Normalized defining polynomial
\( x^{42} + 126 x^{40} + 7371 x^{38} + 265734 x^{36} - 337 x^{35} + 6608385 x^{34} - 35385 x^{33} + \cdots + 16\!\cdots\!33 \)
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: |
\(-291\!\cdots\!491\)
\(\medspace = -\,7^{77}\cdot 13^{21}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(127.74\) | 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: | $7^{11/6}13^{1/2}\approx 127.73740314016263$ | ||
| Ramified primes: |
\(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-91}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $42$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(637=7^{2}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{637}(1,·)$, $\chi_{637}(363,·)$, $\chi_{637}(261,·)$, $\chi_{637}(129,·)$, $\chi_{637}(12,·)$, $\chi_{637}(493,·)$, $\chi_{637}(272,·)$, $\chi_{637}(402,·)$, $\chi_{637}(534,·)$, $\chi_{637}(285,·)$, $\chi_{637}(545,·)$, $\chi_{637}(547,·)$, $\chi_{637}(326,·)$, $\chi_{637}(38,·)$, $\chi_{637}(92,·)$, $\chi_{637}(274,·)$, $\chi_{637}(558,·)$, $\chi_{637}(584,·)$, $\chi_{637}(53,·)$, $\chi_{637}(311,·)$, $\chi_{637}(443,·)$, $\chi_{637}(181,·)$, $\chi_{637}(194,·)$, $\chi_{637}(454,·)$, $\chi_{637}(417,·)$, $\chi_{637}(456,·)$, $\chi_{637}(183,·)$, $\chi_{637}(79,·)$, $\chi_{637}(467,·)$, $\chi_{637}(599,·)$, $\chi_{637}(90,·)$, $\chi_{637}(220,·)$, $\chi_{637}(352,·)$, $\chi_{637}(144,·)$, $\chi_{637}(103,·)$, $\chi_{637}(636,·)$, $\chi_{637}(235,·)$, $\chi_{637}(365,·)$, $\chi_{637}(625,·)$, $\chi_{637}(376,·)$, $\chi_{637}(508,·)$, $\chi_{637}(170,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{1048576}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{97}a^{21}-\frac{34}{97}a^{19}-\frac{45}{97}a^{17}-\frac{1}{97}a^{15}-\frac{23}{97}a^{14}+\frac{5}{97}a^{13}+\frac{4}{97}a^{12}+\frac{5}{97}a^{11}-\frac{31}{97}a^{10}-\frac{14}{97}a^{9}-\frac{42}{97}a^{8}+\frac{27}{97}a^{7}+\frac{37}{97}a^{6}+\frac{36}{97}a^{5}-\frac{23}{97}a^{4}+\frac{39}{97}a^{3}+\frac{7}{97}a^{2}-\frac{45}{97}a+\frac{42}{97}$, $\frac{1}{97}a^{22}-\frac{34}{97}a^{20}-\frac{45}{97}a^{18}-\frac{1}{97}a^{16}-\frac{23}{97}a^{15}+\frac{5}{97}a^{14}+\frac{4}{97}a^{13}+\frac{5}{97}a^{12}-\frac{31}{97}a^{11}-\frac{14}{97}a^{10}-\frac{42}{97}a^{9}+\frac{27}{97}a^{8}+\frac{37}{97}a^{7}+\frac{36}{97}a^{6}-\frac{23}{97}a^{5}+\frac{39}{97}a^{4}+\frac{7}{97}a^{3}-\frac{45}{97}a^{2}+\frac{42}{97}a$, $\frac{1}{97}a^{23}-\frac{37}{97}a^{19}+\frac{21}{97}a^{17}-\frac{23}{97}a^{16}-\frac{29}{97}a^{15}-\frac{2}{97}a^{14}-\frac{19}{97}a^{13}+\frac{8}{97}a^{12}-\frac{38}{97}a^{11}-\frac{29}{97}a^{10}+\frac{36}{97}a^{9}-\frac{33}{97}a^{8}-\frac{16}{97}a^{7}-\frac{26}{97}a^{6}+\frac{2}{97}a^{5}+\frac{1}{97}a^{4}+\frac{20}{97}a^{3}-\frac{11}{97}a^{2}+\frac{22}{97}a-\frac{27}{97}$, $\frac{1}{97}a^{24}-\frac{37}{97}a^{20}+\frac{21}{97}a^{18}-\frac{23}{97}a^{17}-\frac{29}{97}a^{16}-\frac{2}{97}a^{15}-\frac{19}{97}a^{14}+\frac{8}{97}a^{13}-\frac{38}{97}a^{12}-\frac{29}{97}a^{11}+\frac{36}{97}a^{10}-\frac{33}{97}a^{9}-\frac{16}{97}a^{8}-\frac{26}{97}a^{7}+\frac{2}{97}a^{6}+\frac{1}{97}a^{5}+\frac{20}{97}a^{4}-\frac{11}{97}a^{3}+\frac{22}{97}a^{2}-\frac{27}{97}a$, $\frac{1}{15\!\cdots\!61}a^{25}-\frac{420984484639620}{15\!\cdots\!61}a^{24}+\frac{75}{15\!\cdots\!61}a^{23}+\frac{262245016778707}{15\!\cdots\!61}a^{22}+\frac{2475}{15\!\cdots\!61}a^{21}+\frac{34\!\cdots\!35}{15\!\cdots\!61}a^{20}+\frac{47250}{15\!\cdots\!61}a^{19}-\frac{16\!\cdots\!49}{15\!\cdots\!61}a^{18}+\frac{70\!\cdots\!97}{15\!\cdots\!61}a^{17}+\frac{51\!\cdots\!40}{15\!\cdots\!61}a^{16}-\frac{57\!\cdots\!28}{15\!\cdots\!61}a^{15}-\frac{32\!\cdots\!83}{15\!\cdots\!61}a^{14}-\frac{62\!\cdots\!07}{15\!\cdots\!61}a^{13}-\frac{14\!\cdots\!91}{15\!\cdots\!61}a^{12}+\frac{64\!\cdots\!52}{15\!\cdots\!61}a^{11}-\frac{54\!\cdots\!31}{15\!\cdots\!61}a^{10}-\frac{57\!\cdots\!18}{15\!\cdots\!61}a^{9}-\frac{17\!\cdots\!21}{15\!\cdots\!61}a^{8}-\frac{40\!\cdots\!00}{15\!\cdots\!61}a^{7}-\frac{16\!\cdots\!00}{15\!\cdots\!61}a^{6}-\frac{12\!\cdots\!59}{15\!\cdots\!61}a^{5}-\frac{17\!\cdots\!88}{15\!\cdots\!61}a^{4}+\frac{74\!\cdots\!48}{15\!\cdots\!61}a^{3}+\frac{46\!\cdots\!02}{15\!\cdots\!61}a^{2}-\frac{62\!\cdots\!82}{15\!\cdots\!61}a-\frac{31\!\cdots\!99}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{26}+\frac{78}{15\!\cdots\!61}a^{24}-\frac{346158541388053}{15\!\cdots\!61}a^{23}+\frac{2691}{15\!\cdots\!61}a^{22}+\frac{251740573828038}{15\!\cdots\!61}a^{21}+\frac{54054}{15\!\cdots\!61}a^{20}+\frac{71\!\cdots\!60}{15\!\cdots\!61}a^{19}+\frac{700245}{15\!\cdots\!61}a^{18}+\frac{26\!\cdots\!62}{15\!\cdots\!61}a^{17}-\frac{19\!\cdots\!14}{15\!\cdots\!61}a^{16}+\frac{10\!\cdots\!89}{15\!\cdots\!61}a^{15}-\frac{77\!\cdots\!72}{15\!\cdots\!61}a^{14}-\frac{33\!\cdots\!61}{15\!\cdots\!61}a^{13}-\frac{45\!\cdots\!40}{15\!\cdots\!61}a^{12}-\frac{11\!\cdots\!27}{15\!\cdots\!61}a^{11}+\frac{57\!\cdots\!34}{15\!\cdots\!61}a^{10}+\frac{25\!\cdots\!40}{15\!\cdots\!61}a^{9}-\frac{33\!\cdots\!33}{15\!\cdots\!61}a^{8}-\frac{48\!\cdots\!82}{15\!\cdots\!61}a^{7}+\frac{35\!\cdots\!23}{15\!\cdots\!61}a^{6}+\frac{65\!\cdots\!60}{15\!\cdots\!61}a^{5}-\frac{61\!\cdots\!92}{15\!\cdots\!61}a^{4}+\frac{73\!\cdots\!88}{15\!\cdots\!61}a^{3}-\frac{48\!\cdots\!10}{15\!\cdots\!61}a^{2}-\frac{59\!\cdots\!13}{15\!\cdots\!61}a+\frac{24\!\cdots\!41}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{27}+\frac{308391354364047}{15\!\cdots\!61}a^{24}-\frac{3159}{15\!\cdots\!61}a^{23}+\frac{715085204078761}{15\!\cdots\!61}a^{22}-\frac{138996}{15\!\cdots\!61}a^{21}+\frac{220031103815314}{16\!\cdots\!13}a^{20}-\frac{2985255}{15\!\cdots\!61}a^{19}-\frac{109667600055047}{16\!\cdots\!13}a^{18}+\frac{37\!\cdots\!91}{15\!\cdots\!61}a^{17}+\frac{62\!\cdots\!60}{15\!\cdots\!61}a^{16}-\frac{33\!\cdots\!41}{15\!\cdots\!61}a^{15}+\frac{71\!\cdots\!78}{15\!\cdots\!61}a^{14}-\frac{64\!\cdots\!28}{15\!\cdots\!61}a^{13}-\frac{28\!\cdots\!80}{15\!\cdots\!61}a^{12}-\frac{32\!\cdots\!60}{15\!\cdots\!61}a^{11}+\frac{22\!\cdots\!34}{15\!\cdots\!61}a^{10}-\frac{14\!\cdots\!77}{15\!\cdots\!61}a^{9}+\frac{66\!\cdots\!64}{15\!\cdots\!61}a^{8}-\frac{54\!\cdots\!55}{15\!\cdots\!61}a^{7}-\frac{45\!\cdots\!69}{15\!\cdots\!61}a^{6}-\frac{38\!\cdots\!20}{15\!\cdots\!61}a^{5}-\frac{69\!\cdots\!28}{15\!\cdots\!61}a^{4}+\frac{28\!\cdots\!63}{15\!\cdots\!61}a^{3}-\frac{49\!\cdots\!20}{15\!\cdots\!61}a^{2}-\frac{45\!\cdots\!68}{15\!\cdots\!61}a-\frac{31\!\cdots\!54}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{28}-\frac{3402}{15\!\cdots\!61}a^{24}+\frac{113301561072018}{15\!\cdots\!61}a^{23}-\frac{156492}{15\!\cdots\!61}a^{22}-\frac{124955144443974}{15\!\cdots\!61}a^{21}-\frac{3536379}{15\!\cdots\!61}a^{20}-\frac{55\!\cdots\!45}{15\!\cdots\!61}a^{19}-\frac{48866328}{15\!\cdots\!61}a^{18}+\frac{17\!\cdots\!25}{15\!\cdots\!61}a^{17}-\frac{53\!\cdots\!59}{15\!\cdots\!61}a^{16}+\frac{57\!\cdots\!61}{15\!\cdots\!61}a^{15}+\frac{41\!\cdots\!50}{15\!\cdots\!61}a^{14}+\frac{39\!\cdots\!56}{15\!\cdots\!61}a^{13}-\frac{59\!\cdots\!59}{15\!\cdots\!61}a^{12}+\frac{34\!\cdots\!45}{15\!\cdots\!61}a^{11}-\frac{27\!\cdots\!77}{15\!\cdots\!61}a^{10}+\frac{22\!\cdots\!35}{15\!\cdots\!61}a^{9}-\frac{27\!\cdots\!49}{15\!\cdots\!61}a^{8}+\frac{433077470448741}{15\!\cdots\!61}a^{7}-\frac{96\!\cdots\!38}{15\!\cdots\!61}a^{6}+\frac{45\!\cdots\!62}{15\!\cdots\!61}a^{5}+\frac{80\!\cdots\!82}{15\!\cdots\!61}a^{4}-\frac{22\!\cdots\!84}{15\!\cdots\!61}a^{3}-\frac{17\!\cdots\!71}{15\!\cdots\!61}a^{2}+\frac{11\!\cdots\!14}{15\!\cdots\!61}a-\frac{77\!\cdots\!50}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{29}+\frac{33760640237348}{15\!\cdots\!61}a^{24}+\frac{98658}{15\!\cdots\!61}a^{23}+\frac{584546536687438}{15\!\cdots\!61}a^{22}+\frac{4883571}{15\!\cdots\!61}a^{21}+\frac{13\!\cdots\!10}{15\!\cdots\!61}a^{20}+\frac{111878172}{15\!\cdots\!61}a^{19}+\frac{66\!\cdots\!92}{15\!\cdots\!61}a^{18}-\frac{30\!\cdots\!27}{15\!\cdots\!61}a^{17}-\frac{33\!\cdots\!20}{15\!\cdots\!61}a^{16}-\frac{49\!\cdots\!11}{15\!\cdots\!61}a^{15}-\frac{32\!\cdots\!06}{15\!\cdots\!61}a^{14}+\frac{56\!\cdots\!77}{15\!\cdots\!61}a^{13}-\frac{20\!\cdots\!68}{15\!\cdots\!61}a^{12}+\frac{12\!\cdots\!48}{15\!\cdots\!61}a^{11}+\frac{20\!\cdots\!35}{15\!\cdots\!61}a^{10}+\frac{49\!\cdots\!75}{15\!\cdots\!61}a^{9}-\frac{70\!\cdots\!90}{15\!\cdots\!61}a^{8}+\frac{40\!\cdots\!15}{15\!\cdots\!61}a^{7}+\frac{12\!\cdots\!97}{15\!\cdots\!61}a^{6}+\frac{16\!\cdots\!20}{15\!\cdots\!61}a^{5}-\frac{17\!\cdots\!47}{15\!\cdots\!61}a^{4}+\frac{46\!\cdots\!49}{15\!\cdots\!61}a^{3}+\frac{23\!\cdots\!90}{15\!\cdots\!61}a^{2}+\frac{12\!\cdots\!28}{15\!\cdots\!61}a+\frac{49\!\cdots\!19}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{30}+\frac{109620}{15\!\cdots\!61}a^{24}-\frac{338389485806749}{15\!\cdots\!61}a^{23}+\frac{5672835}{15\!\cdots\!61}a^{22}-\frac{418375245448831}{15\!\cdots\!61}a^{21}+\frac{136739988}{15\!\cdots\!61}a^{20}-\frac{38\!\cdots\!70}{15\!\cdots\!61}a^{19}+\frac{1968227100}{15\!\cdots\!61}a^{18}+\frac{49\!\cdots\!60}{15\!\cdots\!61}a^{17}+\frac{32\!\cdots\!26}{15\!\cdots\!61}a^{16}+\frac{25\!\cdots\!20}{15\!\cdots\!61}a^{15}+\frac{20\!\cdots\!99}{15\!\cdots\!61}a^{14}+\frac{59\!\cdots\!95}{15\!\cdots\!61}a^{13}-\frac{70\!\cdots\!32}{15\!\cdots\!61}a^{12}+\frac{29\!\cdots\!77}{15\!\cdots\!61}a^{11}-\frac{53\!\cdots\!93}{15\!\cdots\!61}a^{10}+\frac{69\!\cdots\!56}{15\!\cdots\!61}a^{9}-\frac{30\!\cdots\!47}{15\!\cdots\!61}a^{8}-\frac{68\!\cdots\!76}{15\!\cdots\!61}a^{7}+\frac{24\!\cdots\!70}{15\!\cdots\!61}a^{6}+\frac{36\!\cdots\!87}{15\!\cdots\!61}a^{5}-\frac{33\!\cdots\!93}{15\!\cdots\!61}a^{4}-\frac{50\!\cdots\!41}{15\!\cdots\!61}a^{3}-\frac{32\!\cdots\!20}{15\!\cdots\!61}a^{2}-\frac{27\!\cdots\!52}{15\!\cdots\!61}a+\frac{32\!\cdots\!44}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{31}+\frac{257903302379724}{15\!\cdots\!61}a^{24}-\frac{2548665}{15\!\cdots\!61}a^{23}+\frac{677793625997487}{15\!\cdots\!61}a^{22}-\frac{134569512}{15\!\cdots\!61}a^{21}-\frac{73\!\cdots\!12}{15\!\cdots\!61}a^{20}-\frac{3211317900}{15\!\cdots\!61}a^{19}-\frac{61\!\cdots\!79}{15\!\cdots\!61}a^{18}-\frac{53\!\cdots\!29}{15\!\cdots\!61}a^{17}+\frac{65\!\cdots\!32}{15\!\cdots\!61}a^{16}-\frac{16\!\cdots\!00}{15\!\cdots\!61}a^{15}-\frac{52\!\cdots\!87}{15\!\cdots\!61}a^{14}+\frac{48\!\cdots\!79}{15\!\cdots\!61}a^{13}+\frac{30\!\cdots\!17}{15\!\cdots\!61}a^{12}-\frac{57\!\cdots\!32}{15\!\cdots\!61}a^{11}-\frac{31\!\cdots\!26}{15\!\cdots\!61}a^{10}+\frac{77\!\cdots\!24}{15\!\cdots\!61}a^{9}+\frac{19\!\cdots\!91}{15\!\cdots\!61}a^{8}-\frac{38\!\cdots\!37}{15\!\cdots\!61}a^{7}+\frac{62\!\cdots\!03}{15\!\cdots\!61}a^{6}-\frac{49\!\cdots\!23}{15\!\cdots\!61}a^{5}+\frac{74\!\cdots\!29}{15\!\cdots\!61}a^{4}-\frac{61\!\cdots\!54}{15\!\cdots\!61}a^{3}-\frac{68\!\cdots\!00}{15\!\cdots\!61}a^{2}-\frac{25\!\cdots\!24}{15\!\cdots\!61}a-\frac{21\!\cdots\!58}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{32}-\frac{2912760}{15\!\cdots\!61}a^{24}+\frac{644389891201143}{15\!\cdots\!61}a^{23}-\frac{160784352}{15\!\cdots\!61}a^{22}-\frac{302816744482966}{15\!\cdots\!61}a^{21}-\frac{4037085360}{15\!\cdots\!61}a^{20}+\frac{59\!\cdots\!03}{15\!\cdots\!61}a^{19}-\frac{59769835200}{15\!\cdots\!61}a^{18}+\frac{32\!\cdots\!59}{15\!\cdots\!61}a^{17}+\frac{65\!\cdots\!33}{15\!\cdots\!61}a^{16}+\frac{33\!\cdots\!16}{15\!\cdots\!61}a^{15}+\frac{51\!\cdots\!76}{15\!\cdots\!61}a^{14}+\frac{65\!\cdots\!66}{15\!\cdots\!61}a^{13}+\frac{25\!\cdots\!84}{15\!\cdots\!61}a^{12}-\frac{50\!\cdots\!22}{15\!\cdots\!61}a^{11}-\frac{49\!\cdots\!43}{15\!\cdots\!61}a^{10}-\frac{64\!\cdots\!39}{15\!\cdots\!61}a^{9}+\frac{17\!\cdots\!43}{15\!\cdots\!61}a^{8}-\frac{52\!\cdots\!46}{15\!\cdots\!61}a^{7}-\frac{45\!\cdots\!64}{15\!\cdots\!61}a^{6}-\frac{47\!\cdots\!60}{15\!\cdots\!61}a^{5}+\frac{22\!\cdots\!42}{15\!\cdots\!61}a^{4}+\frac{60\!\cdots\!07}{15\!\cdots\!61}a^{3}-\frac{25\!\cdots\!84}{15\!\cdots\!61}a^{2}-\frac{36\!\cdots\!42}{15\!\cdots\!61}a-\frac{62\!\cdots\!97}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{33}-\frac{717953379991494}{15\!\cdots\!61}a^{24}+\frac{57672648}{15\!\cdots\!61}a^{23}-\frac{235700556631137}{15\!\cdots\!61}a^{22}+\frac{3171995640}{15\!\cdots\!61}a^{21}+\frac{32\!\cdots\!26}{15\!\cdots\!61}a^{20}+\frac{77858074800}{15\!\cdots\!61}a^{19}+\frac{15\!\cdots\!47}{15\!\cdots\!61}a^{18}-\frac{19\!\cdots\!56}{15\!\cdots\!61}a^{17}-\frac{23\!\cdots\!28}{15\!\cdots\!61}a^{16}+\frac{10059263264160}{15\!\cdots\!61}a^{15}+\frac{37\!\cdots\!49}{15\!\cdots\!61}a^{14}+\frac{77\!\cdots\!00}{15\!\cdots\!61}a^{13}+\frac{25\!\cdots\!06}{15\!\cdots\!61}a^{12}+\frac{46\!\cdots\!37}{15\!\cdots\!61}a^{11}-\frac{67\!\cdots\!66}{15\!\cdots\!61}a^{10}-\frac{68\!\cdots\!59}{15\!\cdots\!61}a^{9}+\frac{54\!\cdots\!80}{15\!\cdots\!61}a^{8}+\frac{74\!\cdots\!98}{15\!\cdots\!61}a^{7}-\frac{71\!\cdots\!55}{15\!\cdots\!61}a^{6}+\frac{16\!\cdots\!90}{15\!\cdots\!61}a^{5}+\frac{69\!\cdots\!29}{15\!\cdots\!61}a^{4}+\frac{53\!\cdots\!53}{15\!\cdots\!61}a^{3}+\frac{18\!\cdots\!82}{15\!\cdots\!61}a^{2}+\frac{59\!\cdots\!91}{15\!\cdots\!61}a+\frac{68\!\cdots\!78}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{34}+\frac{67616208}{15\!\cdots\!61}a^{24}+\frac{510107097602784}{15\!\cdots\!61}a^{23}+\frac{3887931960}{15\!\cdots\!61}a^{22}+\frac{326160765680764}{15\!\cdots\!61}a^{21}+\frac{100410068880}{15\!\cdots\!61}a^{20}+\frac{14\!\cdots\!64}{15\!\cdots\!61}a^{19}+\frac{1517561268300}{15\!\cdots\!61}a^{18}+\frac{35\!\cdots\!17}{15\!\cdots\!61}a^{17}-\frac{77\!\cdots\!24}{15\!\cdots\!61}a^{16}-\frac{43\!\cdots\!10}{15\!\cdots\!61}a^{15}+\frac{95528313894816}{15\!\cdots\!61}a^{14}-\frac{72\!\cdots\!32}{15\!\cdots\!61}a^{13}-\frac{28\!\cdots\!86}{15\!\cdots\!61}a^{12}+\frac{21\!\cdots\!34}{15\!\cdots\!61}a^{11}+\frac{31\!\cdots\!67}{15\!\cdots\!61}a^{10}+\frac{68\!\cdots\!95}{15\!\cdots\!61}a^{9}+\frac{37\!\cdots\!86}{15\!\cdots\!61}a^{8}-\frac{18\!\cdots\!49}{15\!\cdots\!61}a^{7}-\frac{13\!\cdots\!30}{15\!\cdots\!61}a^{6}-\frac{37\!\cdots\!03}{15\!\cdots\!61}a^{5}+\frac{65\!\cdots\!84}{15\!\cdots\!61}a^{4}+\frac{17\!\cdots\!61}{15\!\cdots\!61}a^{3}+\frac{56\!\cdots\!45}{15\!\cdots\!61}a^{2}-\frac{58\!\cdots\!40}{15\!\cdots\!61}a-\frac{74\!\cdots\!78}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{35}+\frac{352525692605662}{15\!\cdots\!61}a^{24}-\frac{1183283640}{15\!\cdots\!61}a^{23}+\frac{244083573036371}{15\!\cdots\!61}a^{22}-\frac{66940045920}{15\!\cdots\!61}a^{21}+\frac{41\!\cdots\!79}{15\!\cdots\!61}a^{20}-\frac{1677304559700}{15\!\cdots\!61}a^{19}-\frac{57\!\cdots\!10}{15\!\cdots\!61}a^{18}+\frac{15\!\cdots\!13}{15\!\cdots\!61}a^{17}-\frac{33\!\cdots\!33}{15\!\cdots\!61}a^{16}+\frac{15\!\cdots\!26}{15\!\cdots\!61}a^{15}+\frac{29\!\cdots\!68}{15\!\cdots\!61}a^{14}+\frac{43\!\cdots\!04}{15\!\cdots\!61}a^{13}-\frac{63\!\cdots\!78}{15\!\cdots\!61}a^{12}-\frac{31\!\cdots\!88}{15\!\cdots\!61}a^{11}+\frac{54\!\cdots\!17}{15\!\cdots\!61}a^{10}-\frac{15\!\cdots\!13}{15\!\cdots\!61}a^{9}+\frac{19\!\cdots\!57}{15\!\cdots\!61}a^{8}-\frac{45\!\cdots\!95}{15\!\cdots\!61}a^{7}-\frac{18\!\cdots\!93}{15\!\cdots\!61}a^{6}-\frac{29\!\cdots\!87}{15\!\cdots\!61}a^{5}-\frac{40\!\cdots\!04}{15\!\cdots\!61}a^{4}+\frac{39\!\cdots\!67}{15\!\cdots\!61}a^{3}-\frac{86\!\cdots\!43}{15\!\cdots\!61}a^{2}-\frac{76\!\cdots\!91}{15\!\cdots\!61}a+\frac{47\!\cdots\!30}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{36}-\frac{1419940368}{15\!\cdots\!61}a^{24}-\frac{449551447477671}{15\!\cdots\!61}a^{23}-\frac{83979330336}{15\!\cdots\!61}a^{22}-\frac{687757472802863}{15\!\cdots\!61}a^{21}-\frac{2214042018804}{15\!\cdots\!61}a^{20}-\frac{87\!\cdots\!44}{15\!\cdots\!61}a^{19}-\frac{33993372409920}{15\!\cdots\!61}a^{18}+\frac{19\!\cdots\!27}{15\!\cdots\!61}a^{17}-\frac{38\!\cdots\!68}{15\!\cdots\!61}a^{16}+\frac{91\!\cdots\!70}{15\!\cdots\!61}a^{15}+\frac{10\!\cdots\!92}{15\!\cdots\!61}a^{14}+\frac{19\!\cdots\!96}{15\!\cdots\!61}a^{13}+\frac{61\!\cdots\!68}{15\!\cdots\!61}a^{12}-\frac{10\!\cdots\!57}{15\!\cdots\!61}a^{11}-\frac{66\!\cdots\!28}{15\!\cdots\!61}a^{10}-\frac{47\!\cdots\!10}{15\!\cdots\!61}a^{9}+\frac{64\!\cdots\!96}{15\!\cdots\!61}a^{8}-\frac{52\!\cdots\!14}{15\!\cdots\!61}a^{7}-\frac{57\!\cdots\!01}{15\!\cdots\!61}a^{6}+\frac{53\!\cdots\!33}{15\!\cdots\!61}a^{5}+\frac{13\!\cdots\!02}{15\!\cdots\!61}a^{4}+\frac{59\!\cdots\!73}{15\!\cdots\!61}a^{3}-\frac{47\!\cdots\!09}{15\!\cdots\!61}a^{2}+\frac{51\!\cdots\!01}{15\!\cdots\!61}a+\frac{20\!\cdots\!73}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{37}-\frac{358565933151935}{15\!\cdots\!61}a^{24}+\frac{22516197264}{15\!\cdots\!61}a^{23}-\frac{573248422577523}{15\!\cdots\!61}a^{22}+\frac{1300310391996}{15\!\cdots\!61}a^{21}-\frac{51\!\cdots\!11}{15\!\cdots\!61}a^{20}+\frac{33098809978080}{15\!\cdots\!61}a^{19}+\frac{62\!\cdots\!45}{15\!\cdots\!61}a^{18}-\frac{59\!\cdots\!37}{15\!\cdots\!61}a^{17}+\frac{53\!\cdots\!81}{15\!\cdots\!61}a^{16}+\frac{35\!\cdots\!55}{15\!\cdots\!61}a^{15}-\frac{41\!\cdots\!82}{15\!\cdots\!61}a^{14}-\frac{11\!\cdots\!16}{15\!\cdots\!61}a^{13}+\frac{30\!\cdots\!86}{15\!\cdots\!61}a^{12}-\frac{80\!\cdots\!65}{15\!\cdots\!61}a^{11}+\frac{15\!\cdots\!13}{15\!\cdots\!61}a^{10}-\frac{56\!\cdots\!70}{15\!\cdots\!61}a^{9}-\frac{24\!\cdots\!70}{15\!\cdots\!61}a^{8}+\frac{10\!\cdots\!15}{15\!\cdots\!61}a^{7}+\frac{69\!\cdots\!86}{15\!\cdots\!61}a^{6}+\frac{26\!\cdots\!73}{15\!\cdots\!61}a^{5}+\frac{38\!\cdots\!84}{15\!\cdots\!61}a^{4}+\frac{31\!\cdots\!02}{15\!\cdots\!61}a^{3}-\frac{66\!\cdots\!57}{15\!\cdots\!61}a^{2}-\frac{63\!\cdots\!59}{15\!\cdots\!61}a-\frac{52\!\cdots\!64}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{38}+\frac{27600499872}{15\!\cdots\!61}a^{24}+\frac{573404638906994}{15\!\cdots\!61}a^{23}+\frac{1666380179772}{15\!\cdots\!61}a^{22}+\frac{487183121875867}{15\!\cdots\!61}a^{21}+\frac{44630008293024}{15\!\cdots\!61}a^{20}-\frac{55\!\cdots\!28}{15\!\cdots\!61}a^{19}+\frac{693793765282464}{15\!\cdots\!61}a^{18}+\frac{58\!\cdots\!56}{15\!\cdots\!61}a^{17}-\frac{57\!\cdots\!40}{15\!\cdots\!61}a^{16}+\frac{40\!\cdots\!29}{15\!\cdots\!61}a^{15}-\frac{17\!\cdots\!39}{15\!\cdots\!61}a^{14}+\frac{38\!\cdots\!44}{15\!\cdots\!61}a^{13}+\frac{80\!\cdots\!88}{15\!\cdots\!61}a^{12}-\frac{63\!\cdots\!50}{15\!\cdots\!61}a^{11}+\frac{72\!\cdots\!76}{15\!\cdots\!61}a^{10}-\frac{42\!\cdots\!39}{15\!\cdots\!61}a^{9}+\frac{20\!\cdots\!33}{15\!\cdots\!61}a^{8}+\frac{15\!\cdots\!55}{15\!\cdots\!61}a^{7}+\frac{26\!\cdots\!11}{15\!\cdots\!61}a^{6}-\frac{57\!\cdots\!64}{15\!\cdots\!61}a^{5}-\frac{13\!\cdots\!50}{15\!\cdots\!61}a^{4}+\frac{73\!\cdots\!64}{15\!\cdots\!61}a^{3}+\frac{50\!\cdots\!91}{15\!\cdots\!61}a^{2}-\frac{24\!\cdots\!17}{15\!\cdots\!61}a-\frac{52\!\cdots\!12}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{39}+\frac{666078424142389}{15\!\cdots\!61}a^{24}-\frac{403657310628}{15\!\cdots\!61}a^{23}+\frac{723247677596655}{15\!\cdots\!61}a^{22}-\frac{23681228890176}{15\!\cdots\!61}a^{21}-\frac{44\!\cdots\!83}{15\!\cdots\!61}a^{20}-\frac{610329853669536}{15\!\cdots\!61}a^{19}-\frac{15\!\cdots\!74}{15\!\cdots\!61}a^{18}+\frac{47\!\cdots\!35}{15\!\cdots\!61}a^{17}+\frac{92\!\cdots\!13}{15\!\cdots\!61}a^{16}+\frac{68\!\cdots\!23}{15\!\cdots\!61}a^{15}+\frac{72\!\cdots\!17}{15\!\cdots\!61}a^{14}+\frac{58\!\cdots\!02}{15\!\cdots\!61}a^{13}+\frac{65\!\cdots\!64}{15\!\cdots\!61}a^{12}+\frac{14\!\cdots\!62}{15\!\cdots\!61}a^{11}+\frac{37\!\cdots\!62}{15\!\cdots\!61}a^{10}+\frac{71\!\cdots\!48}{15\!\cdots\!61}a^{9}+\frac{74\!\cdots\!48}{15\!\cdots\!61}a^{8}+\frac{27\!\cdots\!79}{15\!\cdots\!61}a^{7}-\frac{58\!\cdots\!53}{15\!\cdots\!61}a^{6}-\frac{17\!\cdots\!19}{15\!\cdots\!61}a^{5}-\frac{50\!\cdots\!70}{15\!\cdots\!61}a^{4}+\frac{51\!\cdots\!43}{15\!\cdots\!61}a^{3}+\frac{66\!\cdots\!13}{15\!\cdots\!61}a^{2}+\frac{61\!\cdots\!75}{15\!\cdots\!61}a+\frac{46\!\cdots\!72}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{40}-\frac{504571638285}{15\!\cdots\!61}a^{24}+\frac{649837721431783}{15\!\cdots\!61}a^{23}-\frac{30947060481480}{15\!\cdots\!61}a^{22}-\frac{570376623764082}{15\!\cdots\!61}a^{21}-\frac{839203548795612}{15\!\cdots\!61}a^{20}-\frac{47\!\cdots\!17}{15\!\cdots\!61}a^{19}-\frac{13\!\cdots\!00}{15\!\cdots\!61}a^{18}-\frac{77\!\cdots\!66}{15\!\cdots\!61}a^{17}+\frac{46\!\cdots\!93}{15\!\cdots\!61}a^{16}+\frac{12\!\cdots\!59}{15\!\cdots\!61}a^{15}-\frac{64\!\cdots\!75}{15\!\cdots\!61}a^{14}-\frac{53\!\cdots\!83}{15\!\cdots\!61}a^{13}+\frac{12\!\cdots\!09}{15\!\cdots\!61}a^{12}-\frac{40\!\cdots\!46}{15\!\cdots\!61}a^{11}+\frac{62\!\cdots\!69}{15\!\cdots\!61}a^{10}-\frac{55\!\cdots\!21}{15\!\cdots\!61}a^{9}-\frac{96\!\cdots\!46}{15\!\cdots\!61}a^{8}+\frac{11\!\cdots\!34}{15\!\cdots\!61}a^{7}+\frac{54\!\cdots\!23}{15\!\cdots\!61}a^{6}-\frac{46\!\cdots\!45}{15\!\cdots\!61}a^{5}-\frac{26\!\cdots\!67}{15\!\cdots\!61}a^{4}+\frac{14\!\cdots\!02}{15\!\cdots\!61}a^{3}+\frac{70\!\cdots\!64}{15\!\cdots\!61}a^{2}+\frac{48\!\cdots\!16}{15\!\cdots\!61}a-\frac{58\!\cdots\!55}{15\!\cdots\!61}$, $\frac{1}{15\!\cdots\!61}a^{41}+\frac{313333331137426}{15\!\cdots\!61}a^{24}+\frac{6895812389895}{15\!\cdots\!61}a^{23}-\frac{662012795354091}{15\!\cdots\!61}a^{22}+\frac{409611255959763}{15\!\cdots\!61}a^{21}-\frac{70\!\cdots\!13}{15\!\cdots\!61}a^{20}+\frac{10\!\cdots\!50}{15\!\cdots\!61}a^{19}+\frac{47\!\cdots\!34}{15\!\cdots\!61}a^{18}+\frac{51\!\cdots\!04}{15\!\cdots\!61}a^{17}+\frac{67\!\cdots\!55}{15\!\cdots\!61}a^{16}-\frac{52\!\cdots\!72}{15\!\cdots\!61}a^{15}-\frac{54\!\cdots\!45}{15\!\cdots\!61}a^{14}+\frac{77\!\cdots\!36}{15\!\cdots\!61}a^{13}+\frac{25\!\cdots\!97}{15\!\cdots\!61}a^{12}-\frac{34\!\cdots\!12}{15\!\cdots\!61}a^{11}+\frac{59\!\cdots\!37}{15\!\cdots\!61}a^{10}+\frac{14\!\cdots\!49}{15\!\cdots\!61}a^{9}-\frac{62\!\cdots\!10}{15\!\cdots\!61}a^{8}+\frac{54\!\cdots\!23}{15\!\cdots\!61}a^{7}-\frac{24\!\cdots\!36}{15\!\cdots\!61}a^{6}+\frac{42\!\cdots\!68}{15\!\cdots\!61}a^{5}-\frac{30\!\cdots\!31}{15\!\cdots\!61}a^{4}-\frac{59\!\cdots\!17}{15\!\cdots\!61}a^{3}-\frac{24\!\cdots\!76}{15\!\cdots\!61}a^{2}+\frac{13\!\cdots\!68}{15\!\cdots\!61}a-\frac{44\!\cdots\!63}{15\!\cdots\!61}$
| 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: |
\( -1 \)
(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
| A cyclic group of order 42 |
| The 42 conjugacy class representatives for $C_{42}$ |
| Character table for $C_{42}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-91}) \), \(\Q(\zeta_{7})^+\), 6.0.36924979.1, 7.7.13841287201.1, 14.0.84150067079150835865691353219.1, \(\Q(\zeta_{49})^+\) |
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 | $42$ | $42$ | $21^{2}$ | R | $42$ | R | $42$ | ${\href{/padicField/19.3.0.1}{3} }^{14}$ | $21^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{14}$ | $42$ | ${\href{/padicField/41.7.0.1}{7} }^{6}$ | ${\href{/padicField/43.7.0.1}{7} }^{6}$ | $21^{2}$ | $21^{2}$ | $21^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| Deg $42$ | $42$ | $1$ | $77$ | |||
|
\(13\)
| 13.14.7.2 | $x^{14} + 14480427 x^{2} - 690233687$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
| 13.14.7.2 | $x^{14} + 14480427 x^{2} - 690233687$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ | |
| 13.14.7.2 | $x^{14} + 14480427 x^{2} - 690233687$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |