Normalized defining polynomial
\( x^{42} - 11 x^{41} + 31 x^{40} + 15 x^{39} + 149 x^{38} - 2171 x^{37} + 5622 x^{36} - 5813 x^{35} + \cdots + 9042834947 \)
Invariants
| Degree: | $42$ |
| |
| Signature: | $[0, 21]$ |
| |
| Discriminant: |
\(-233\!\cdots\!507\)
\(\medspace = -\,7^{28}\cdot 43^{39}\)
|
| |
| Root discriminant: | \(120.28\) |
| |
| Galois root discriminant: | $7^{2/3}43^{13/14}\approx 120.2792830317974$ | ||
| Ramified primes: |
\(7\), \(43\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-43}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{42}$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(301=7\cdot 43\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{301}(128,·)$, $\chi_{301}(1,·)$, $\chi_{301}(2,·)$, $\chi_{301}(4,·)$, $\chi_{301}(65,·)$, $\chi_{301}(8,·)$, $\chi_{301}(137,·)$, $\chi_{301}(11,·)$, $\chi_{301}(130,·)$, $\chi_{301}(256,·)$, $\chi_{301}(16,·)$, $\chi_{301}(274,·)$, $\chi_{301}(22,·)$, $\chi_{301}(151,·)$, $\chi_{301}(260,·)$, $\chi_{301}(156,·)$, $\chi_{301}(32,·)$, $\chi_{301}(39,·)$, $\chi_{301}(170,·)$, $\chi_{301}(44,·)$, $\chi_{301}(176,·)$, $\chi_{301}(51,·)$, $\chi_{301}(183,·)$, $\chi_{301}(64,·)$, $\chi_{301}(193,·)$, $\chi_{301}(204,·)$, $\chi_{301}(78,·)$, $\chi_{301}(207,·)$, $\chi_{301}(211,·)$, $\chi_{301}(85,·)$, $\chi_{301}(214,·)$, $\chi_{301}(88,·)$, $\chi_{301}(219,·)$, $\chi_{301}(226,·)$, $\chi_{301}(102,·)$, $\chi_{301}(107,·)$, $\chi_{301}(113,·)$, $\chi_{301}(242,·)$, $\chi_{301}(247,·)$, $\chi_{301}(121,·)$, $\chi_{301}(254,·)$, $\chi_{301}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{1048576}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{337}a^{37}+\frac{83}{337}a^{36}+\frac{119}{337}a^{35}-\frac{44}{337}a^{34}-\frac{149}{337}a^{33}+\frac{41}{337}a^{32}-\frac{157}{337}a^{31}+\frac{73}{337}a^{30}+\frac{167}{337}a^{29}+\frac{106}{337}a^{28}-\frac{140}{337}a^{27}-\frac{167}{337}a^{26}+\frac{150}{337}a^{25}+\frac{137}{337}a^{24}+\frac{83}{337}a^{23}-\frac{35}{337}a^{22}+\frac{141}{337}a^{21}+\frac{47}{337}a^{20}+\frac{114}{337}a^{19}+\frac{113}{337}a^{18}+\frac{75}{337}a^{17}-\frac{91}{337}a^{16}+\frac{166}{337}a^{15}+\frac{41}{337}a^{14}-\frac{152}{337}a^{13}+\frac{117}{337}a^{12}-\frac{23}{337}a^{11}-\frac{46}{337}a^{10}-\frac{123}{337}a^{9}+\frac{96}{337}a^{8}+\frac{94}{337}a^{7}+\frac{117}{337}a^{6}-\frac{47}{337}a^{5}-\frac{145}{337}a^{4}-\frac{35}{337}a^{3}+\frac{110}{337}a^{2}-\frac{129}{337}a+\frac{71}{337}$, $\frac{1}{84587}a^{38}-\frac{22}{84587}a^{37}-\frac{10281}{84587}a^{36}+\frac{4985}{84587}a^{35}+\frac{32442}{84587}a^{34}+\frac{7598}{84587}a^{33}-\frac{22323}{84587}a^{32}-\frac{14446}{84587}a^{31}+\frac{24854}{84587}a^{30}-\frac{17092}{84587}a^{29}-\frac{42274}{84587}a^{28}+\frac{8467}{84587}a^{27}+\frac{5216}{84587}a^{26}+\frac{9325}{84587}a^{25}+\frac{5244}{84587}a^{24}-\frac{36721}{84587}a^{23}-\frac{22470}{84587}a^{22}+\frac{17257}{84587}a^{21}+\frac{34608}{84587}a^{20}-\frac{17923}{84587}a^{19}+\frac{20899}{84587}a^{18}+\frac{39214}{84587}a^{17}+\frac{14439}{84587}a^{16}+\frac{38890}{84587}a^{15}+\frac{28906}{84587}a^{14}+\frac{20121}{84587}a^{13}-\frac{5568}{84587}a^{12}+\frac{15512}{84587}a^{11}-\frac{31015}{84587}a^{10}+\frac{7956}{84587}a^{9}+\frac{13604}{84587}a^{8}+\frac{28328}{84587}a^{7}-\frac{22779}{84587}a^{6}+\frac{11193}{84587}a^{5}-\frac{33675}{84587}a^{4}-\frac{933}{84587}a^{3}-\frac{20778}{84587}a^{2}+\frac{22715}{84587}a+\frac{14787}{84587}$, $\frac{1}{17\cdots 49}a^{39}+\frac{6002678012}{17\cdots 49}a^{38}+\frac{1261619063676}{17\cdots 49}a^{37}+\frac{407043669912976}{17\cdots 49}a^{36}+\frac{2819756335677}{17\cdots 49}a^{35}-\frac{355293810027673}{17\cdots 49}a^{34}-\frac{87455754300957}{17\cdots 49}a^{33}+\frac{808897253512992}{17\cdots 49}a^{32}-\frac{258991936282850}{17\cdots 49}a^{31}+\frac{426966805727161}{17\cdots 49}a^{30}-\frac{237512242123769}{17\cdots 49}a^{29}+\frac{557763975630923}{17\cdots 49}a^{28}+\frac{320451456512828}{17\cdots 49}a^{27}-\frac{618538654617230}{17\cdots 49}a^{26}+\frac{5066411768737}{17\cdots 49}a^{25}+\frac{366576915917628}{17\cdots 49}a^{24}-\frac{187488780099930}{17\cdots 49}a^{23}+\frac{804708238194396}{17\cdots 49}a^{22}+\frac{354795532741723}{17\cdots 49}a^{21}-\frac{869287891762369}{17\cdots 49}a^{20}-\frac{225587811249473}{17\cdots 49}a^{19}+\frac{128430945426788}{17\cdots 49}a^{18}+\frac{395079880039544}{17\cdots 49}a^{17}+\frac{504514888289626}{17\cdots 49}a^{16}-\frac{821956349092599}{17\cdots 49}a^{15}-\frac{738953494198687}{17\cdots 49}a^{14}-\frac{755660907837977}{17\cdots 49}a^{13}-\frac{854105875346853}{17\cdots 49}a^{12}+\frac{555191232186713}{17\cdots 49}a^{11}+\frac{675984304095620}{17\cdots 49}a^{10}+\frac{252622714445955}{17\cdots 49}a^{9}-\frac{854986080701802}{17\cdots 49}a^{8}-\frac{558655499254597}{17\cdots 49}a^{7}-\frac{681540335262451}{17\cdots 49}a^{6}+\frac{335373309336563}{17\cdots 49}a^{5}-\frac{692466245559277}{17\cdots 49}a^{4}+\frac{14901982856660}{17\cdots 49}a^{3}+\frac{138969006411012}{17\cdots 49}a^{2}-\frac{831437809048532}{17\cdots 49}a+\frac{161158348300817}{17\cdots 49}$, $\frac{1}{94\cdots 31}a^{40}+\frac{384}{94\cdots 31}a^{39}+\frac{8315592999579}{94\cdots 31}a^{38}+\frac{13\cdots 58}{94\cdots 31}a^{37}-\frac{32\cdots 88}{94\cdots 31}a^{36}+\frac{12\cdots 84}{94\cdots 31}a^{35}+\frac{21\cdots 96}{94\cdots 31}a^{34}+\frac{24\cdots 02}{94\cdots 31}a^{33}+\frac{23\cdots 21}{94\cdots 31}a^{32}-\frac{14\cdots 61}{94\cdots 31}a^{31}+\frac{61\cdots 76}{94\cdots 31}a^{30}+\frac{22\cdots 09}{94\cdots 31}a^{29}-\frac{16\cdots 25}{94\cdots 31}a^{28}+\frac{45\cdots 50}{94\cdots 31}a^{27}+\frac{43\cdots 96}{94\cdots 31}a^{26}-\frac{34\cdots 58}{94\cdots 31}a^{25}-\frac{32\cdots 90}{94\cdots 31}a^{24}-\frac{11\cdots 95}{94\cdots 31}a^{23}+\frac{22\cdots 99}{94\cdots 31}a^{22}-\frac{29\cdots 48}{94\cdots 31}a^{21}-\frac{39\cdots 62}{94\cdots 31}a^{20}-\frac{35\cdots 30}{94\cdots 31}a^{19}+\frac{69\cdots 30}{37\cdots 81}a^{18}+\frac{19\cdots 22}{94\cdots 31}a^{17}+\frac{39\cdots 14}{94\cdots 31}a^{16}+\frac{21\cdots 92}{94\cdots 31}a^{15}+\frac{35\cdots 23}{94\cdots 31}a^{14}-\frac{20\cdots 24}{94\cdots 31}a^{13}+\frac{53\cdots 81}{94\cdots 31}a^{12}+\frac{38\cdots 85}{94\cdots 31}a^{11}-\frac{26\cdots 46}{94\cdots 31}a^{10}+\frac{51\cdots 12}{11\cdots 89}a^{9}+\frac{39\cdots 97}{94\cdots 31}a^{8}-\frac{18\cdots 83}{94\cdots 31}a^{7}+\frac{18\cdots 27}{94\cdots 31}a^{6}-\frac{34\cdots 22}{94\cdots 31}a^{5}-\frac{19\cdots 37}{94\cdots 31}a^{4}+\frac{29\cdots 19}{94\cdots 31}a^{3}-\frac{41\cdots 45}{94\cdots 31}a^{2}+\frac{32\cdots 09}{94\cdots 31}a-\frac{45\cdots 32}{94\cdots 31}$, $\frac{1}{52\cdots 61}a^{41}-\frac{61\cdots 78}{52\cdots 61}a^{40}+\frac{49\cdots 18}{52\cdots 61}a^{39}-\frac{22\cdots 50}{52\cdots 61}a^{38}+\frac{70\cdots 18}{52\cdots 61}a^{37}-\frac{22\cdots 41}{52\cdots 61}a^{36}-\frac{56\cdots 69}{52\cdots 61}a^{35}+\frac{19\cdots 70}{52\cdots 61}a^{34}-\frac{10\cdots 93}{52\cdots 61}a^{33}-\frac{19\cdots 71}{52\cdots 61}a^{32}-\frac{97\cdots 73}{52\cdots 61}a^{31}-\frac{12\cdots 52}{52\cdots 61}a^{30}+\frac{13\cdots 75}{52\cdots 61}a^{29}+\frac{22\cdots 53}{52\cdots 61}a^{28}-\frac{19\cdots 77}{66\cdots 59}a^{27}-\frac{21\cdots 39}{52\cdots 61}a^{26}-\frac{23\cdots 64}{52\cdots 61}a^{25}-\frac{24\cdots 66}{52\cdots 61}a^{24}-\frac{21\cdots 09}{52\cdots 61}a^{23}+\frac{12\cdots 13}{52\cdots 61}a^{22}-\frac{20\cdots 18}{52\cdots 61}a^{21}+\frac{83\cdots 95}{52\cdots 61}a^{20}+\frac{12\cdots 46}{52\cdots 61}a^{19}+\frac{26\cdots 48}{52\cdots 61}a^{18}-\frac{10\cdots 74}{52\cdots 61}a^{17}+\frac{86\cdots 33}{52\cdots 61}a^{16}+\frac{89\cdots 20}{20\cdots 11}a^{15}+\frac{23\cdots 52}{52\cdots 61}a^{14}+\frac{52\cdots 62}{52\cdots 61}a^{13}+\frac{24\cdots 48}{52\cdots 61}a^{12}+\frac{10\cdots 56}{52\cdots 61}a^{11}+\frac{17\cdots 30}{52\cdots 61}a^{10}-\frac{96\cdots 20}{52\cdots 61}a^{9}+\frac{19\cdots 94}{52\cdots 61}a^{8}-\frac{14\cdots 54}{52\cdots 61}a^{7}-\frac{45\cdots 16}{52\cdots 61}a^{6}-\frac{41\cdots 89}{52\cdots 61}a^{5}-\frac{39\cdots 40}{52\cdots 61}a^{4}-\frac{12\cdots 25}{52\cdots 61}a^{3}+\frac{25\cdots 48}{52\cdots 61}a^{2}+\frac{22\cdots 29}{52\cdots 61}a+\frac{25\cdots 25}{52\cdots 61}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | not computed |
| |
| Narrow class group: | not computed |
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| Relative class number: | data not computed |
Unit group
| Rank: | $20$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: | not computed |
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| Regulator: | not computed |
|
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 = \mathstrut &\frac{2^{0}\cdot(2\pi)^{21}\cdot R \cdot h}{2\cdot\sqrt{2333538056680443170216809877092138324063458978840788486681774071870758537049156298101507}}\cr\mathstrut & \text{
Galois group
| A cyclic group of order 42 |
| The 42 conjugacy class representatives for $C_{42}$ |
| Character table for $C_{42}$ |
Intermediate fields
| \(\Q(\sqrt{-43}) \), \(\Q(\zeta_{7})^+\), 6.0.190896307.1, 7.7.6321363049.1, 14.0.1718264124282290785243.1, 21.21.171318696215827426793735775028238670573001.1 |
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 | $42$ | $42$ | $42$ | R | $21^{2}$ | ${\href{/padicField/13.7.0.1}{7} }^{6}$ | $21^{2}$ | $42$ | $21^{2}$ | ${\href{/padicField/29.14.0.1}{14} }^{3}$ | $21^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{7}$ | ${\href{/padicField/41.7.0.1}{7} }^{6}$ | R | $21^{2}$ | $21^{2}$ | $21^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.3.4a1.2 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ |
| 7.2.3.4a1.2 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ | |
| 7.2.3.4a1.2 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ | |
| 7.2.3.4a1.2 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ | |
| 7.2.3.4a1.2 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ | |
| 7.2.3.4a1.2 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ | |
| 7.2.3.4a1.2 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ | |
|
\(43\)
| 43.1.14.13a1.1 | $x^{14} + 43$ | $14$ | $1$ | $13$ | $C_{14}$ | $$[\ ]_{14}$$ |
| 43.1.14.13a1.1 | $x^{14} + 43$ | $14$ | $1$ | $13$ | $C_{14}$ | $$[\ ]_{14}$$ | |
| 43.1.14.13a1.1 | $x^{14} + 43$ | $14$ | $1$ | $13$ | $C_{14}$ | $$[\ ]_{14}$$ |