22}-\frac{1}{11}a^{21}+\frac{2}{11}a^{20}-\frac{1}{11}a^{19}-\frac{3}{11}a^{18}+\frac{3}{11}a^{17}+\frac{1}{11}a^{16}+\frac{3}{11}a^{15}-\frac{2}{11}a^{14}+\frac{5}{11}a^{13}+\frac{2}{11}a^{12}+\frac{3}{11}a^{11}+\frac{4}{11}a^{10}+\frac{4}{11}a^{9}-\frac{3}{11}a^{8}-\frac{1}{11}a^{7}-\frac{3}{11}a^{6}+\frac{2}{11}a^{5}-\frac{5}{11}a^{4}-\frac{4}{11}a^{3}-\frac{3}{11}a^{2}-\frac{3}{11}a+\frac{4}{11}$, $\frac{1}{5357}a^{39}-\frac{50}{5357}a^{38}+\frac{159}{5357}a^{37}+\frac{57}{5357}a^{36}+\frac{175}{5357}a^{35}+\frac{135}{5357}a^{34}+\frac{106}{5357}a^{33}-\frac{163}{5357}a^{32}+\frac{217}{5357}a^{31}+\frac{10}{487}a^{30}+\frac{112}{5357}a^{29}+\frac{221}{5357}a^{28}-\frac{118}{5357}a^{27}-\frac{400}{5357}a^{26}-\frac{1420}{5357}a^{25}-\frac{2213}{5357}a^{24}-\frac{2586}{5357}a^{23}+\frac{494}{5357}a^{22}+\frac{1123}{5357}a^{21}+\frac{167}{487}a^{20}+\frac{97}{5357}a^{19}-\frac{2469}{5357}a^{18}+\frac{393}{5357}a^{17}-\frac{1536}{5357}a^{16}-\frac{2526}{5357}a^{15}-\frac{1975}{5357}a^{14}-\frac{124}{5357}a^{13}+\frac{2490}{5357}a^{12}+\frac{1238}{5357}a^{11}+\frac{1532}{5357}a^{10}+\frac{2159}{5357}a^{9}+\frac{142}{5357}a^{8}-\frac{1365}{5357}a^{7}-\frac{821}{5357}a^{6}+\frac{226}{5357}a^{5}-\frac{2660}{5357}a^{4}+\frac{1021}{5357}a^{3}+\frac{996}{5357}a^{2}-\frac{1173}{5357}a+\frac{2311}{5357}$, $\frac{1}{5357}a^{40}+\frac{94}{5357}a^{38}+\frac{215}{5357}a^{37}+\frac{103}{5357}a^{36}+\frac{119}{5357}a^{35}+\frac{38}{5357}a^{34}-\frac{20}{487}a^{33}-\frac{141}{5357}a^{32}-\frac{241}{5357}a^{31}-\frac{232}{5357}a^{30}-\frac{23}{5357}a^{29}+\frac{218}{5357}a^{28}+\frac{31}{5357}a^{27}-\frac{1940}{5357}a^{26}-\frac{1137}{5357}a^{25}+\frac{2670}{5357}a^{24}-\frac{725}{5357}a^{23}+\frac{499}{5357}a^{22}-\frac{1427}{5357}a^{21}-\frac{2044}{5357}a^{20}+\frac{1894}{5357}a^{19}+\frac{2102}{5357}a^{18}+\frac{582}{5357}a^{17}+\frac{2003}{5357}a^{16}-\frac{239}{487}a^{15}+\frac{474}{5357}a^{14}+\frac{673}{5357}a^{13}-\frac{395}{5357}a^{12}+\frac{2557}{5357}a^{11}-\frac{2083}{5357}a^{10}-\frac{1970}{5357}a^{9}+\frac{865}{5357}a^{8}+\frac{2518}{5357}a^{7}+\frac{1058}{5357}a^{6}+\frac{2309}{5357}a^{5}-\frac{1950}{5357}a^{4}-\frac{1524}{5357}a^{3}-\frac{1047}{5357}a^{2}+\frac{640}{5357}a-\frac{843}{5357}$, $\frac{1}{5357}a^{41}+\frac{45}{5357}a^{38}-\frac{233}{5357}a^{37}+\frac{118}{5357}a^{36}+\frac{146}{5357}a^{35}+\frac{239}{5357}a^{34}+\frac{122}{5357}a^{33}-\frac{16}{5357}a^{32}-\frac{16}{487}a^{31}-\frac{136}{5357}a^{30}-\frac{83}{5357}a^{29}+\frac{18}{487}a^{28}-\frac{101}{5357}a^{27}-\frac{549}{5357}a^{26}-\frac{1184}{5357}a^{25}-\frac{2113}{5357}a^{24}-\frac{1865}{5357}a^{23}-\frac{624}{5357}a^{22}-\frac{953}{5357}a^{21}-\frac{1308}{5357}a^{20}+\frac{1750}{5357}a^{19}-\frac{1092}{5357}a^{18}+\frac{612}{5357}a^{17}-\frac{1423}{5357}a^{16}-\frac{2173}{5357}a^{15}+\frac{1750}{5357}a^{14}-\frac{1401}{5357}a^{13}-\frac{2613}{5357}a^{12}-\frac{1088}{5357}a^{11}-\frac{365}{5357}a^{10}+\frac{511}{5357}a^{9}+\frac{2319}{5357}a^{8}-\frac{1635}{5357}a^{7}+\frac{2537}{5357}a^{6}+\frac{2617}{5357}a^{5}+\frac{1120}{5357}a^{4}-\frac{2543}{5357}a^{3}-\frac{2402}{5357}a^{2}+\frac{331}{5357}a-\frac{32}{5357}$, $\frac{1}{35297273}a^{42}-\frac{178}{35297273}a^{41}+\frac{3248}{35297273}a^{40}+\frac{2773}{35297273}a^{39}+\frac{560496}{35297273}a^{38}+\frac{876594}{35297273}a^{37}+\frac{336231}{35297273}a^{36}-\frac{1160972}{35297273}a^{35}-\frac{1524040}{35297273}a^{34}-\frac{404268}{35297273}a^{33}+\frac{1043657}{35297273}a^{32}+\frac{36803}{3208843}a^{31}-\frac{896851}{35297273}a^{30}+\frac{894975}{35297273}a^{29}-\frac{527267}{35297273}a^{28}+\frac{1423280}{35297273}a^{27}+\frac{15947754}{35297273}a^{26}+\frac{12907434}{35297273}a^{25}+\frac{11650186}{35297273}a^{24}-\frac{9815380}{35297273}a^{23}-\frac{14567455}{35297273}a^{22}+\frac{11876003}{35297273}a^{21}+\frac{10282823}{35297273}a^{20}+\frac{9472315}{35297273}a^{19}-\frac{2818260}{35297273}a^{18}-\frac{16852426}{35297273}a^{17}+\frac{17332497}{35297273}a^{16}+\frac{8849870}{35297273}a^{15}+\frac{8130244}{35297273}a^{14}-\frac{2365549}{35297273}a^{13}-\frac{2978735}{35297273}a^{12}-\frac{6258720}{35297273}a^{11}-\frac{13591778}{35297273}a^{10}+\frac{6083219}{35297273}a^{9}+\frac{8244671}{35297273}a^{8}-\frac{17596710}{35297273}a^{7}+\frac{12198558}{35297273}a^{6}+\frac{17326196}{35297273}a^{5}+\frac{10605256}{35297273}a^{4}+\frac{11550007}{35297273}a^{3}-\frac{11487849}{35297273}a^{2}-\frac{6286856}{35297273}a+\frac{4688731}{35297273}$, $\frac{1}{35297273}a^{43}-\frac{2080}{35297273}a^{41}+\frac{1085}{35297273}a^{40}-\frac{150}{35297273}a^{39}+\frac{564561}{35297273}a^{38}-\frac{869933}{35297273}a^{37}-\frac{23368}{3208843}a^{36}+\frac{1043461}{35297273}a^{35}+\frac{244642}{35297273}a^{34}-\frac{545527}{35297273}a^{33}-\frac{259976}{35297273}a^{32}-\frac{43900}{35297273}a^{31}+\frac{623640}{35297273}a^{30}+\frac{510503}{35297273}a^{29}-\frac{572997}{35297273}a^{28}+\frac{75398}{3208843}a^{27}-\frac{9197785}{35297273}a^{26}+\frac{550398}{3208843}a^{25}-\frac{1793073}{35297273}a^{24}+\frac{16949332}{35297273}a^{23}+\frac{83407}{35297273}a^{22}-\frac{15843131}{35297273}a^{21}+\frac{17422600}{35297273}a^{20}-\frac{3754216}{35297273}a^{19}-\frac{679785}{35297273}a^{18}-\frac{8625495}{35297273}a^{17}-\frac{5661879}{35297273}a^{16}-\frac{7842985}{35297273}a^{15}+\frac{15578126}{35297273}a^{14}+\frac{11143815}{35297273}a^{13}+\frac{11737839}{35297273}a^{12}+\frac{5986923}{35297273}a^{11}+\frac{16064912}{35297273}a^{10}-\frac{10056493}{35297273}a^{9}+\frac{9474137}{35297273}a^{8}+\frac{791957}{3208843}a^{7}+\frac{7914779}{35297273}a^{6}-\frac{8243503}{35297273}a^{5}+\frac{984133}{3208843}a^{4}-\frac{12277063}{35297273}a^{3}-\frac{3130998}{35297273}a^{2}-\frac{12288908}{35297273}a-\frac{3539860}{35297273}$, $\frac{1}{11\!\cdots\!83}a^{44}+\frac{10\!\cdots\!59}{10\!\cdots\!53}a^{43}+\frac{48\!\cdots\!14}{11\!\cdots\!83}a^{42}+\frac{89\!\cdots\!14}{11\!\cdots\!83}a^{41}-\frac{29\!\cdots\!38}{11\!\cdots\!83}a^{40}-\frac{47\!\cdots\!29}{11\!\cdots\!83}a^{39}+\frac{80\!\cdots\!08}{11\!\cdots\!83}a^{38}-\frac{62\!\cdots\!05}{11\!\cdots\!83}a^{37}-\frac{39\!\cdots\!79}{10\!\cdots\!53}a^{36}-\frac{20\!\cdots\!99}{11\!\cdots\!83}a^{35}+\frac{51\!\cdots\!34}{11\!\cdots\!83}a^{34}+\frac{17\!\cdots\!92}{11\!\cdots\!83}a^{33}+\frac{39\!\cdots\!92}{11\!\cdots\!83}a^{32}-\frac{19\!\cdots\!73}{11\!\cdots\!83}a^{31}+\frac{67\!\cdots\!49}{11\!\cdots\!83}a^{30}-\frac{39\!\cdots\!36}{11\!\cdots\!83}a^{29}-\frac{35\!\cdots\!77}{11\!\cdots\!83}a^{28}+\frac{25\!\cdots\!20}{11\!\cdots\!83}a^{27}+\frac{56\!\cdots\!91}{11\!\cdots\!83}a^{26}+\frac{19\!\cdots\!24}{11\!\cdots\!83}a^{25}+\frac{10\!\cdots\!49}{10\!\cdots\!53}a^{24}-\frac{38\!\cdots\!45}{11\!\cdots\!83}a^{23}+\frac{53\!\cdots\!41}{11\!\cdots\!83}a^{22}+\frac{13\!\cdots\!70}{10\!\cdots\!53}a^{21}-\frac{57\!\cdots\!56}{11\!\cdots\!83}a^{20}+\frac{17\!\cdots\!70}{11\!\cdots\!83}a^{19}-\frac{11\!\cdots\!06}{11\!\cdots\!83}a^{18}-\frac{18\!\cdots\!15}{11\!\cdots\!83}a^{17}-\frac{33\!\cdots\!38}{11\!\cdots\!83}a^{16}-\frac{19\!\cdots\!38}{11\!\cdots\!83}a^{15}+\frac{88\!\cdots\!41}{11\!\cdots\!83}a^{14}-\frac{36\!\cdots\!39}{11\!\cdots\!83}a^{13}-\frac{42\!\cdots\!03}{11\!\cdots\!83}a^{12}-\frac{38\!\cdots\!84}{11\!\cdots\!83}a^{11}+\frac{84\!\cdots\!78}{96\!\cdots\!23}a^{10}-\frac{39\!\cdots\!41}{10\!\cdots\!53}a^{9}+\frac{41\!\cdots\!63}{11\!\cdots\!83}a^{8}-\frac{12\!\cdots\!57}{11\!\cdots\!83}a^{7}-\frac{42\!\cdots\!31}{11\!\cdots\!83}a^{6}+\frac{12\!\cdots\!84}{11\!\cdots\!83}a^{5}+\frac{49\!\cdots\!22}{11\!\cdots\!83}a^{4}-\frac{28\!\cdots\!34}{11\!\cdots\!83}a^{3}-\frac{17\!\cdots\!65}{11\!\cdots\!83}a^{2}+\frac{35\!\cdots\!47}{11\!\cdots\!83}a+\frac{20\!\cdots\!04}{11\!\cdots\!83}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $44$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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
$C_3\times C_{15}$ (as 45T2):
| An abelian group of order 45 |
| The 45 conjugacy class representatives for $C_3\times C_{15}$ |
| Character table for $C_3\times C_{15}$ is not computed |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $15^{3}$ | R | $15^{3}$ | $15^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{15}$ | ${\href{/padicField/13.3.0.1}{3} }^{15}$ | $15^{3}$ | $15^{3}$ | $15^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{15}$ | $15^{3}$ | ${\href{/padicField/37.5.0.1}{5} }^{9}$ | $15^{3}$ | $15^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{15}$ | ${\href{/padicField/53.5.0.1}{5} }^{9}$ | $15^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.15.20.65 | $x^{15} + 30 x^{14} + 360 x^{13} + 2175 x^{12} + 6840 x^{11} + 11016 x^{10} + 13050 x^{9} + 21060 x^{8} + 9720 x^{7} + 24084 x^{6} + 55728 x^{5} + 167184 x^{4} + 79137 x^{3} + 474822 x^{2} + 138024$ | $3$ | $5$ | $20$ | $C_{15}$ | $[2]^{5}$ |
| 3.15.20.65 | $x^{15} + 30 x^{14} + 360 x^{13} + 2175 x^{12} + 6840 x^{11} + 11016 x^{10} + 13050 x^{9} + 21060 x^{8} + 9720 x^{7} + 24084 x^{6} + 55728 x^{5} + 167184 x^{4} + 79137 x^{3} + 474822 x^{2} + 138024$ | $3$ | $5$ | $20$ | $C_{15}$ | $[2]^{5}$ | |
| 3.15.20.65 | $x^{15} + 30 x^{14} + 360 x^{13} + 2175 x^{12} + 6840 x^{11} + 11016 x^{10} + 13050 x^{9} + 21060 x^{8} + 9720 x^{7} + 24084 x^{6} + 55728 x^{5} + 167184 x^{4} + 79137 x^{3} + 474822 x^{2} + 138024$ | $3$ | $5$ | $20$ | $C_{15}$ | $[2]^{5}$ | |
|
\(61\)
| Deg $45$ | $15$ | $3$ | $42$ |